Contagion modeling between the financial and insurance markets with time changed processes

Abstract

This study analyzes the impact of contagion between financial and non-life insurance markets on the asset–liability management policy of an insurance company. The indirect dependence between these markets is modeled by assuming that the assets return and non-life insurance claims are led respectively by time-changed Brownian and jump processes, for which stochastic clocks are integrals of mutually self-exciting processes. This model exhibits delayed co-movements between financial and non-life insurance markets, caused by events like natural disasters, epidemics, or economic recessions.

1. Introduction

Non life insurance claims, by nature, are not correlated to financial markets, excepted in case of events like natural disasters, epidemics, or serious economic recession. For example, in 2003, the severe acute respiratory syndrome (SARS) spread across several countries and affected with a delay the insurance industry in different ways. Some areas of impacted insurance operations are clear—event cancellations coverage, travel insurance and life and health policies. This epidemic also slowed down economic exchanges and indirectly caused turmoil in financial markets. More recently, during the financial crisis of 2008, the number of claims covered by credit insurances surged in US, as underlined in a recent report from the IMF (2016). As last example, we mention climate changes. It is already affecting and will over time significantly affect the incidence of natural conditions such as: tropical cyclones; winter storms; wild-fires; hail storms; lightning strikes; droughts and floods. These events are expected to affect significantly property claims to non-life insurers. In parallel, climate change will have a huge economic and social impact and will lead to financial instability. These observations motivate us to study the influence of a potential contagion between the insurance and financial markets on the asset–liability management policy of insurers.

The literature about the modeling and management of non-life insurance company is vast. The starting point of research in this field is the classical Cramer–Lundberg (1903) risk model, in which the arrival of claims is modeled by a Poisson process. Since then, many extensions have been developed and proposed bounds on the insurer’s ruin probability in various frameworks. Later, Björk and Grandell (1988) and Embrechts et al. (1993) introduced a Cox process in the Cramer–Lundberg model, for the modeling of claim arrivals. Albrecher and Asmussen (2006) studied a Cox process with shot noise intensity. Dassios and Zhao, 2011, Dassios and Zhao, 2012 analyzed the clustering phenomenon of claims, caused by a self-exciting process. Another strand of the literature focuses on the optimization of investment, reinsurance and dividend policies, in a Cramer–Lundberg approach. For example, Browne (1995) showed in a one-dimensional diffusion model that the strategy maximizing the expected exponential utility of terminal wealth also minimizes the ruin probability. Asmussen and Taksar (1997) studied the optimal dividend policy for an insurer. Hipp and Plum (2000) optimized the investment policy of a non life insurer’s surplus, in a Brownian setting. Schmidli, 2002, Schmidli, 2006 instead of maximizing the utility of the surplus or dividends, adapted the investment and reinsurance strategies to minimize the probability of ruin. Kaluszka, 2001, Kaluszka, 2004 examined the optimal reinsurance problem under various mean–variance premium principles. Yuen et al. (2015) considered the optimal proportional reinsurance strategy in a risk model with multiple dependent classes of insurance business. And recently, Yin and Yuen (2015) studied the optimal dividend problems for a jump diffusion model with capital injections and proportional transaction costs. Whereas Zheng et al. (2016) investigated the robust optimal portfolio and reinsurance problem for an ambiguity-averse insurer.

This work studies the optimal proportional reinsurance, dividends and asset allocation for a non-life insurer, in presence of a contagion risk between financial and insurance activities. Drawing on the theoretical and empirical background regarding the market time scale, as in Ané and Geman (2000) or Salmon and Tham (2007), we build time-changed dynamics with chronometers that are integrated positive Hawkes processes. This approach, inspired from Hainaut (2016d) introduces a nonlinear dependence between assets and liabilities. Hawkes processes developed by Hawkes, 1971a, Hawkes, 1971b and Hawkes and Oakes (1974), are parsimonious self-exciting point processes for which the intensity jumps in response and reverts to a target level in the absence of event. This dynamics is increasingly used in finance to model the clustering of shocks. Empirical analysis conducted in Aït-Sahalia et al., 2015, Aït-Sahalia et al., 2014 or in Embrechts et al. (2011) emphasizes the importance of this effect in stocks or CDS markets. They also underline that clustering is not characterized by a single jump but by the amplification of this movement that takes place over days. Recently, Hainaut, 2016a, Hainaut, 2016b detects self-excitation in interest rate markets. And the paper of (Hainaut, 2016c) analyzes the impact of the clustering of jumps on prices and risk of variable annuities. In this work, Hawkes processes determine the pace of market clocks.

This research contributes to the literature in several directions. Firstly, it is an elegant method to introduce dependence between a geometric Brownian motion and a risk process. In this framework, we find the main features of clocks: means, variances and their joint moment generating function (mgf).

Secondly, we show that the linear dependence between log-prices and claims is proportional to the risk premium of stocks and to the insurer’s average profit. In particular, when the insurer does not charge any fee above the pure premium, the correlation is null despite an evident dependence by construction. When the insurer’s margin is positive, the linear correlation is induced by incomes from the insurance activity, reinvested in the financial market. From an economic point of view, the linear dependence between insurance and financial markets in a time-changed model find its origin in the existence of a risk premium in both segments.

Thirdly, we prove that the insurer’s ruin probability is still below the Cramer–Lundberg bound, if the surplus is not invested in the stocks market. Fourthly, we determine the optimal investment, reinsurance and dividend policies that maximize the exponential utility drawn from dividends and terminal surplus. Surprisingly, Optimal reinsurance and investment strategies are independent from markets clocks. Whereas, the optimal dividend is a linear function of the wealth and of intensities of chronometers. Finally, we compare with optimal strategies when the claim process is approached by a Brownian motion.

2. Stochastic clocks of financial and insurance markets

Papers of Ané and Geman (2000) and Salmon and Tham (2007) provide pieces of evidence that the time scale of financial markets is not chronological but rather driven by traded volumes. Starting from this observation, we respectively model financial returns and insurance claims by a Brownian motion and a jump process, observed on distinct random scales of time. This approach allows us to replicate clustering of shocks observed in financial and in insurance markets. It also introduces contagion and random correlation between assets and liabilities. The chronometers measuring the time scales of financial and insurance markets are respectively denoted by ${\tau }_t^S$ and ${\tau }_t^L$. They are positive increasing processes defined as the integrals of two processes ${\left({\lambda }_t^S\right)}_t$ and ${\left({\lambda }_t^L\right)}_t$ on a probability space $\Omega$, endowed with a probability measure $P$ and a natural filtration denoted by ${\left({\mathcal{G}}_t\right)}_t$:

$${\tau }_t^S≔{\int }_0^t{\lambda }_s^{S}ds$$ (1)

$${\tau }_t^L≔{\int }_0^t{\lambda }_s^{L}ds\text{.}$$

By construction, the sample paths of random clocks are continuous and $d{\tau }_t^S={\lambda }_t^{S}dt$, $d{\tau }_t^L={\lambda }_t^{L}dt$. ${\lambda }_t^S$ and ${\lambda }_t^L$ are non-homogeneous processes that may be interpreted as the frequencies at which economic or actuarial information flows. Their dynamics is ruled by two auxiliary jump processes ${\left(Z_t^S\right)}_t$ and ${\left(Z_t^L\right)}_t$ such that $Z_t^S={\sum }_{k=1}^{N_t^S}{J}_k^S$ and $Z_t^L={\sum }_{k=1}^{N_t^L}{J}_k^L$. Where ${\left(N_t^S\right)}_t$ and ${\left(N_t^L\right)}_t$ are point processes with random jumps $J_k^S$ and $J_k^L$. The intensities of jump arrivals are assumed equal to the frequencies of information flows: ${\lambda }_t^S$ and ${\lambda }_t^L$.

For the sake of simplicity, jumps are exponential random variables with densities ${\nu }^S\left(z\right)={\rho }_S{e}^{-{\rho }_{S}z}{1}_{\{z\ge 0\}}$ and ${\nu }^L\left(z\right)={\rho }_L{e}^{-{\rho }_{L}z}{1}_{\{z\ge 0\}}$ where ${\rho }_S$ and ${\rho }_L$ are positive and constant. The average sizes of jumps are equal to ${\mu }_S=\frac{1}{{\rho }_S}$ and ${\mu }_L=\frac{1}{{\rho }_L}$. Whereas the moment generating functions of jumps are respectively ${\varphi }^{J_S}\left(\omega \right)≔\mathbb{E}\left(e^{\omega J^S}\right)$ and ${\varphi }^{J_L}\left(\omega \right)≔\mathbb{E}\left(e^{\omega J^L}\right)$. We assume that the couple of intensities ${\lambda }_t^S$ and ${\lambda }_t^L$ obeys to the next dynamics:

$$\left(\begin{array}{c}\hfill d{\lambda }_t^S\hfill \\ \hfill d{\lambda }_t^L\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\alpha }_S({\theta }_S-{\lambda }_t^S)\hfill \\ \hfill {\alpha }_L({\theta }_L-{\lambda }_t^L)\hfill \end{array}\right)dt+\underset{\Xi }{\underbrace{\left(\begin{array}{cc}\hfill {\eta }_{SS}\hfill & \hfill {\eta }_{SL}\hfill \\ \hfill {\eta }_{LS}\hfill & \hfill {\eta }_{LL}\hfill \end{array}\right)}}\left(\begin{array}{c}\hfill d\underset{t}{\overset{S}{Z}}\hfill \\ \hfill d{Z}_t^L\hfill \end{array}\right)\text{.}$$ (2)

These processes revert respectively at speeds ${\alpha }_S$ and ${\alpha }_L$ toward ${\theta }_S$ or ${\theta }_L$. The parameters ${\eta }_{LS},{\eta }_{SS},{\eta }_{SL},{\eta }_{LL}$ are constant and positive. This last relation underlines the main features of our approach: contagion, mutual and self-excitation. Indeed, when the clock of the financial (resp. insurance) market speeds up due to a jump of $Z_t^S$ (resp. $Z_t^L$), the chronometer of the insurance (resp. financial) market accelerates proportionally. This also raises the volatility as longer periods, measured on the market time scale, are observed on the same invariable chronological time scale. Another consequence of a jump is an instantaneous increase in the probability of observing a new financial or actuarial shock as ${\lambda }_t^S$ and ${\lambda }_t^L$ are the intensities of point processes $Z_t^S$ and $Z_t^L$. We check by direct differentiation that intensities are the sum of a deterministic function and of two jump processes,

$${\lambda }_t^S={\theta }_S+e^{-{\alpha }_{S}t}({\lambda }_0^S-{\theta }_S)+{\eta }_{SS}{\int }_0^t{e}^{-{\alpha }_S(t-s)}d{Z}_s^S+{\eta }_{SL}{\int }_0^t{e}^{-{\alpha }_S(t-s)}d{Z}_s^L\text{,}$$ (3)

$${\lambda }_t^L={\theta }_L+e^{-{\alpha }_{L}t}({\lambda }_0^L-{\theta }_L)+{\eta }_{LS}{\int }_0^t{e}^{-{\alpha }_L(t-s)}d{Z}_s^S+{\eta }_{LL}{\int }_0^t{e}^{-{\alpha }_L(t-s)}d{Z}_s^L\text{.}$$

These expressions are useful to find closed form expressions of expected intensities, from which we will infer the conditions guaranteeing the stability of jump processes.

Proposition 2.1

The expectations of ${\lambda }_t^S$ and ${\lambda }_t^L$ are equal to

$$\left(\begin{array}{c}\hfill m^S\left(t\right)\hfill \\ \hfill m_0^S\left(t\right)\hfill \end{array}\right)≔\left(\begin{array}{c}\hfill \mathbb{E}\left({\lambda }_t^S\right|{\mathcal{G}}_0)\hfill \\ \hfill \mathbb{E}\left({\lambda }_t^L\right|{\mathcal{G}}_0)\hfill \end{array}\right)=V\left(\begin{array}{cc}\hfill \frac{1}{{\gamma }_1}(e^{{\gamma }_{1}t}-1)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{{\gamma }_2}(e^{{\gamma }_{2}t}-1)\hfill \end{array}\right)V^{-1}\left(\begin{array}{c}\hfill {\alpha }_S{\theta }_S\hfill \\ \hfill {\alpha }_L{\theta }_L\hfill \end{array}\right)+V\left(\begin{array}{cc}\hfill e^{{\gamma }_{1}t}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill e^{{\gamma }_{2}t}\hfill \end{array}\right)V^{-1}\left(\begin{array}{c}\hfill {\lambda }_0^S\hfill \\ \hfill {\lambda }_0^L\hfill \end{array}\right)$$ (4)

where $V$ is the following matrix

$$V=\left(\begin{array}{cc}\hfill -\frac{{\eta }_{SL}}{{\rho }_L}\hfill & \hfill -\frac{{\eta }_{SL}}{{\rho }_L}\hfill \\ \hfill \frac{{\eta }_{SS}}{{\rho }_S}-{\alpha }_S-{\gamma }_1\hfill & \hfill \frac{{\eta }_{SS}}{{\rho }_S}-{\alpha }_S-{\gamma }_2\hfill \end{array}\right)\text{.}$$ (5)

If we note

$$\Delta ={((\frac{{\eta }_{SS}}{{\rho }_S}-{\alpha }_S)-(\frac{{\eta }_{LL}}{{\rho }_L}-{\alpha }_L))}^2+4\frac{{\eta }_{LS}}{{\rho }_S}\frac{{\eta }_{SL}}{{\rho }_L}$$

then ${\gamma }_1$ , ${\gamma }_2$ are constant and defined by the following relations

$${\gamma }_1=\frac{1}{2}(\frac{{\eta }_{SS}}{{\rho }_S}+\frac{{\eta }_{LL}}{{\rho }_L}-({\alpha }_S+{\alpha }_L))+\frac{1}{2}\sqrt{\Delta }\text{,}$$ (6)

$${\gamma }_2=\frac{1}{2}(\frac{{\eta }_{SS}}{{\rho }_S}+\frac{{\eta }_{LL}}{{\rho }_L}-({\alpha }_S+{\alpha }_L))-\frac{1}{2}\sqrt{\Delta }\text{.}$$

Proof

Consider the functions $g^S={\lambda }_t^S$ and $g^L={\lambda }_t^L$. According to Eqs. (3) and if the infinitesimal generators of these functions are denoted by $\mathcal{A}{g}^S$ and $\mathcal{A}{g}^L$, their expectations satisfy the relation

$$\mathbb{E}\left(\mathcal{A}{g}^S\right)={\alpha }_S({\theta }_S-\mathbb{E}\left({\lambda }_t^S\right))+\mathbb{E}\left({\lambda }_t^S\right){\int }_{-\infty }^{+\infty }{\eta }_{SS}zd{\nu }_S\left(z\right)+\mathbb{E}\left({\lambda }_t^L\right){\int }_{-\infty }^{+\infty }{\eta }_{SL}zd{\nu }_L\left(z\right)={\alpha }_S({\theta }_S-\mathbb{E}\left({\lambda }_t^S\right))+\mathbb{E}\left({\lambda }_t^S\right){\eta }_{SS}{\mu }_S+\mathbb{E}\left({\lambda }_t^L\right){\eta }_{SL}{\mu }_L$$

$$\mathbb{E}\left(\mathcal{A}{g}^L\right)={\alpha }_L({\theta }_L-\mathbb{E}\left({\lambda }_t^L\right))+\mathbb{E}\left({\lambda }_t^S\right){\int }_{-\infty }^{+\infty }{\eta }_{LS}zd{\nu }_S\left(z\right)+\mathbb{E}\left({\lambda }_t^L\right){\int }_{-\infty }^{+\infty }{\eta }_{LL}zd{\nu }_L\left(z\right)={\alpha }_L({\theta }_L-\mathbb{E}\left({\lambda }_t^L\right))+\mathbb{E}\left({\lambda }_t^S\right){\eta }_{LS}{\mu }_S+\mathbb{E}\left({\lambda }_t^L\right){\eta }_{LL}{\mu }_L\text{.}$$

The first moments $m_S\left(t\right)$ and $m_L\left(t\right)$ are then solutions of a system of ordinary differential equations (ODE’s) with respect to time:

$$\frac{\partial }{\partial t}\left(\begin{array}{c}\hfill m_S\hfill \\ \hfill m_L\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\alpha }_S{\theta }_S\hfill \\ \hfill {\alpha }_L{\theta }_L\hfill \end{array}\right)+\left(\begin{array}{cc}\hfill ({\eta }_{SS}{\mu }_S-{\alpha }_S)\hfill & \hfill {\eta }_{SL}{\mu }_L\hfill \\ \hfill {\eta }_{LS}{\mu }_S\hfill & \hfill ({\eta }_{LL}{\mu }_L-{\alpha }_L)\hfill \end{array}\right)\left(\begin{array}{c}\hfill m_S\hfill \\ \hfill m_L\hfill \end{array}\right)\text{.}$$ (7)

Finding a solution requires to determine eigenvalues $\gamma$ and eigenvectors $(v_1,v_2)$ of the matrix present in the right term of this system:

$$\left(\begin{array}{cc}\hfill ({\eta }_{SS}{\mu }_S-{\alpha }_S)\hfill & \hfill {\eta }_{SL}{\mu }_L\hfill \\ \hfill {\eta }_{LS}{\mu }_S\hfill & \hfill ({\eta }_{LL}{\mu }_L-{\alpha }_L)\hfill \end{array}\right)\left(\begin{array}{c}\hfill v_1\hfill \\ \hfill v_2\hfill \end{array}\right)=\gamma \left(\begin{array}{c}\hfill v_1\hfill \\ \hfill v_2\hfill \end{array}\right)\text{.}$$

Eigenvalues cancel the determinant of the following matrix:

$$det\left(\begin{array}{cc}\hfill ({\eta }_{SS}{\mu }_S-{\alpha }_S)-\gamma \hfill & \hfill {\eta }_{SL}{\mu }_L\hfill \\ \hfill {\eta }_{LS}{\mu }_S\hfill & \hfill ({\eta }_{LL}{\mu }_L-{\alpha }_L)-\gamma \hfill \end{array}\right)=0$$

and are solutions of the second order equation:

$${\gamma }^2-\gamma (({\eta }_{SS}{\mu }_S-{\alpha }_S)+({\eta }_{LL}{\mu }_L-{\alpha }_L))+({\eta }_{SS}{\mu }_S-{\alpha }_S)({\eta }_{LL}{\mu }_L-{\alpha }_L)-{\eta }_{SL}{\eta }_{LS}{\mu }_L{\mu }_S=0$$

which has for discriminant:

$$\Delta ={(({\eta }_{SS}{\mu }_S-{\alpha }_S)-({\eta }_{LL}{\mu }_L-{\alpha }_L))}^2+4{\eta }_{SL}{\eta }_{LS}{\mu }_L{\mu }_S\text{.}$$

Then ${\gamma }_1$, ${\gamma }_2$ are constants defined by the following relations

$${\gamma }_1=\frac{1}{2}(\frac{{\eta }_{SS}}{{\rho }_S}+\frac{{\eta }_{LL}}{{\rho }_L}-({\alpha }_S+{\alpha }_L))+\frac{1}{2}\sqrt{\Delta }\text{,}$$ (8)

$${\gamma }_2=\frac{1}{2}(\frac{{\eta }_{SS}}{{\rho }_S}+\frac{{\eta }_{LL}}{{\rho }_L}-({\alpha }_S+{\alpha }_L))-\frac{1}{2}\sqrt{\Delta }\text{.}$$

An eigenvector is orthogonal to each rows of the matrix:

$$\left(\begin{array}{cc}\hfill ({\eta }_{SS}{\mu }_S-{\alpha }_S)-\gamma \hfill & \hfill {\eta }_{SL}{\mu }_L\hfill \\ \hfill {\eta }_{LS}{\mu }_S\hfill & \hfill ({\eta }_{LL}{\mu }_L-{\alpha }_L)-\gamma \hfill \end{array}\right)\left(\begin{array}{c}\hfill v_1\hfill \\ \hfill v_2\hfill \end{array}\right)=0\text{,}$$

then necessary,

$$\left(\begin{array}{c}\hfill v_1^i\hfill \\ \hfill v_2^i\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -{\eta }_{SL}{\mu }_L\hfill \\ \hfill ({\eta }_{SS}{\mu }_S-{\alpha }_S)-{\gamma }_i\hfill \end{array}\right)\text{for}i=1,2\text{.}$$

Let $D=\mathrm{diag}({\gamma }_1,{\gamma }_2)$. The matrix in the right term of Eq. (7) admits the representation:

$$\left(\begin{array}{cc}\hfill ({\eta }_{SS}{\mu }_S-{\alpha }_S)\hfill & \hfill {\eta }_{SL}{\mu }_L\hfill \\ \hfill {\eta }_{LS}{\mu }_S\hfill & \hfill ({\eta }_{LL}{\mu }_L-{\alpha }_L)\hfill \end{array}\right)=VD{V}^{-1}\text{,}$$

where $V$ is the matrix of eigenvectors, as defined in Eq. (5). If two new variables are defined as follows:

$$\left(\begin{array}{c}\hfill u_S\hfill \\ \hfill u_L\hfill \end{array}\right)=V^{-1}\left(\begin{array}{c}\hfill m_S\hfill \\ \hfill m_L\hfill \end{array}\right)\text{.}$$

The system (7) is decoupled into two independent ODE’s:

$$\frac{\partial }{\partial t}\left(\begin{array}{c}\hfill u_S\hfill \\ \hfill u_L\hfill \end{array}\right)=V^{-1}\left(\begin{array}{c}\hfill {\alpha }_S{\theta }_S\hfill \\ \hfill {\alpha }_L{\theta }_L\hfill \end{array}\right)+\left(\begin{array}{cc}\hfill {\gamma }_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\gamma }_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill u_1\hfill \\ \hfill u_2\hfill \end{array}\right)\text{.}$$ (9)

And introducing the following notations

$$V^{-1}\left(\begin{array}{c}\hfill {\alpha }_S{\theta }_S\hfill \\ \hfill {\alpha }_L{\theta }_L\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {ϵ}_1\hfill \\ \hfill {ϵ}_2\hfill \end{array}\right)\text{,}$$

leads to the solutions for the system (9):

$$u_S\left(t\right)=\frac{{ϵ}_1}{{\gamma }_1}(e^{{\gamma }_{1}t}-1)+d_1{e}^{{\gamma }_{1}t}$$

$$u_L\left(t\right)=\frac{{ϵ}_2}{{\gamma }_2}(e^{{\gamma }_{2}t}-1)+d_2{e}^{{\gamma }_{2}t}$$

where $d={(d_1,d_2)}^{\prime }$ is such that $d=V^{-1}({\lambda }_0^S,{\lambda }_0^L)$ or in matrix form,

$$\left(\begin{array}{c}\hfill u_S\hfill \\ \hfill u_L\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill \frac{1}{{\gamma }_1}(e^{{\gamma }_{1}t}-1)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{{\gamma }_2}(e^{{\gamma }_{2}t}-1)\hfill \end{array}\right)V^{-1}\left(\begin{array}{c}\hfill {\alpha }_S{\theta }_S\hfill \\ \hfill {\alpha }_L{\theta }_L\hfill \end{array}\right)+\left(\begin{array}{cc}\hfill e^{{\gamma }_{1}t}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill e^{{\gamma }_{2}t}\hfill \end{array}\right)V^{-1}\left(\begin{array}{c}\hfill {\lambda }_0^S\hfill \\ \hfill {\lambda }_0^L\hfill \end{array}\right)\text{.}$$

 □

According to this last result, intensities are stable, in the sense that the limits of ${\lambda }_t^S$ and ${\lambda }_t^L$ exist when $t\to +\infty$ if and only if ${\gamma }_1$ and ${\gamma }_2$ are negative. In this case, the asymptotic expectations are equal to

$$\underset{t\to \infty }{lim}\left(\begin{array}{c}\hfill \mathbb{E}\left(\underset{t}{\overset{S}{\lambda }}\right|{\mathcal{G}}_0)\hfill \\ \hfill \mathbb{E}\left(\underset{t}{\overset{L}{\lambda }}\right|{\mathcal{G}}_0)\hfill \end{array}\right)=V\left(\begin{array}{cc}\hfill -\frac{1}{{\gamma }_1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{1}{{\gamma }_2}\hfill \end{array}\right)V^{-1}\left(\begin{array}{c}\hfill {\alpha }_S{\theta }_S\hfill \\ \hfill {\alpha }_L{\theta }_L\hfill \end{array}\right)\text{.}$$ (10)

If ${\gamma }_1>0$ or ${\gamma }_2>0$, the frequency of claim arrivals and the volatility of stocks are not bounded when $t\to +\infty$. For this reason, we assume in the remainder of this work that ${\gamma }_1\le 0$ or ${\gamma }_2\le 0$. The next corollary links the expected values of chronometers to the chronological time. And it may be proven by direct integration of expressions (4).

Corollary 2.2

The expected values of ${\tau }_t^S$ and ${\tau }_t^L$ are given by

$$\left(\begin{array}{c}\hfill \mathbb{E}\left({\tau }_t^S\right|{\mathcal{G}}_0)\hfill \\ \hfill \mathbb{E}\left({\tau }_t^S\right|{\mathcal{G}}_0)\hfill \end{array}\right)=V\left(\begin{array}{cc}\hfill \frac{1}{{\gamma }_1^2}(e^{{\gamma }_{1}t}-1)-\frac{t}{{\gamma }_1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{{\gamma }_2^2}(e^{{\gamma }_{2}t}-1)-\frac{t}{{\gamma }_2}\hfill \end{array}\right)V^{-1}\left(\begin{array}{c}\hfill {\alpha }_S{\theta }_S\hfill \\ \hfill {\alpha }_L{\theta }_L\hfill \end{array}\right)+V\left(\begin{array}{cc}\hfill \frac{1}{{\gamma }_1}(e^{{\gamma }_{1}t}-1)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{{\gamma }_2}(e^{{\gamma }_{2}t}-1)\hfill \end{array}\right)V^{-1}\left(\begin{array}{c}\hfill {\lambda }_0^S\hfill \\ \hfill {\lambda }_0^L\hfill \end{array}\right)\text{.}$$ (11)

The variances of intensities do not admit any closed form expression. However, we can compute them numerically by solving a system of ordinary differential equations:

Proposition 2.3

Let us denote the variances of ${\lambda }^S$ and ${\lambda }^L$ by $V_S\left(t\right)=\mathbb{E}\left({\left({\lambda }_t^S\right)}^2\right)-{\left(\mathbb{E}\left({\lambda }_t^S\right)\right)}^2$ and $V_L\left(t\right)=\mathbb{E}\left({\left({\lambda }_t^L\right)}^2\right)-{\left(\mathbb{E}\left({\lambda }_t^L\right)\right)}^2$ . Their covariance is $V_{SL}\left(t\right)=\mathbb{E}\left({\lambda }_t^S{\lambda }_t^L\right)-\mathbb{E}\left({\lambda }_t^S\right)\mathbb{E}\left({\lambda }_t^L\right)$ . $V_S$ , $V_L$ and $V_{SL}$ are solutions of the following system of ODE’s:

$$\frac{\partial }{\partial t}\left(\begin{array}{c}\hfill V_S\hfill \\ \hfill V_L\hfill \\ \hfill V_{SL}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {\eta }_{SS}^2\frac{2}{{\rho }_S^2}\hfill & \hfill {\eta }_{SL}^2\frac{2}{{\rho }_L^2}\hfill \\ \hfill {\eta }_{LS}^2\frac{2}{{\rho }_S^2}\hfill & \hfill {\eta }_{LL}^2\frac{2}{{\rho }_L^2}\hfill \\ \hfill {\eta }_{SS}{\eta }_{LS}\frac{2}{{\rho }_S^2}\hfill & \hfill {\eta }_{LL}{\eta }_{SL}\frac{2}{{\rho }_L^2}\hfill \end{array}\right)\left(\begin{array}{c}\hfill m_S\left(t\right)\hfill \\ \hfill m_L\left(t\right)\hfill \end{array}\right)+\left(\begin{array}{ccc}\hfill 2({\eta }_{SS}\frac{1}{{\rho }_S}-{\alpha }_S)\hfill & \hfill 0\hfill & \hfill 2{\eta }_{SL}\frac{1}{{\rho }_L}\hfill \\ \hfill 0\hfill & \hfill 2({\eta }_{LL}\frac{1}{{\rho }_L}-{\alpha }_L)\hfill & \hfill 2{\eta }_{LS}\frac{1}{{\rho }_S}\hfill \\ \hfill {\eta }_{LS}\frac{1}{{\rho }_S}\hfill & \hfill {\eta }_{SL}\frac{1}{{\rho }_L}\hfill & \hfill ({\eta }_{SS}\frac{1}{{\rho }_S}+{\eta }_{LL}\frac{1}{{\rho }_L})-{\alpha }_S-{\alpha }_L\hfill \end{array}\right)\left(\begin{array}{c}\hfill V_S\hfill \\ \hfill V_L\hfill \\ \hfill V_{SL}\hfill \end{array}\right)\text{,}$$ (12)

and satisfy the initial conditions

$$V_i\left(0\right)=0\text{for}i=S,L,SL\text{.}$$

Proof

Let us introduce the notations: $g_S={\left({\lambda }_t^S\right)}^2$, $g_L={\left({\lambda }_t^L\right)}^2$ and $g_{SL}={\lambda }_t^S{\lambda }_t^L$. Infinitesimal generators of these functions are:

$$\mathcal{A}{g}_S=2{\lambda }_t^S{\alpha }_S{\theta }_S-2{\left({\lambda }_t^S\right)}^2{\alpha }_S+{\lambda }_t^S{\int }_{-\infty }^{+\infty }2{\lambda }_t^S{\eta }_{SS}z+{\left({\eta }_{SS}z\right)}^{2}d{\nu }_S\left(z\right)+{\lambda }_t^L{\int }_{-\infty }^{+\infty }2{\lambda }_t^S{\eta }_{SL}z+{\left({\eta }_{SL}z\right)}^{2}d{\nu }_L\left(z\right)\text{,}$$

$$\mathcal{A}{g}_L=2{\lambda }_t^L{\alpha }_L{\theta }_L-2{\left({\lambda }_t^L\right)}^2{\alpha }_L+{\lambda }_t^L{\int }_{-\infty }^{+\infty }2{\lambda }_t^L{\eta }_{LL}z+{\left({\eta }_{LL}z\right)}^{2}d{\nu }_L\left(z\right)+{\lambda }_t^S{\int }_{-\infty }^{+\infty }2{\lambda }_t^L{\eta }_{LS}z+{\left({\eta }_{LS}z\right)}^{2}d{\nu }_S\left(z\right)\text{,}$$

$$\mathcal{A}{g}_{SL}={\alpha }_S({\theta }_S-{\lambda }_t^S){\lambda }_t^L+{\alpha }_L({\theta }_L-{\lambda }_t^L){\lambda }_t^S+{\lambda }_t^S{\int }_{-\infty }^{+\infty }({\lambda }_t^S+{\eta }_{SS}z)({\lambda }_t^L+{\eta }_{LS}z)-{\lambda }_t^S{\lambda }_t^{L}d{\nu }_S\left(z\right)+{\lambda }_t^L{\int }_{-\infty }^{+\infty }({\lambda }_t^S+{\eta }_{SL}z)({\lambda }_t^L+{\eta }_{LL}z)-{\lambda }_t^S{\lambda }_t^{L}d{\nu }_L\left(z\right)$$

we denote $v_S=\mathbb{E}\left({\left({\lambda }_t^S\right)}^2\right)$, $v_L=\mathbb{E}\left({\left({\lambda }_t^L\right)}^2\right)$, $v_{SL}=\mathbb{E}\left({\lambda }_t^S{\lambda }_t^L\right)$, and ${\xi }_S=\mathbb{E}\left({\left(J^S\right)}^2\right)$. ${\nu }_S$, ${\nu }_L$ and ${\nu }_{SL}$ are solutions of a system of ODE’s:

$$\frac{\partial }{\partial t}{v}_S=2m_S\left(t\right){\alpha }_S{\theta }_S-2v_S\left(t\right){\alpha }_S+2v_L\left(t\right){\eta }_{SS}{\mu }_S+m_S\left(t\right){\eta }_{SS}^2\mathbb{E}\left({\left(J^S\right)}^2\right)+2v_{SL}\left(t\right){\eta }_{SL}{\mu }_L+m_L\left(t\right){\eta }_{SL}^2\mathbb{E}\left({\left(J^L\right)}^2\right)$$

$$\frac{\partial }{\partial t}{v}_L=2m_L\left(t\right){\alpha }_L{\theta }_L-2v_L\left(t\right){\alpha }_L+2v_L\left(t\right){\eta }_{LL}{\mu }_L+m_L\left(t\right){\eta }_{LL}^2\mathbb{E}\left({\left(J^L\right)}^2\right)+2v_3\left(t\right){\eta }_{LS}{\mu }_S+m_1\left(t\right){\eta }_{LS}^2\mathbb{E}\left({\left(J^S\right)}^2\right)$$ (13)

$$\frac{\partial }{\partial t}{v}_{SL}=m_L\left(t\right){\alpha }_S{\theta }_S-{\alpha }_S{v}_{SL}\left(t\right)+m_S\left(t\right){\alpha }_L{\theta }_L-{\alpha }_L{v}_{SL}\left(t\right)+v_S\left(t\right){\eta }_{LS}{\mu }_S+v_{SL}\left(t\right){\eta }_{SS}{\mu }_S+m_S\left(t\right){\eta }_{SS}{\eta }_{LS}\mathbb{E}\left({\left(J^S\right)}^2\right)+v_L\left(t\right){\eta }_{SL}{\mu }_L+v_{SL}\left(t\right){\delta }_{LL}{\mu }_L+m_L\left(t\right){\eta }_{LL}{\eta }_{SL}\mathbb{E}\left({\left(J^L\right)}^2\right)\text{.}$$

As centered second moments $V_i\left(t\right)$, are linked to non centered ones, $v_i$ by the next differential equations

$$\{\begin{array}{cc}\frac{\partial }{\partial t}{V}_i=\frac{\partial }{\partial t}{v}_i-2m_i\frac{\partial }{\partial t}{m}_i& i=S,L\\ \frac{\partial }{\partial t}{V}_{SL}=\frac{\partial }{\partial t}{v}_{SL}-m_S\frac{\partial }{\partial t}{m}_L-m_L\frac{\partial }{\partial t}{m}_S\text{.}\end{array}$$

Combining Eqs. (7), (13) allows us to conclude. □

The variances and correlation of intensities are then computable by an Euler’s method applied to ODE’s (12). The next proposition allows us to calculate numerically the joint moments of ${\tau }_t^S$ and ${\tau }_t^L$. This result is used later to evaluate covariances and correlations.

Proposition 2.4

The generating function of ${\tau }_t^S$ and ${\tau }_t^L$ is an exponential affine function of intensities

$$\mathbb{E}(e^{{\omega }_S{\tau }_s^S+{\omega }_L{\tau }_s^L}\mid {\mathcal{G}}_t)=e^{{\omega }_S{\tau }_t^S+{\omega }_L{\tau }_t^L}\times exp(A(t,s)+{\left(\begin{array}{c}\hfill B_S(t,s)\hfill \\ \hfill B_L(t,s)\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right))\text{,}$$

where $A(.),B_S(.)$ and $B_L(.)$ satisfies the system of ODE’s

$$\frac{\partial }{\partial t}A=-{\alpha }_S{\theta }_S{B}_S-{\alpha }_L{\theta }_L{B}_L$$

$$\frac{\partial }{\partial t}{B}_S=-{\omega }_S+{\alpha }_S{B}_S-({\varphi }^{J_S}(B_S{\eta }_{SS}+B_L{\eta }_{LS})-1)$$

$$\frac{\partial }{\partial t}{B}_L=-{\omega }_L+{\alpha }_L{B}_L-({\varphi }^{J_L}(B_S{\eta }_{SL}+B_L{\eta }_{LL})-1)$$

with the terminal conditions $B_S(s,s)=B_L(s,s)=0$ and $A(s,s)=0$ .

Proof

The generating function of ${\tau }_s^S$ and ${\tau }_s^L$ for $s\ge t$ is denoted by $f(t,{\tau }_t^S,{\tau }_t^L,{\lambda }_t^S,{\lambda }_t^L)$. and is solution of an Itô SDE:

$$0=f_t+f_{{\tau }^S}{\lambda }_t^S+{\alpha }_S({\theta }_S-{\lambda }_t^S)f_{{\lambda }^S}+f_{{\tau }^L}{\lambda }_t^L+{\alpha }_L({\theta }_L-{\lambda }_t^L)f_{{\lambda }^L}+{\lambda }_t^S{\int }_0^{+\infty }f(t,{\tau }_t^S,{\tau }_t^L,{\lambda }_t^S+{\eta }_{SS}z,{\lambda }_t^L+{\eta }_{LS}z)-f{\nu }^S\left(dz\right)+{\lambda }_t^L{\int }_0^{+\infty }f(t,{\tau }_t^S,{\tau }_t^L,{\lambda }_t^S+{\eta }_{SL}z,{\lambda }_t^L+{\eta }_{LL}z)-f{\nu }^L\left(dz\right)\text{.}$$ (14)

Let us assume that $f$ is an exponential affine function of ${\tau }_t^S,{\tau }_t^L$ and ${\lambda }_t^S,{\lambda }_t^L$:

$$f=exp(A(t,s)+{\left(\begin{array}{c}\hfill B_S(t,s)\hfill \\ \hfill B_L(t,s)\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right)+{\left(\begin{array}{c}\hfill C_S(t,s)\hfill \\ \hfill C_L(t,s)\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\tau }_t^S\hfill \\ \hfill {\tau }_t^L\hfill \end{array}\right))\text{,}$$

where coefficients are functions of time. Under this assumption, the partial derivatives of $f$ are given by:

$$f_t=(\frac{\partial }{\partial t}A+{\left(\begin{array}{c}\hfill \frac{\partial }{\partial t}{B}_S\hfill \\ \hfill \frac{\partial }{\partial t}{B}_L\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right)+{\left(\begin{array}{c}\hfill \frac{\partial }{\partial t}{C}_S\hfill \\ \hfill \frac{\partial }{\partial t}{C}_L\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\tau }_t^S\hfill \\ \hfill {\tau }_t^L\hfill \end{array}\right))\times f\text{,}$$

$$f_{{\tau }^S}=C_{S}f{f}_{{\tau }^L}=C_{L}f{f}_{{\lambda }^S}=B_{S}f{f}_{{\lambda }^L}=B_{L}f\text{.}$$

And integrands in Eq. (14) are rewritten as follows:

$$f(t,{\tau }_t^S,{\tau }_t^L,{\lambda }_t^S+{\eta }_{SS}z,{\lambda }_t^L+{\eta }_{LS}z)-f=f[exp\left((B_S{\eta }_{SS}+B_L{\eta }_{LS})z\right)-1]$$

$$f(t,{\tau }_t^S,{\tau }_t^L,{\lambda }_t^S+{\eta }_{SL}z,{\lambda }_t^L+{\eta }_{LL}z)-f=f[exp\left((B_S{\eta }_{SL}+B_L{\eta }_{LL})z\right)-1]\text{.}$$

If ${\varphi }^{J_S}\left(z\right)=\frac{{\rho }_S}{{\rho }_S-z}$ and ${\varphi }^{J_L}\left(z\right)=\frac{{\rho }_L}{{\rho }_L-z}$ are the moment generating functions of jumps, we obtain that

$$0=\frac{\partial }{\partial t}A+{\left(\begin{array}{c}\hfill \frac{\partial }{\partial t}{B}_S\hfill \\ \hfill \frac{\partial }{\partial t}{B}_L\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right)+{\left(\begin{array}{c}\hfill \frac{\partial }{\partial t}{C}_S\hfill \\ \hfill \frac{\partial }{\partial t}{C}_L\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\tau }_t^S\hfill \\ \hfill {\tau }_t^L\hfill \end{array}\right)+C_S{\lambda }_t^S+{\alpha }_S({\theta }_S-{\lambda }_t^S)B_S+C_L{\lambda }_t^L+{\alpha }_L({\theta }_L-{\lambda }_t^L)B_L+{\lambda }_t^S({\varphi }^{J_S}(B_S{\eta }_{SS}+B_L{\eta }_{LS})-1)+{\lambda }_t^L({\varphi }^{J_L}(B_S{\eta }_{SL}+B_L{\eta }_{LL})-1)\text{.}$$

Then necessarily $C_S={\omega }_S$ and $C_L={\omega }_L$. The Itô equation becomes then:

$$0=\frac{\partial }{\partial t}A+{\alpha }_S{\theta }_S{B}_S+{\alpha }_L{\theta }_L{B}_L+{\left(\begin{array}{c}\hfill \frac{\partial }{\partial t}{B}_S\hfill \\ \hfill \frac{\partial }{\partial t}{B}_L\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right)+{\omega }_S{\lambda }_t^S-{\alpha }_S{B}_S{\lambda }_t^S+{\omega }_L{\lambda }_t^L-{\alpha }_L{B}_L{\lambda }_t^L+{\lambda }_t^S({\varphi }^{J_S}(B_S{\eta }_{SS}+B_L{\eta }_{LS})-1)+{\lambda }_t^L({\varphi }^{J_L}(B_S{\eta }_{SL}+B_L{\eta }_{LL})-1)$$

and regrouping terms allows us to conclude that

$$\frac{\partial }{\partial t}A=-{\alpha }_S{\theta }_S{B}_S-{\alpha }_L{\theta }_L{B}_L$$

$$\frac{\partial }{\partial t}{B}_S=-{\omega }_S+{\alpha }_S{B}_S-({\varphi }^{J_S}(B_S{\eta }_{SS}+B_L{\eta }_{LS})-1)$$

$$\frac{\partial }{\partial t}{B}_L=-{\omega }_L+{\alpha }_L{B}_L-({\varphi }^{J_L}(B_S{\eta }_{SL}+B_L{\eta }_{LL})-1)\text{.}$$

 □

From this last proposition, we infer that the first cross moment of chronometers is equal to

$$\mathbb{E}({\tau }_t^S{\tau }_t^L\mid {\mathcal{G}}_0)={\frac{\partial }{\partial {\omega }_S}\frac{\partial }{\partial {\omega }_L}\mathbb{E}(e^{{\omega }_S{\tau }_s^S+{\omega }_L{\tau }_s^L}\mid {\mathcal{G}}_t)|}_{{\omega }_S={\omega }_L=0}={\frac{\partial }{\partial {\omega }_S}\frac{\partial }{\partial {\omega }_L}(A(t,s)+{\left(\begin{array}{c}\hfill B_S(t,s)\hfill \\ \hfill B_L(t,s)\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right))|}_{{\omega }_S={\omega }_L=0}$$

that unfortunately does not admit any closed form expression. However, this cross-moment is computable numerically by a finite difference method. To understand the influence of mutual and self excitation between intensities on standard deviations and correlations, five numerical tests are conducted. The sets of parameters considered for this exercise and results are reported in Table 1.

Table 1.

This table reports the one year expectations, standard deviations of ${\tau }_t^S$, ${\tau }_t^L$ and their correlation, for five sets of parameters.

Parameters	Test 1	Test 2	Test 3	Test 4	Test 5
${\epsilon }_1$	0.00	0.04	0.08	0.12	0.16
${\alpha }_S,{\alpha }_L$	19.91	19.91	19.91	19.91	19.91
${\theta }_S,{\theta }_L$	0.48	0.48	0.48	0.48	0.48
${\rho }_S,{\rho }_L$	0.49	0.49	0.49	0.49	0.49
${\epsilon }_2$	5.72	5.72	5.72	5.72	5.72
$\mathbb{E}\left({\tau }_1^S\right)=\mathbb{E}\left({\tau }_1^L\right)$	1.07	1.13	1.18	1.24	1.30
$\mathrm{std}\left({\tau }_1^S\right)=\mathrm{std}\left({\tau }_1^L\right)$	4.12	4.26	4.46	4.70	4.98
$\mathrm{corr}({\tau }_1^S,{\tau }_1^L)$	0.00	0.03	0.07	0.10	0.12

Speeds and levels of reversion for ${\lambda }_t^1$ and ${\lambda }_t^2$ are assumed equal. Whereas the matrix $\Xi$ of mutual excitation weights, is parametrized as follows:

$$\Xi =\left(\begin{array}{cc}\hfill {\eta }_{SS}\hfill & \hfill {\eta }_{SL}\hfill \\ \hfill {\eta }_{LS}\hfill & \hfill {\eta }_{LL}\hfill \end{array}\right)={ϵ}_1\left(\begin{array}{cc}\hfill cos\left({ϵ}_2\right)\hfill & \hfill sin\left({ϵ}_2\right)\hfill \\ \hfill sin\left({ϵ}_2\right)\hfill & \hfill cos\left({ϵ}_2\right)\hfill \end{array}\right)\text{.}$$

When ${ϵ}_2$ is null, ${\eta }_{SL}$ and ${\eta }_{LS}$ are null and stochastic clocks are independent. Increasing ${ϵ}_2$ raises the correlation between ${\tau }_1^S$ and ${\tau }_1^L$ from 0% to 12%. It also speeds up on average the clocks and standard deviations slightly move up. This confirms that a large range of positive correlations may be achieved by introducing mutual excitation in the dynamics of intensities. Notice also that with this parametrization of $\Xi$, expectations and standard deviations of ${\tau }_t^S$ and ${\tau }_t^L$ are equal and independent from ${ϵ}_2$. The left plot of Fig. 1 shows the term structure of standard deviations of ${\tau }_t^S$ and ${\tau }_t^L$, for different initial values of intensities. It emphasizes that just after a jump of ${\lambda }_t^S$ or ${\lambda }_t^L$, the standard deviations of ${\tau }_t^S$ and ${\tau }_t^L$ step up to a higher level. At longer term, the growth of these standard deviations becomes nearly linear. The right plot of the same figure presents the curve of correlations between ${\tau }_t^S$ and ${\tau }_t^L$ for three different sets of initial values of ${\lambda }_t^S$ and ${\lambda }_t^L$. When ${\lambda }_0^S$ and ${\lambda }_0^L$ take their lowest values, that are ${\theta }_S$ and ${\theta }_L$, the term structure of correlation is an increasing function of time that reverts at medium term to a constant level. When ${\lambda }_0^S$ and ${\lambda }_0^L$ are significantly higher than ${\theta }_S$ and ${\theta }_L$, the curve of correlations is a humped function of time, that also reverts to a constant level.

Fig. 1.

Left plot: evolution of variances of ${\tau }_t^S$, ${\tau }_t^L$ with respect to time. Right plot: correlation between ${\tau }_t^S$, ${\tau }_t^L$ with respect to time. Parameters used are those reported in Table 1, in the column labeled “Test 3”. Three different sets of values for ${\lambda }_0^S$ and ${\lambda }_0^L$ are considered: (1) ${\lambda }_0^S={\lambda }_0^L={\theta }_S={\theta }_L$ (2) ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(\frac{2}{{\rho }_S},\frac{2}{{\rho }_L})}^{\top }$ (3) ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(\frac{3}{{\rho }_S},0)}^{\top }$.

3. Financial assets and the insurance claims process

We develop now the dynamics of markets in which the insurance company operates. We assume that the financial market is composed of two assets: cash and stocks. The risk free rate earned by the cash account is denoted by $r$ and is constant. The stock price, noted $S_t$, is a time-changed geometric Brownian motion:

$$\frac{d{S}_t}{S_t}=rdt+(\mu -r)d{\tau }_t^S+\sigma d{W}_{{\tau }_t^S}^S$$ (15)

where $\mu$ and $\sigma$ are positive constants and $W_t^S$ is a Brownian motion. In this approach, only the risk premium depends upon the market time scale. The Brownian motion being a stable process, this relation is equivalent to the following one which emphasizes that the drift and the volatility of stock prices are random and proportional to the intensity of the financial clock:

$$\frac{d{S}_t}{S_t}=(r+(\mu -r){\lambda }_t^S)dt+\sigma \sqrt{{\lambda }_t^S}d{W}_t^S\text{.}$$

In a certain way, the dynamics of stock prices is comparable to these stochastic volatility models, with a random drift. In our approach, the volatility is indeed mean reverting like in the Heston’s model, except that it is driven by a bivariate self-exciting jump process. Using the Itô’s lemma leads to the following relation for the log-price

$$dln{S}_t=(r+(\mu -r-\frac{{\sigma }^2}{2}){\lambda }_t^S)dt+\sigma \sqrt{{\lambda }_t^S}d{W}_t^S$$

and from which we infer the stock price

$$S_t=S_{0}exp(rt+(\mu -r-\frac{{\sigma }^2}{2}){\int }_0^t{\lambda }_s^{S}ds+\sigma {\int }_0^t\sqrt{{\lambda }_s^S}d{W}_s^S),=S_{0}exp(rt+(\mu -r-\frac{{\sigma }^2}{2}){\tau }_t^S+\sigma W_{{\tau }_t^S}^S)\text{.}$$ (16)

On the side of liabilities, we consider a time-changed version of a risk process that is the difference accumulated premiums and aggregated claims:

$$L_t=c{\tau }_t^L-\sum _{i=1}^{N_{{\tau }_t^L}}{J}_i$$

before reinsurance. $c$ is the premium rate, $J_i$ are the random claims and $N_t$ is a Poisson process with a constant intensity $\lambda$. The statistical distribution of $J$ is not specified but we assume that its moment generating function, denoted by ${\varphi }^J\left(\omega \right)=\mathbb{E}\left(e^{\omega J}\right)$, exists. Its probability density function is written ${\nu }^J(.)$. The premium rate is assumed strictly bigger than the expected claims, $c>\lambda \mathbb{E}\left(J\right)$. If this condition is not fulfilled, the ruin occurs with certainty. Notice also that the dynamics of liabilities is equivalent to the following one:

$$d{L}_t=cd{\tau }_t^L-Jd{N}_{{\tau }_t^L}=c{\lambda }_t^{L}dt-Jd{N}_t^b$$ (17)

where $N_t^b$ is a point process with marks of size $J$ and an intensity equal to $\left(\lambda {\int }_0^t{\lambda }_s^{L}ds\right)$. Indeed by definition, the probability of observing $k$ claims till (chronological) time $t$ is equal to

$$P(N_{{\tau }_t^L}=k)=\frac{{\left({\int }_0^{{\tau }_t^L}\lambda ds\right)}^k}{k!}{e}^{-\lambda {\tau }_t^L}=\frac{{\left({\int }_0^t\lambda {\lambda }_s^{L}ds\right)}^k}{k!}{e}^{-{\int }_0^t\lambda {\lambda }_s^{L}ds}=P(N_t^b=k)\text{.}$$

The natural filtration associated to $S_t$ and $L_t$ is denoted by ${\left({\mathcal{H}}_t\right)}_t$ and the augmented filtration carrying all the information is ${\mathcal{F}}_t≔{\mathcal{H}}_t\vee {\mathcal{G}}_t$ (remember that ${\mathcal{G}}_t$ is the natural filtration of chronometers). In the specification of our model, the claims process is not directly correlated to the financial market. However a dependence appears through the correlation that exists between stochastic clocks driving insurance and financial markets. This dependence is measured by the next proposition that has also an interesting economic interpretation.

Proposition 3.1

The covariance between the logarithm of stocks prices and the risk process is given by

$$(\mathbb{E}(ln\left(S_t\right)L_t|{\mathcal{F}}_0)-\mathbb{E}(ln\left(S_t\right)|{\mathcal{F}}_0)\mathbb{E}\left(L_t\right)|{\mathcal{F}}_0)=(c-\lambda \mathbb{E}\left(J\right))(\mu -r-\frac{1}{2}{\sigma }^2)[\mathbb{E}\left({\tau }_t^S{\tau }_t^L\right|{\mathcal{G}}_0)-\mathbb{E}\left({\tau }_t^S\right|{\mathcal{G}}_0)\mathbb{E}\left({\tau }_t^L\right|{\mathcal{G}}_0)]\text{.}$$

Proof

To prove this result, it is sufficient to remember the expressions of log prices and of the risk process:

$$ln{S}_t=ln{S}_0+rt+(\mu -r-\frac{{\sigma }^2}{2}){\tau }_t^S+\sigma W_{{\tau }_t}^S$$

$$L_t=L_0+c{\tau }_t^L-\sum _{i=1}^{N_{{\tau }_t^L}}{J}_i\text{.}$$

A direct calculation leads to the following expressions for the cross expectation and product of expectations (we momentarily forget the filtration with respect to which we calculate these quantities to lighten notations):

$$\mathbb{E}(ln\left(S_t\right)L_t)=ln{S}_0{L}_0+ln{S}_{0}c\mathbb{E}\left({\tau }_t^L\right)-ln{S}_0\mathbb{E}\left(\sum _{i=1}^{N_{{\tau }_t^L}}{J}_i\right)+r{L}_{0}t+rtc\mathbb{E}\left({\tau }_t^L\right)-rt\mathbb{E}\left(\sum _{i=1}^{N_{{\tau }_t^L}}{J}_i\right)+L_0(\mu -r-\frac{1}{2}{\sigma }^2)\mathbb{E}\left({\tau }_t^S\right)+c(\mu -r-\frac{1}{2}{\sigma }^2)\mathbb{E}\left({\tau }_t^S{\tau }_t^L\right)-(\mu -r-\frac{{\sigma }^2}{2})\mathbb{E}\left({\tau }_t^S\sum _{i=1}^{N_{{\tau }_t^L}}{J}_i\right)$$

and

$$\mathbb{E}(ln{S}_t)\mathbb{E}\left(L_t\right)=ln{S}_0\mathbb{E}\left({\tau }_t^S\right)+cln{S}_0\mathbb{E}\left({\tau }_t^L\right)-ln{S}_0\mathbb{E}\left({\tau }_t^S\sum _{i=1}^{N_{{\tau }_t^L}}{J}_i\right)+r{L}_{0}t+rtc\mathbb{E}\left({\tau }_t^L\right)-rt\mathbb{E}\left(\sum _{i=1}^{N_{{\tau }_t^L}}{J}_i\right)+L_0(\mu -r-\frac{1}{2}{\sigma }^2)\mathbb{E}\left({\tau }_t^S\right)+c(\mu -r-\frac{{\sigma }^2}{2})\mathbb{E}\left({\tau }_t^S\right)\mathbb{E}\left({\tau }_t^L\right)-(\mu -r-\frac{{\sigma }^2}{2})\mathbb{E}\left({\tau }_t^S\right)\mathbb{E}\left(\sum _{i=1}^{N_{{\tau }_t^L}}{J}_i\right)\text{.}$$

The covariance is then

$$(\mathbb{E}(ln\left(S_t\right)L_t)-\mathbb{E}(ln\left(S_t\right))\mathbb{E}\left(L_t\right))=c(\mu -r-\frac{1}{2}{\sigma }^2)[\mathbb{E}\left({\tau }_t^S{\tau }_t^L\right|{\mathcal{G}}_0)-\mathbb{E}\left({\tau }_t^S\right|{\mathcal{G}}_0)\mathbb{E}\left({\tau }_t^L\right|{\mathcal{G}}_0)]-(\mu -r-\frac{{\sigma }^2}{2})[\mathbb{E}\left(J\right)\mathbb{E}\left({\tau }_t^S{N}_{{\tau }_t^L}\right)-\lambda \mathbb{E}\left(J\right)\mathbb{E}\left({\tau }_t^S\right|{\mathcal{G}}_0)\mathbb{E}\left({\tau }_t^L\right|{\mathcal{G}}_0)]\text{.}$$

Finally, nesting conditional expectations leads to

$$\mathbb{E}\left({\tau }_t^S{N}_{{\tau }_t^L}\right|{\mathcal{F}}_0)=\mathbb{E}\left(\mathbb{E}\left({\tau }_t^S{N}_{{\tau }_t^L}\right|{\mathcal{F}}_0\vee {\mathcal{G}}_t)\right|{\mathcal{F}}_0)=\lambda \mathbb{E}\left(\left({\tau }_t^S\right){\tau }_t^L\right|{\mathcal{F}}_0)\text{.}$$

 □

This result appeals several comments. Firstly, the correlation between log-prices and claims is proportional to the one of stochastic clocks. Secondly, it depends on the product of the average risk premium of stocks, ($\mu -r-\frac{1}{2}{\sigma }^2$), and of the insurer’s average profit, $(c-\lambda \mathbb{E}\left(J\right))$, that may also be interpreted as an insurance risk premium. If $c$ is equal to the pure premium rate, $c=\lambda \mathbb{E}\left(J\right)$, the expected surplus and the covariance are both null. In this case, we do not observe any linear dependence between assets and liabilities. And this despite the fact that the asset price and risk process are clearly not independent by construction. The covariance differs from zero only if the insurance business is profitable on average. The linear dependence is in this case induced by insurer’s incomes that are next reinvested in the financial market. From an economic point of view, the correlation between insurance and financial markets comes then exclusively from the existence of a risk premium in both segments.

The next proposition introduces the joint mgf of log-prices and liabilities.

Proposition 3.2

The joint moment generating function of $ln{S}_t$ and $L_t$ is given by

$$\mathbb{E}(e^{{\omega }_{S}ln{S}_s+{\omega }_L{L}_s}\mid {\mathcal{F}}_t)=S_t^{{\omega }_S}{e}^{{\omega }_{S}r(s-t)}{e}^{{\omega }_L{L}_t}\mathbb{E}(exp({\omega }_S^b({\tau }_s^L-{\tau }_t^L)+{\omega }_L^b({\tau }_s^L-{\tau }_t^L))\mid {\mathcal{G}}_t)$$ (18)

where

$${\omega }_S^b=(\mu -r-\frac{{\sigma }^2}{2}){\omega }_S+\frac{1}{2}{\omega }_S^2{\sigma }^2$$

$${\omega }_L^b=[{\omega }_{L}c+\lambda ({\varphi }^J(-{\omega }_L)-1)]$$

and

$$\mathbb{E}(exp({\omega }_S^b({\tau }_s^L-{\tau }_t^L)+{\omega }_L^b({\tau }_s^L-{\tau }_t^L))\mid {\mathcal{G}}_t)=exp(A(t,s)+{\left(\begin{array}{c}\hfill B_S(t,s)\hfill \\ \hfill B_L(t,s)\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right))\text{,}$$

with $A$ , $B_S$ and $B_L$ defined in   Proposition  2.4 .

Proof

To prove this result, we use again nested conditional expectations and the fact that conditionally to filtration ${\mathcal{G}}_s$, log prices and liabilities are independent

$$\mathbb{E}(e^{{\omega }_{S}ln{S}_s+{\omega }_L{L}_s}\mid {\mathcal{F}}_t)=\mathbb{E}(\mathbb{E}(e^{{\omega }_{S}ln{S}_s+{\omega }_L{L}_s}\mid {\mathcal{F}}_t\vee {\mathcal{G}}_s)\mid {\mathcal{F}}_t)\mathbb{E}(\mathbb{E}(e^{{\omega }_{S}ln{S}_s}\mid {\mathcal{F}}_t\vee {\mathcal{G}}_s)\mathbb{E}(e^{{\omega }_L{L}_s}\mid {\mathcal{F}}_t\vee {\mathcal{G}}_s)\mid {\mathcal{F}}_t)\text{.}$$

If we remind the expression (16) for $S_s$, the first term in this last product is equal to

$$\mathbb{E}(S_s^{{\omega }_S}\mid {\mathcal{F}}_t\vee {\mathcal{G}}_s)=\mathbb{E}(e^{{\omega }_{S}ln{S}_s}\mid {\mathcal{F}}_t\vee {\mathcal{G}}_s)=S_t^{{\omega }_S}{e}^{{\omega }_{S}r(s-t)}exp\left(((\mu -r-\frac{{\sigma }^2}{2}){\omega }_S+\frac{1}{2}{\omega }_S^2{\sigma }^2)({\tau }_s^S-{\tau }_t^S)\right)\text{.}$$

On the other hand, the moment generating function of the claims process, in absence of any time change, is equal to

$$\mathbb{E}\left(e^{\omega (cs-\sum _{i=1}^{N_s}{J}_i)}\right)=e^{\omega \left(cs\right)}\mathbb{E}\left(e^{-\omega \left(\sum _{i=1}^{N_s}{J}_i\right)}\right)=exp\left([\omega c+\lambda ({\varphi }^J(-\omega )-1)]s\right)\text{.}$$

Then

$$\mathbb{E}(e^{{\omega }_L{L}_s}\mid {\mathcal{F}}_t\vee {\mathcal{G}}_s)=e^{{\omega }_L{L}_t}exp\left([{\omega }_{L}c+\lambda ({\varphi }^J(-{\omega }_L)-1)]({\tau }_s^L-{\tau }_t^L)\right)$$

and we can conclude with Proposition 2.4. □

The generating function of cross-moments is derived numerically to calculate the correlation between $S_t$ and $L_t$ reported in Table 2 . We consider the five sets of parameters of Table 1, that were used to analyze the impact of mutual excitation on the correlation between stochastic clocks. In our example, the one year correlation between assets and liabilities is comparable to the correlation between stochastic clocks. For example, a correlation of 7% between ${\tau }_1^S$ and ${\tau }_1^L$ induces a correlation of 8% between $S_1$ and $L_1$. The correlation also depends upon the time horizon that is considered. This point is illustrated in Fig. 2 that shows the term structure of correlations between $ln{S}_t$ and $L_t$, for different values of ${\lambda }_0^S$ and ${\lambda }_0^L$. The correlation between $ln{S}_t$ and $L_t$ is clearly negligible at short term and next reverts at medium term to a constant level. This means that a delay is induced between the occurrence of an event in one market and the reaction of the other market. In other words, there is well contagion between the insurance and financial markets but the dependence is not instantaneous. Some time is needed to assimilate the information flow from a market and to eventually cause a shock in the other market.

Table 2.

This table reports the 1 year expectations, standard deviations and correlation of assets and liabilities. ${\lambda }_0^S={\lambda }_0^L={\theta }_S={\theta }_L$. Parameters defining ${\lambda }_t^S$ and ${\lambda }_t^L$, are those presented in Table 1. Others parameters defining the asset and risk processes are: $\mu =5\%$, $r=0\%$, $\sigma =20\%$, $\lambda =200$. Claims are exponential random variables with a parameter $\rho =1$. The premium rate is $c=\frac{\lambda }{\rho }1.10$.

	Test 1	Test 2	Test 3	Test 4	Test 5
$\mathrm{corr}({\tau }_1^S,{\tau }_1^L)$	0.00	0.03	0.07	0.10	0.12
$\mathbb{E}(ln{S}_1)$	0.04	0.04	0.05	0.05	0.05
$\mathbb{E}\left(L_1\right)$	22.43	23.50	24.63	25.82	27.06
$\mathrm{std}(ln{S}_1)$	0.21	0.22	0.23	0.23	0.24
$\mathrm{std}\left(L_1\right)$	42.23	43.46	45.16	47.33	49.97
$\mathrm{corr}(S_1,L_1)$	0.00	0.04	0.08	0.12	0.15

Fig. 2.

Evolution of the correlation between $S_t$ and $L_t$ with respect to time. Parameters used are those reported in Table 1, in the column labeled “Test 3”. Three different sets of values for ${\lambda }_0^S$ and ${\lambda }_0^L$ are considered: (1) ${\lambda }_0^S={\lambda }_0^L={\theta }_S={\theta }_L$ (2) ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(\frac{2}{{\rho }_S},\frac{2}{{\rho }_L})}^{\top }$ (3) ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(\frac{3}{{\rho }_S},0)}^{\top }$.

In practice, the calibration of this model is a challenging exercise given that assets and liabilities depend on two hidden state variables. This point is detailed in a study of Hainaut (2016d) that proposes an approached method to calibrate similar Lévy process, time-changed by an integrated self-excited subordinator. Here, we only summarize how to adapt this procedure to our ALM framework and refer to the original article for details. The method is based on the premise that intensities remain constant and equal to their asymptotic averages ${\lambda }_{\infty }^S$ and ${\lambda }_{\infty }^L$:

$$\left(\begin{array}{c}\hfill {\lambda }_{\infty }^S\hfill \\ \hfill {\lambda }_{\infty }^L\hfill \end{array}\right)≔\underset{t\to \infty }{lim}\left(\begin{array}{c}\hfill \mathbb{E}\left(\underset{t}{\overset{S}{\lambda }}\right|{\mathcal{G}}_0)\hfill \\ \hfill \mathbb{E}\left(\underset{t}{\overset{L}{\lambda }}\right|{\mathcal{G}}_0)\hfill \end{array}\right)$$ (19)

as defined by Eq. (10). Under this assumption, $ln{S}_t$ and $L_t$ become stationary processes. The probability density function (pdf) for a given set of parameters $\Theta$ may then be computed by inverting numerically the moment generating function (18) with a two dimensions Discrete Fourier Transform (2D DFT). The set of parameters $\Theta$ is next fitted by maximization of the log-likelihood calculated with the numerical pdf. The 2D DFT algorithm is provided in Hainaut (2016d). This study also reveals that despite the bias introduced by the hypothesis $({\lambda }_t^S,{\lambda }_t^L)=({\lambda }_{\infty }^S,{\lambda }_{\infty }^L)$, the fit is of good quality. An alternative approach consists firstly to develop a particle filter to retrieve the sample path of ${\lambda }_t^S$ and ${\lambda }_t^L$ (again we refer to Hainaut (2016d) for a presentation of this filter). And secondly to combine it with a Particle Markov Chain Monte-Carlo method, as explained in Andrieu et al. (2010).

Before investigating the problem of optimal asset allocation, we propose a bound on the ruin probability when insurer’s earnings are not invested in financial markets. We look for an upper bound of $P({\tau }^{\ast }\le \infty )$, where ${\tau }^{\ast }$ is the first time such that $L_t<0$. To infer this bound, that is a common indicator of risk in the actuarial literature, we first determine the conditions under which an exponential combination of processes of the form

$$M_t(g_S,g_L,g_R,\xi )≔exp(g_S{\lambda }_t^S+g_L{\lambda }_t^L+g_R{L}_t-\xi t)$$ (20)

is a local martingale.

Proposition 3.3

If for any $\xi \in {\mathbb{R}}^{+}$ there exists a suitable triplet $(g_S,g_L,g_R)$ solution of the system of equations

$$0=-\xi +g_S{\alpha }_S{\theta }_S+g_L{\alpha }_L{\theta }_L$$ (21)

$$0=-g_S{\alpha }_S+({\varphi }^{J_S}(g_S{\eta }_{SS}+g_L{\eta }_{LS})-1)$$

$$0=-g_L{\alpha }_L+({\varphi }^{J_L}(g_S{\eta }_{SL}+g_L{\eta }_{LL})-1)+g_{R}c+\lambda ({\varphi }^J(-g_R)-1)$$

then $M_t$ is a local martingale.

Proof

$M_t$ is a local martingale if and only if its infinitesimal generator

$$\mathcal{A}{M}_t=-\xi M+g_S{\alpha }_S({\theta }_S-{\lambda }_t^S)M+g_L{\alpha }_L({\theta }_L-{\lambda }_t^L)M+g_{R}c{\lambda }_t^{L}M+{\lambda }_t^{S}M\int (e^{(g_S{\eta }_{SS}+g_L{\eta }_{LS})z}-1){\nu }^S\left(dz\right)+{\lambda }_t^{L}M\int (e^{(g_S{\eta }_{SL}+g_L{\eta }_{LL})z}-1){\nu }^L\left(dz\right)+\lambda {\lambda }_t^{L}M\int (e^{-g_{R}z}-1){\nu }^J\left(dz\right)$$

is null. Regrouping terms leads to the system (21). □

Notice that at this stage, we made assumptions on the distribution of $J_S$ and $J_L$ but not on the claims size $J$. The only constraint is that its mgf exists. We finally have the following expression for the asymptotic probability of ruin:

Proposition 3.4

If $\xi \ge 0$ and $g_R\left(\xi \right)\le 0$ then

$$P({\tau }^{\ast }\le \infty )=\frac{M_0(g_S,g_L,g_R,\xi )}{\underset{T\to \infty }{lim}\mathbb{E}\left(M_{{\tau }^{\ast }}(g_S,g_L,g_R,\xi )\right|{\mathcal{F}}_0,{\tau }^{\ast }\le T)}\text{.}$$ (22)

Proof

For any given time horizon $T$, we have that $exp(g_S{\lambda }_T^S+g_L{\lambda }_T^L+g_R{L}_T-\xi T)$ is a local martingale if conditions (21) are fulfilled. On the other hand, the minimum between ${\tau }^{\ast }$ and $T$ is a stopping time and according to the optional stopping theorem, we infer that

$$\mathbb{E}\left(M_{{\tau }^{\ast }\wedge T}\right|{\mathcal{F}}_0)=P({\tau }^{\ast }>T)\mathbb{E}\left(M_T\right|{\mathcal{F}}_0,{\tau }^{\ast }>T)+P({\tau }^{\ast }\le T)\mathbb{E}\left(M_{{\tau }^{\ast }}\right|{\mathcal{F}}_0,{\tau }^{\ast }\le T)=M_0\text{.}$$

If $g_R<0$ and $\xi \ge 0$ and as processes ${\lambda }_T^S$ and ${\lambda }_T^L$ do not explode, then the first term of the previous equation converges to zero:

$$\underset{T\to \infty }{lim}\mathbb{E}\left(M_T\right|{\mathcal{F}}_0,{\tau }^{\ast }>T)=\underset{T\to \infty }{lim}exp(-\xi T)\mathbb{E}(exp(g_S\underset{T}{\overset{S}{\lambda }}+g_L\underset{T}{\overset{L}{\lambda }}+g_R{L}_T)|{\mathcal{F}}_0,{\tau }^{\ast }>T)=0$$

and $M_0$ is then equal to

$$M_0=P({\tau }^{\ast }\le \infty )\underset{T\to \infty }{lim}\mathbb{E}\left(M_{{\tau }^{\ast }}\right|{\mathcal{F}}_0,{\tau }^{\ast }\le T)\text{.}$$

On the other hand

$$\underset{T\to \infty }{lim}\mathbb{E}\left(M_{{\tau }^{\ast }}\right|{\mathcal{F}}_0,{\tau }^{\ast }\le T)=\underset{T\to \infty }{lim}\mathbb{E}(exp(g_S\underset{{\tau }^{\ast }}{\overset{S}{\lambda }}+g_L\underset{{\tau }^{\ast }}{\overset{L}{\lambda }}+g_R{L}_{{\tau }^{\ast }}-\xi {\tau }^{\ast })|{\mathcal{F}}_0,{\tau }^{\ast }\le T)$$

and we can conclude. □

Finally, we infer the following upper bound on the asymptotic probability of ruin:

Corollary 3.5

If $g_R$ defined as the solution of the nonlinear equation

$$0=g_{R}c+\lambda ({\varphi }^J(-g_R)-1)$$ (23)

is negative, the asymptotic probability of ruin admits the following bound:

$$P({\tau }^{\ast }\le \infty )\le M_0(0,0,g_R,0)=exp\left(g_R{L}_0\right)\text{.}$$

Proof

If $\xi \ge 0$, the asymptotic probability of ruin admits indeed the representation (22). And if $\xi =0$, $g_L\left(\xi \right)=0$ and $g_S\left(\xi \right)=0$ satisfy the first equation of the system of Eqs. (21). $M_t$ is a martingale if $g_R$ is solution of Eq. (23). As $L_{{\tau }^{\ast }}<0$, it follows that

$$\mathbb{E}\left(M_{{\tau }^{\ast }}(g_S,g_L,g_R,\xi )\right|{\mathcal{F}}_0,{\tau }^{\ast }\le T)=\mathbb{E}(exp\left(g_R{L}_{{\tau }^{\ast }}\right)|{\mathcal{F}}_0,{\tau }^{\ast }\le T)>1\text{.}$$

 □

If claims sizes $J$ are exponential random variables of parameter $\rho$, then Eq. (23) admits the solution: $g_R=(\frac{\lambda }{c}-\rho )$. As the premium rate is assumed strictly bigger than the pure premium, $c>\lambda \frac{1}{\rho }$, $g_R<0$ and the asymptotic probability of ruin is bounded by $P({\tau }^{\ast }\le \infty )\le e^{(\frac{\lambda }{c}-\rho )L_0}$. This upper bound is identical to the bound on the asymptotic ruin probability, in a Cramer–Lundberg model.

So as to evaluate the accuracy of this Cramer–Lundberg bound and to validate numerically Eq. (22), simulations are performed. Parameters used to simulate stochastic clocks are those reported in the third column of Table 1. Claims are exponential random variables with $\rho =1$ whereas the frequency of claims is set to $\lambda =200$. The premium includes a safety margin, from 2.5% to 20% and the premium rate is such that $c=\frac{\lambda }{\rho }(1+safetymargin)$. The initial provision $L_0$, is equal to 5. We use an Euler discretized version of Eqs. (2), (17) to simulate 5000 sample paths of $L_t$ over a period of 100 years, with time step of 0.005. The results reported in Table 3 emphasizes that the gap between the real ruin probabilities and the upper bound varies between 5.52% for a margin of 2.5% to 11.60% for a margin of 12.5%. According to Eq. (22), the asymptotic probability of ruin when $\xi =0$, $g_L\left(\xi \right)=0$ and $g_S\left(\xi \right)=0$ is given by

$$P({\tau }^{\ast }\le \infty )=\frac{exp\left(g_R{L}_0\right)}{\mathbb{E}(exp\left(g_R{L}_{{\tau }^{\ast }}\right)|{\mathcal{F}}_0,{\tau }^{\ast }\le T)}\text{.}$$

To check this relation, we evaluate numerically the expectation present in the denominator of this quotient. The probabilities of ruin computed by this way are reported in the third column of Table 3 and relatively close to real ones. This confirms the validity of Eq. (22). The spread between these probabilities of default varies from 1.74% to 3.49%, depending upon the level of safety margin. This spread is due to the limited number of simulations, to the time horizon and to numerical errors generated by the Euler discretization.

Table 3.

Comparison of simulated ruin probabilities with the Cramer–Lundberg upper bound and the ruin probabilities computed with Eq. (22).

Safety margin	Simulated ruin probabilities (%)	Cramer–Lundberg bound (%)	Numerical approximation of Eq. (22) (%)
2.5%	83.0	88.5	85.2
5.0%	70.6	78.8	73.3
7.5%	60.1	70.6	63.6
10%	52.1	63.5	55.3
12.5%	45.8	57.4	48.8
15.0%	40.5	52.1	43.3
17.5%	36.5	47.5	38.7
20.0%	32.2	43.5	34.0

4. Optimal asset allocation, reinsurance and dividends

This section focuses on the asset–liability management (ALM) policy of an insurance company investing in financial markets the incomes from the insurance activity. We denote by ${\pi }_t$ the percentage of the total asset managed by the insurer that is invested in stocks. Furthermore, we assume that the claims process is proportionally reinsured. A such reinsurance treaty foresees the transfer of a fraction, $1-q_t$ of collected premiums to the reinsurer, in exchange of the covering of the same fraction of claims. Finally, the insurer also distributes to shareholders a continuous dividend that is noted $d_t$.

In this framework, the economic value of the insurance company also called surplus and denoted by $X_t$, is the difference between the assets and liabilities. It obeys to the next dynamics:

$$d{X}_t=(1-{\pi }_t)X_{t}rdt+{\pi }_t{X}_t\frac{d{S}_t}{S_t}-d_t+d{L}_t-(1-q_t)cd{\tau }_t+(1-q_t)Jd{N}_{{\tau }_t}\text{.}$$

The first line is related to financial operations whereas the second line is the income from insurance activities. If we replace in this last equation $\frac{d{S}_t}{S_t}$ and $d{L}_t$ by their expressions (15), (17), we obtain the following SDE for the surplus:

$$d{X}_t=(r{X}_t+{\pi }_t(\mu -r){\lambda }_t^S{X}_t-d_t+c{q}_t{\lambda }_t^L)dt+{\pi }_t{X}_t\sigma \sqrt{{\lambda }_t^S}d{W}_t^S-q_{t}Jd{N}_t^b$$

where $N_t^b$ is a point process with an intensity equal to $\left(\lambda {\int }_0^t{\lambda }_s^{L}ds\right)$. If the horizon of management is noted $T$, the insurer optimizes the investment, reinsurance and dividend policies. The criteria of optimization are the discounted utility of dividends distributed over this period and the discounted utility of the terminal surplus. If these utilities are respectively denoted by $U_1(.)$ and $U_2(.)$, and if the discount rate is $\beta$, the value function of the insurer is defined by the following relation:

$$V(t,X_t,{\lambda }_t^S,{\lambda }_t^L,{\tau }_t^S,{\tau }_t^L)=\underset{{\pi }_t,d_t{q}_t}{max}\mathbb{E}({\int }_t^T{e}^{-\beta (s-t)}{U}_1\left(d_s\right)ds+e^{-\beta (T-t)}{U}_2\left(X_T\right)|{\mathcal{F}}_t)\text{.}$$

If we refer to the theory of stochastic optimal control, the value function of this optimization problem is the solution of a Hamilton Jacobi Bellman equation (HJB):

$$0=V_t-\beta V+\underset{\pi ,d,q}{max}[(rX+\pi (\mu -r){\lambda }^{S}X-d+cq{\lambda }^L)V_X+U_1\left(d\right)+\frac{1}{2}\left({\pi }^2{X}^2{\sigma }^2{\lambda }^S\right)V_{XX}+V_{\tau S}{\lambda }^S+V_{\tau L}{\lambda }^L+\lambda \underset{t}{\overset{L}{\lambda }}(\int V(X-qz)-V{\nu }^J\left(dz\right))+{\alpha }^S({\theta }^S-{\lambda }^S)V_{\lambda S}+{\lambda }^S\int V({\lambda }_t^S+{\eta }_{SS}z,{\lambda }^L+{\eta }_{LS}z)-V{\nu }^S\left(dz\right)+{\alpha }^L({\theta }^L-{\lambda }^L)V_{\lambda L}+{\lambda }^L\int V({\lambda }_t^S+{\eta }_{SL}z,{\lambda }^L+{\eta }_{LL}z)-V{\nu }^L\left(dz\right)]$$ (24)

where $V_t$, $V_X$, $V_{XX}$, $V_{\lambda S}$ and $V_{\lambda L}$ are the partial derivatives of the value function with respect to driving stochastic processes. In the previous equation, we momentarily forget the index related to time so as to lighten notations. The terminal condition is $V(T,X_T,{\lambda }_T^S,{\lambda }_T^L,{\tau }_T^S,{\tau }_T^L)=U_2\left(X_T\right)$. If we derive the HJB equation with respect to $\pi$, we infer that the optimal investment policy is:

$${\pi }^{\ast }=-\frac{(\mu -r)V_X}{{\sigma }^{2}X{V}_{XX}}\text{.}$$ (25)

Using the same approach allows us to infer the optimal dividend policy:

$$d^{\ast }=U_1^{\prime -1}\left(V_X\right)\text{.}$$ (26)

On the other hand, the optimal reinsurance satisfies the following relation

$$0=c{V}_X-\lambda \int V_X(X-q^{\ast }z)z{\nu }^J\left(dz\right)$$ (27)

and if we insert these results into the relation (24), the HJB equation may be rewritten as follows:

$$0=V_t-\beta V+(rX-\frac{1}{2}\frac{{(\mu -r)}^2{V}_X}{{\sigma }^2{V}_{XX}}{\lambda }^S+cq{\lambda }^L-U_1^{\prime -1}\left(V_X\right))V_X+U_1\left(U_1^{\prime -1}\left(V_X\right)\right)+V_{\tau S}{\lambda }^S+V_{\tau L}{\lambda }^L+\lambda {\lambda }^L(\int V(X-qz)-V{\nu }^J\left(dz\right))+{\alpha }^S({\theta }^S-{\lambda }^S)V_{\lambda S}+{\lambda }^S\int V({\lambda }_t^S+{\eta }_{SS}z,{\lambda }^L+{\eta }_{LS}z)-V{\nu }^S\left(dz\right)+{\alpha }^L({\theta }^L-{\lambda }^L)V_{\lambda L}+{\lambda }^L\int V({\lambda }_t^S+{\eta }_{SL}z,{\lambda }^L+{\eta }_{LL}z)-V{\nu }^L\left(dz\right)\text{.}$$ (28)

Utility functions are assumed exponential: $U_1\left(y\right)=-\frac{1}{{\gamma }_1}{e}^{-{\gamma }_{1}y}$ and $U_2\left(y\right)=-\frac{1}{{\gamma }_2}{e}^{-{\gamma }_{2}y}$. In this particular case, it is possible to infer a semi-closed form expression for the value function:

Proposition 4.1

The value function solving the HJB equation   (24)   is an exponential affine function of the wealth and of chronometers intensities:

$$V(T,X_T,{\lambda }_T^S,{\lambda }_T^L)=-\frac{1}{{\gamma }_2}exp(A(t,T)+{\left(\begin{array}{c}\hfill B_S(t,T)\hfill \\ \hfill B_L(t,T)\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right)+C(t,T)X_t)\text{,}$$ (29)

where $A(.)$ , $B_S(.)$ , $B_L(.)$ and $C(.)$ are deterministic functions of time, solutions of the next ODE’s:

$$\frac{\partial }{\partial t}A(t,T)=\beta -\frac{1}{{\gamma }_1}(ln(-\frac{1}{{\gamma }_2}C(t,T))+A(t,T)-1)C(t,T)-{\alpha }^S{\theta }^S{B}_S(t,T)-{\alpha }^L{\theta }^L{B}_L(t,T)$$ (30)

$$\frac{\partial }{\partial t}C(t,T)=-rC(t,T)-\frac{1}{{\gamma }_1}C{(t,T)}^2$$

$$\frac{\partial }{\partial t}{B}_S(t,T)=-(\frac{1}{{\gamma }_1}C(t,T)-{\alpha }^S)B_S(t,T)+\frac{1}{2}\frac{{(\mu -r)}^2}{{\sigma }^2}-({\varphi }^S({\eta }_{SS}{B}_S(t,T)+{\eta }_{LS}{B}_L(t,T))-1)$$

$$\frac{\partial }{\partial t}{B}_L(t,T)=-(\frac{1}{{\gamma }_1}C(t,T)-{\alpha }^L)B_L(t,T)-cC(t,T)q_t^{\ast }-\lambda ({\varphi }^J(-C(t,T)q_t^{\ast })-1)-({\varphi }^L({\eta }_{SL}{B}_S(t,T)+{\eta }_{LL}{B}_L(t,T))-1)$$

with the terminal conditions $A(T,T)=0$ , $C(T,T)=-{\gamma }_2$ , $B_S(T,T)=B_L(T,T)=0$ . $q_t^{\ast }$ is here the optimal reinsurance solution of the next relation:

$${\int }_0^{+\infty }z{e}^{-Cqz}{\nu }^J\left(dz\right)=\frac{c}{\lambda }\text{.}$$ (31)

Proof

To prove this, we use a verification argument. Under the assumption that the value function has an exponential form of the type (29), the partial derivatives of $V$ with respect to risk factors and time are given by

$$V_t=(A_t+C_{t}X+B_{St}{\lambda }^S+B_{Lt}{\lambda }^L)V$$

$$V_X=CV{V}_{XX}=C^{2}V{V}_{\lambda S}=B_{S}V{V}_{\lambda L}=B_{L}V$$

$$V({\lambda }_t^S+{\eta }_{SS}z,{\lambda }_t^L+{\eta }_{LS}z)-V=V(e^{({\eta }_{SS}{B}_S+{\eta }_{LS}{B}_L)z}-1)$$

$$V({\lambda }_t^S+{\eta }_{SL}z,{\lambda }_t^L+{\eta }_{LL}z)-V=V(e^{({\eta }_{SL}{B}_S+{\eta }_{LL}{B}_L)z}-1)$$

$$V(X-qz)-V=V(e^{-Cqz}-1)\text{.}$$

On the other hand, $U_1^{\prime }\left(y\right)=e^{-{\gamma }_{1}y}$ and $U_1^{\prime -1}\left(y\right)=-\frac{1}{{\gamma }_1}ln\left(y\right)$ then

$$U_1^{\prime -1}\left(V_X\right)-\frac{1}{{\gamma }_1}ln\left(CV\right)=-\frac{1}{{\gamma }_1}[ln(-\frac{1}{{\gamma }_2}C)+A+CX+B_S{\lambda }^S+B_L{\lambda }^L]$$

$$U_1\left(U_1^{\prime -1}\left(V_X\right)\right)=-\frac{1}{{\gamma }_1}{e}^{ln\left(V_X\right)}=-\frac{1}{{\gamma }_1}CV\text{.}$$

If we insert these intermediate results in the HJB equation, we obtain that

$$0=(A_t+C_{t}X+B_{St}{\lambda }^S+B_{Lt}{\lambda }^L)-\beta +(rCX-\frac{1}{2}\frac{{(\mu -r)}^2}{{\sigma }^2}{\lambda }^S+cC{q}^{\ast }{\lambda }^L)+\frac{1}{{\gamma }_1}[Cln(-\frac{1}{{\gamma }_2}C)+AC+C^{2}X+B_{S}C{\lambda }^S+B_{L}C{\lambda }^L]-\frac{1}{{\gamma }_1}C+{\lambda }^L\left[\lambda ({\varphi }^J(-Cq)-1)\right]+{\alpha }^S({\theta }^S-{\lambda }^S)B_S+{\lambda }^S\left[({\varphi }^S({\eta }_{SS}{B}_S+{\eta }_{LS}{B}_L)-1)\right]+{\alpha }^L({\theta }^L-{\lambda }^L)B_L+{\lambda }^L\left[({\varphi }^L({\eta }_{SL}{B}_S+{\eta }_{LL}{B}_L)-1)\right]$$ (32)

and the optimal reinsurance $q^{\ast }$ is solution of the next equation:

$$c-\lambda (\int z{e}^{-C{q}^{\ast }z}{\nu }^J\left(dz\right))=0\text{.}$$

The relation (32) is satisfied whatever the value of risk factors if and only if the relations (30) hold. □

By direct differentiation, we can check that $C(t,T)$ admits the following closed form expression:

$$C(t,T)={(-\frac{1}{{\gamma }_1}\frac{1}{r}(1-e^{-r(T-t)})-\frac{1}{{\gamma }_2}{e}^{-r(T-t)})}^{-1}\text{.}$$

We will see that this function plays a crucial role in the determination of the optimal reinsurance and investment policy. The optimal ratio of reinsurance is indeed solution of a nonlinear equation that does not admit any closed form expression. However, if we approach the exponential in Eq. (31) by a first order Taylor’s development, we infer that the optimal reinsurance rate satisfies approximately the next relation:

$${\int }_0^{+\infty }z{e}^{-Cqz}{\nu }^J\left(dz\right)\approx {\int }_0^{+\infty }(z-Cq{z}^2){\nu }^J\left(dz\right)\mathbb{E}\left(J\right)-Cq\mathbb{E}\left(J^2\right)=\frac{c}{\lambda }\text{,}$$

from which we obtain finally that

$$q_t^{\ast }\approx \frac{1}{C(t,T)\mathbb{E}\left(J^2\right)}(\mathbb{E}\left(J\right)-\frac{c}{\lambda }),=(\frac{1}{{\gamma }_1}\frac{1}{r}(1-e^{-r(T-t)})+\frac{1}{{\gamma }_2}{e}^{-r(T-t)})\frac{1}{\mathbb{E}\left(J^2\right)}(\frac{c}{\lambda }-\mathbb{E}\left(J\right))\text{.}$$ (33)

As $C(t,T)$ is strictly negative, the optimal reinsurance rate is proportional to the safety margin on claims size ($\frac{c}{\lambda }-\mathbb{E}\left(J\right)$), embedded in the premium rate. Notice that the reinsurance ratio exclusively depends on frequency of claims measured on the insurance market scale and not on the chronological time scale. From this last relation, we also infer that no claim is re-insured if $c$ is equal to the pure premium rate, $c=\lambda \mathbb{E}\left(J\right)$. If the premium rate is higher or lower than this pure premium, the reinsurance rate is inversely proportional to the second moment of the claims size. Finally, the optimal reinsurance is independent from the insurer’s wealth. It is a pure deterministic function of time that converges toward $\frac{1}{{\gamma }_2\mathbb{E}\left(J^2\right)}(\mathbb{E}\left(J\right)-\frac{c}{\lambda })$ at expiry.

On the other hand, the optimal investment policy consists to invest the following time varying percentage of the total asset in stocks:

$${\pi }_t^{\ast }=-\frac{(\mu -r)}{{\sigma }^2}\frac{1}{C(t,T)}\frac{1}{X_t}=\frac{(\mu -r)}{{\sigma }^2}(\frac{1}{{\gamma }_1}\frac{1}{r}(1-e^{-r(T-t)})+\frac{1}{{\gamma }_2}{e}^{-r(T-t)})\frac{1}{X_t}$$ (34)

and is independent from clocks of financial and insurance markets. At the end of the time horizon, the insurance company holds $\frac{(\mu -r)}{{\sigma }^2{\gamma }_2}$ of stocks. As for the reinsurance rate, the optimal investment policy is based on the drift and variance measured on the financial market scale and not on the chronological scale. This is not the case of the optimal dividend that explicitly depends on intensities of clocks:

$$d_t^{\ast }=-\frac{1}{{\gamma }_1}[ln(-\frac{1}{{\gamma }_2}C(t,T))+A(t,T)+C(t,T)X_t+B_S(t,T){\lambda }_t^S+B_L(t,T){\lambda }_t^L]\text{.}$$

However, as $B_S(t,T)$ and $B_L(t,T)$ converge toward zero when $t\to T$, the influence of ${\lambda }_t^S$ and ${\lambda }_t^L$ is lessened with the passage of time. Even if the optimal management policy is determined, we do not have at this stage any information about the distribution of the optimal wealth, $X_t^{\ast }$. However, we can approach numerically its moments. Under the assumption that the optimal reinsurance ratio is close to the one in Eq. (33), the dynamics of the surplus is given by the next SDE

$$d{X}_t^{\ast }={\mu }_1^{\ast }(t,T)dt+{\mu }_2^{\ast }(t,T)X_t^{\ast }dt+{\mu }_3^{\ast }(t,T){\lambda }_t^{S}dt+{\mu }_4^{\ast }(t,T){\lambda }_t^{L}dt-\frac{(\mu -r)}{\sigma C(t,T)}\sqrt{{\lambda }_t^S}d{W}_t^S-\frac{(\mathbb{E}\left(J\right)-\frac{c}{\lambda })}{C(t,T)\mathbb{E}\left(J^2\right)}Jd{N}_t^b$$

with

$${\mu }_1^{\ast }(t,T)=\frac{1}{{\gamma }_1}(ln(-\frac{1}{{\gamma }_2}C(t,T))+A(t,T)){\mu }_2^{\ast }(t,T)=(r+\frac{1}{{\gamma }_1}C(t,T))$$

$${\mu }_3^{\ast }(t,T)=\frac{B_S(t,T)}{{\gamma }_1}-\frac{{(\mu -r)}^2}{{\sigma }^{2}C(t,T)}{\mu }_4^{\ast }(t,T)=\frac{B_L(t,T)}{{\gamma }_1}+\frac{c(\mathbb{E}\left(J\right)-\frac{c}{\lambda })}{C(t,T)\mathbb{E}\left(J^2\right)}$$

and where functions $C(t,T)$, $B_L(t,T)$ and $B_S(t,T)$ are defined in Proposition 4.1 by the system of ODE’s (30).

Proposition 4.2

The mgf of the optimal wealth under the assumption that the optimal reinsurance ratio is approached by   (33)   is given by the following expression

$$\mathbb{E}\left(e^{\omega X_s^{\ast }}\right|{\mathcal{F}}_t)=exp(A^{\ast }(t,s)+{\left(\begin{array}{c}\hfill B_S^{\ast }(t,s)\hfill \\ \hfill B_L^{\ast }(t,s)\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right)+C^{\ast }(t,s)X_t)\text{,}$$ (35)

where the functions $A^{\ast },B_S^{\ast }$ , $B_L^{\ast }$ and $C^{\ast }$ satisfy the next system of ODE’s:

$$\frac{\partial }{\partial t}{A}^{\ast }(t,s)=-{\mu }_1(t,T)C^{\ast }(t,s)-{\alpha }_S{\theta }_S{B}_S^{\ast }(t,s)-{\alpha }_L{\theta }_L{B}_L^{\ast }(t,s)$$

$$\frac{\partial }{\partial t}{C}^{\ast }(t,s)=-{\mu }_2(t,T)C^{\ast }(t,s)$$

$$\frac{\partial }{\partial t}{B}_S^{\ast }(t,s)=-{\mu }_3(t,T)C^{\ast }(t,s)-\left(\frac{1}{2}\frac{{(\mu -r)}^2}{{\sigma }^2}\right){\left(\frac{C^{\ast }(t,s)}{C(t,s)}\right)}^2+{\alpha }_S{B}_S^{\ast }(t,s)-({\varphi }^{J_S}(B_S^{\ast }(t,s){\eta }_{SS}+B_L^{\ast }(t,s){\eta }_{LS})-1)$$

$$\frac{\partial }{\partial t}{B}_L^{\ast }(t,s)=-{\mu }_4(t,T)C^{\ast }(t,s)+{\alpha }_L{B}_L^{\ast }(t,s)-({\varphi }^{J_L}(B_S^{\ast }(t,s){\eta }_{SL}+B_L^{\ast }(t,s){\eta }_{LL})-1)-\lambda ({\varphi }^J(-C^{\ast }(t,s)q_t^{\ast }z)-1)$$

with the terminal conditions $A^{\ast }(s,s)=0$ , $C^{\ast }(s,s)=\omega$ , $B_S^{\ast }(s,s)=0$ and $B_L^{\ast }(s,s)=0$ .

Proof

Let us denote by $f(t,X_t,{\lambda }_t^S,{\lambda }_t^L)$, the moment generating function of $X_s$ for $s\ge t$. This mgf is solution of the Itô SDE:

$$0=f_t+{\mu }_1{f}_X+{\mu }_{2}X{f}_X+{\mu }_3{\lambda }^S{f}_X+{\mu }_4{\lambda }^L{f}_X+\frac{1}{2}\frac{1}{C^2}\frac{{(\mu -r)}^2}{{\sigma }^2}{\lambda }^S{f}_{XX}+{\alpha }_S({\theta }_S-{\lambda }^S)f_{{\lambda }^S}+{\lambda }^S{\int }_0^{+\infty }f(t,X_t,{\lambda }_t^S+{\eta }_{SS}z,{\lambda }_t^L+{\eta }_{LS}z)-f{\nu }^S\left(dz\right)+{\alpha }_L({\theta }_L-{\lambda }^L)f_{{\lambda }^L}+{\lambda }^L{\int }_0^{+\infty }f(t,X_t,{\lambda }_t^S+{\eta }_{SL}z,{\lambda }_t^L+{\eta }_{LL}z)-f{\nu }^L\left(dz\right)+\lambda {\lambda }^L{\int }_0^{+\infty }f(t,X_t-q_t^{\ast }z,{\lambda }_t^S,{\lambda }_t^L)-f{\nu }^J\left(dz\right)\text{.}$$

Assuming that the mgf has the form of Eq. (35) and using the same approach as for the proof of Proposition 4.1, allows us to prove the proposition. □

To conclude this section, we solve numerically the asset–liability management problem. The parameters of stochastic clocks used for this exercise are those reported in the third column of Table 1. The average growth rate and the standard deviation of the asset return, are set to $\mu =5\%$, $\sigma =20\%$ whereas the risk free rate is 2%. Claims are exponential random variables with $\rho =1$. The frequency of claims is equal to $\lambda =200$. The premium rate includes a safety margin of 10%: $c=\frac{\lambda }{\rho }1.10$. The coefficients of risk aversion are respectively ${\gamma }_1=10$ or ${\gamma }_1=$ 20 and ${\gamma }_2=5$. The discount rate used in utility function is set to $\beta =1\%$. Finally the time horizon is 10 years and the initial wealth is $X_0=5$.

The optimal ALM strategy presented in Fig. 3 is obtained with ${\lambda }_0^S={\lambda }_0^L={\theta }_S={\theta }_L$. The upper left graph displays the expected wealth for different maturities. Its analysis must be related to the lower left graph that shows the term structure of expected dividends. We observe that during a first period of 4 years, the richness increases on average by 18% to 36%, depending upon ${\gamma }_1$. Incomes from insurance activities and investments are on average higher than distributed dividends that however increase linearly. After 4 or 5 years, dividends become too high to be financed exclusively by incomes and a part of the surplus is redistributed to shareholders. On the other hand, positions in risky assets and reinsurance are reduced with time. The upper right graph of Fig. 3 presents the optimal reinsurance rate that is exclusively a function of time. This ratio falls nearly linearly from 5.4% or 3.1% for ${\gamma }_1=10$ or ${\gamma }_1=$ 20% to 1%. The optimal amount of stocks (${\pi }_t{X}_t^{\ast }$) is also independent from the size of the surplus and decreases linearly from 0.8 or 0.47 for ${\gamma }_1=10$ or ${\gamma }_1=20$ to 0.15.

Fig. 3.

Upper left plot: expectations and standard deviations of the optimal surplus. Upper right plot: optimal reinsurance ratio. Lower left plot: optimal amount of stocks. Lower right plot: expected dividends.

The graphs in Fig. 4 show the influence of the initial values of ${\lambda }_0^S$ and ${\lambda }_0^L$ on expectations and standard deviations of the future expected wealth. Three scenarii are compared: ${\lambda }_0^S={\lambda }_0^L={\theta }_S={\theta }_L$, ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(\frac{3}{{\rho }_S},0)}^{\top }$ and ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(0,\frac{3}{{\rho }_L})}^{\top }$. Stepping up ${\lambda }_0^S$ or ${\lambda }_0^L$ respectively accelerates the asset and liability clocks. As on average the risk process and investments are profitable, any acceleration of business time increases the gain but also the risk, measured on the chronological time scale. As most of gains are capitalized, high values for ${\lambda }_0^S$ or ${\lambda }_0^L$ raise the expected wealth over the first six months. Fig. 5 presents the term structure of expected dividends, in three scenarii. As high values for ${\lambda }_0^S$ or ${\lambda }_0^L$ generate an extra profit over the first six months, the initial expected dividend is bigger than when ${\lambda }_0^S={\lambda }_0^L={\theta }_S={\theta }_L$.

Fig. 4.

The upper graph compares the expectations and standard deviations of the wealth process in two scenarii: (1) ${\lambda }_0^S={\lambda }_0^L={\theta }_S={\theta }_L$ (2) ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(\frac{3}{{\rho }_S},0)}^{\top }$. The lower graph compares the expectations and standard deviations in two scenarii (1) ${\lambda }_0^S={\lambda }_0^L={\theta }_S={\theta }_L$ and (2) ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(0,\frac{3}{{\rho }_L})}^{\top }$.

Fig. 5.

This graph presents the term structure of expected dividends in three scenarii: (1) ${\lambda }_0^S={\lambda }_0^L={\theta }_S={\theta }_L$ (2) ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(\frac{3}{{\rho }_S},0)}^{\top }$ and (3) ${({\lambda }_0^S,{\lambda }_0^L)}^{\top }={({\theta }_S,{\theta }_L)}^{\top }+\Xi {(0,\frac{3}{{\rho }_L})}^{\top }$.

5. Optimal asset allocation, reinsurance and dividends with a Brownian approximation

In many circumstances, working with Brownian motions rather than jump processes allows to obtain analytical results. On the other hand, approaching a claims process by an equivalent Brownian dynamics is often a good approximation, particularly if the number of claims is high. These reasons motivate us to study the case in which the liabilities of the insurance company are driven by the next SDE:

$$d{L}_t=cd{\tau }_t^L-\lambda \mathbb{E}\left(J\right)d{\tau }_t^L+\sqrt{\lambda \mathbb{E}\left(J^2\right)}d{W}_{{\tau }_t^L}^L$$

where $W_{{\tau }_t^L}^L$ is a Brownian motion. The scaling property of the Brownian motion allows us to rewrite the liability process as follows:

$$d{L}_t=(c-\lambda \mathbb{E}\left(J\right)){\lambda }_t^{L}dt+\sqrt{\lambda \mathbb{E}\left(J^2\right)}\sqrt{{\lambda }_t^L}d{W}_t^L$$ (36)

from which we infer that the risk process at time $t$ is the following sum:

$$L_t=(c-\lambda \mathbb{E}\left(J\right)){\tau }_t^L+\sqrt{\lambda \mathbb{E}\left(J^2\right)}{\int }_0^t\sqrt{{\lambda }_s^L}d{W}_s^L\text{.}$$

This expressions reveals that both the average and variance of $L_t$ are proportional to the chronometer of the insurance market: $(c-\lambda \mathbb{E}\left(J\right))\mathbb{E}\left({\tau }_t^L\right|{\mathcal{F}}_0)$ and $\lambda \mathbb{E}\left(J^2\right)\mathbb{E}\left({\tau }_t^L\right|{\mathcal{F}}_0)$. It is possible to show that the covariance between liabilities and the log prices of stocks is induced by the dependence between clocks of financial and insurance markets. And this covariance is still provided by Proposition 3.1. We will not present all features of this process like the joint mgf of $L_t$ and $log{S}_t$. However, most of proofs presented in previous sections are easily adaptable to the Brownian case. As previously, ${\pi }_t$, $d_t$ and $q_t$ denote respectively the percentage of the stocks hold by the insurer, the dividend and the retention level. In the Brownian framework, the dynamics of the surplus is driven by the next relation:

$$d{X}_t=(1-{\pi }_t)X_{t}rdt+{\pi }_t{X}_t\frac{d{S}_t}{S_t}-d_t+d{L}_t-(1-q_t)cd{\tau }_t+(1-q_t)(\lambda \mathbb{E}\left(J\right){\tau }_t^L-\sqrt{\lambda \mathbb{E}\left(J^2\right)}{\int }_0^t\sqrt{{\lambda }_s^L}d{W}_s^L)\text{.}$$

If we replace in this last equation $\frac{d{S}_t}{S_t}$ by its expression (15) and $d{L}_t$ by its approximation (36), we infer that $X_t$ is now ruled by the SDE:

$$d{X}_t=(r{X}_t+{\pi }_t(\mu -r){\lambda }_t^S{X}_t-d_t+c{q}_t{\lambda }_t^L-\lambda q_t\mathbb{E}\left(J\right){\lambda }_t^L)dt+{\pi }_t{X}_t\sigma \sqrt{{\lambda }_t^S}d{W}_t^S+q_t\sqrt{\lambda \mathbb{E}\left(J^2\right)}\sqrt{{\lambda }_t^L}d{W}_t^L\text{.}$$

By construction, the Brownian motions $W_t^L$ and $W_t^S$ are independent and the correlation is only induced by the stochastic clocks. Then

$${\pi }_t{X}_t\sigma \sqrt{{\lambda }_t^S}d{W}_t^S+q_t\sqrt{\lambda \mathbb{E}\left(J^2\right)}\sqrt{{\lambda }_t^L}d{W}_t^L\sim N(0,\sqrt{{\pi }_t^2{X}_t^2{\sigma }^2{\lambda }_t^S+q_t^2\lambda \mathbb{E}\left(J^2\right){\lambda }_t^L}\sqrt{dt})\text{.}$$

We can then replace these two Brownian motions by a single one $W_t$ defined on the same filtration as follows

$$d{X}_t=((r+{\pi }_t(\mu -r){\lambda }_t^S)X_t-d_t+q_t(c-\lambda \mathbb{E}\left(J\right)){\lambda }_t^L)dt+\sqrt{{\pi }_t^2{X}_t^2{\sigma }^2{\lambda }_t^S+q_t^2\lambda \mathbb{E}\left(J^2\right){\lambda }_t^L}d{W}_t\text{.}$$

The insurer adjusts the investment, dividend and reinsurance policy so as to maximize the following objective:

$$V(t,X_t,{\lambda }_t^S,{\lambda }_t^L,{\tau }_t^S,{\tau }_t^L)=\underset{{\pi }_t,d_t{q}_t}{max}\mathbb{E}({\int }_t^T{e}^{-\beta (s-t)}{U}_1\left(d_s\right)ds+e^{-\beta (T-t)}{U}_2\left(X_T\right)|{\mathcal{F}}_t)$$ (37)

where $U_1\left(y\right)=-\frac{1}{{\gamma }_1}{e}^{-{\gamma }_{1}y}$ and $U_2\left(y\right)=-\frac{1}{{\gamma }_2}{e}^{-{\gamma }_{2}y}$ are the utility from dividends and from the terminal surplus. The value function of this optimization problem solves the next Hamilton Jacobi Bellman equation (HJB):

$$0=V_t-\beta V+\underset{\pi ,d,q}{max}[(rX+\pi (\mu -r){\lambda }^{S}X-d+cq{\lambda }^L-\lambda q\mathbb{E}\left(J\right){\lambda }^L)V_X+U_1\left(d\right)+\frac{1}{2}({\pi }^2{X}^2{\sigma }^2{\lambda }^S+q^2\lambda \mathbb{E}\left(J^2\right)\underset{t}{\overset{L}{\lambda }})V_{XX}+V_{\tau S}{\lambda }^S+V_{\tau L}{\lambda }^L+{\alpha }^S({\theta }^S-\underset{t}{\overset{S}{\lambda }})V_{\lambda S}+\underset{t}{\overset{S}{\lambda }}\int V({\lambda }_t^S+{\eta }_{SS}z,{\lambda }_t^L+{\eta }_{LS}z)-V{\nu }^S\left(dz\right)+{\alpha }^L({\theta }^L-{\lambda }_t^L)V_{\lambda L}+{\lambda }_t^L\int V({\lambda }_t^S+{\eta }_{SL}z,{\lambda }_t^L+{\eta }_{LL}z)-V{\nu }^L\left(dz\right)]$$

with the terminal conditions $V(T,X_T,{\lambda }_T^S,{\lambda }_T^L,{\tau }_T^S,{\tau }_T^L)=U_2\left(X_T\right)$. Using the same approach as for Proposition 4.1 allows us to establish the next result:

Proposition 5.1

The value function defined by Eq.   (37)   in a Brownian setting, is the exponential of an affine function of risk factors

$$V(T,X_T,{\lambda }_T^S,{\lambda }_T^L)=-\frac{1}{{\gamma }_2}exp(A(t,T)+{\left(\begin{array}{c}\hfill B_S(t,T)\hfill \\ \hfill B_L(t,T)\hfill \end{array}\right)}^{\top }\left(\begin{array}{c}\hfill {\lambda }_t^S\hfill \\ \hfill {\lambda }_t^L\hfill \end{array}\right)+C(t,T)X_t)\text{,}$$

where $A(t,T)$ , $B_S(t,T)$ , $B_L(t,T)$ and $C(t,T)$ are functions of time, solutions of the following ODE’s

$$\frac{\partial }{\partial t}A(t,T)=\beta -\frac{1}{{\gamma }_1}C(t,T)(ln(-\frac{1}{{\gamma }_2}C(t,T))+A(t,T)-1)-{\alpha }^S{\theta }^S{B}_S(t,T)-{\alpha }^L{\theta }^L{B}_L(t,T)$$

$$\frac{\partial }{\partial t}C(t,T)=-rC(t,T)-\frac{1}{{\gamma }_1}C{(t,T)}^2$$

$$\frac{\partial }{\partial t}{B}_S(t,T)=\frac{1}{2}\frac{{(\mu -r)}^2}{{\sigma }^2}-(\frac{1}{{\gamma }_1}C(t,T)-{\alpha }^S)B_S(t,T)-\left[({\varphi }^S({\eta }_{SS}{B}_S(t,T)+{\eta }_{LS}{B}_L(t,T))-1)\right]$$

$$\frac{\partial }{\partial t}{B}_L(t,T)=\frac{1}{2}\frac{{(c-\lambda \mathbb{E}\left(J\right))}^2}{\lambda \mathbb{E}\left(J^2\right)}-(\frac{1}{{\gamma }_1}C(t,T)-{\alpha }^L)B_L(t,T)-\left[({\varphi }^L({\eta }_{SL}{B}_S(t,T)+{\eta }_{LL}{B}_L(t,T))-1)\right]$$

with the terminal conditions $A(T,T)=0$ , $C(T,T)=-{\gamma }_2$ , $B_S(T,T)=B_L(T,T)=0$ . The optimal investment policy is given by

$${\pi }_t^{\ast }=-\frac{(\mu -r)}{{\sigma }^{2}C(t,T)}\frac{1}{X_t}\text{.}$$ (38)

The optimal dividend is equal to

$$d_t^{\ast }=-\frac{1}{{\gamma }_1}[ln(-\frac{1}{{\gamma }_2}C)+A+CX+B_S{\lambda }^S+B_L{\lambda }^L]$$ (39)

and the optimal reinsurance ratio is

$$q^{\ast }=-\frac{(c-\lambda \mathbb{E}\left(J\right))}{\lambda \mathbb{E}\left(J^2\right)C(t,T)}\text{.}$$ (40)

This last proposition emphasizes that the investment strategy remains unchanged compared to the one obtained with the original claims process. The expression (39) of the optimal dividend is also identical to the one in the previous model. However, as functions $A$, $B_S$ and $B_L$ differ from those defined in Proposition 4.1, dividends effectively depend upon the claims model. Finally, we notice that the optimal reinsurance rate is equal to the approached ratio proposed in Eq. (33) for the original claims dynamics.

6. Conclusions

This study develops a model in which the contagion between insurance and financial markets is induced by time-changed processes. This framework presents several interesting features. Firstly, the moment generating functions of market clocks, assets and liabilities have a semi-closed form expression. Secondly, the asymptotic probability of ruin for the risk process admits an upper bound. Thirdly, the model may be used for asset–liability management purposes.

Numerical tests emphasize the ability of the model to generate a wide variety of term structures of correlations between assets and liabilities. On the other hand, the correlation is induced by earnings of the insurance business that are reinvested in the financial market. If the insurer does not charge any fee above the pure premium, there is not any linear dependence between the asset and liability despite the fact that the asset price and the risk process are not independent by construction. Another interesting feature is that the short term correlation between markets is negligible. In our approach, a delay is induced between the occurrence of an event in one market and the reaction of the other market. In other words, there is well contagion between the insurance and financial markets but the impact is not instantaneous.

When used in a ALM framework, the model remains analytically tractable. Optimal reinsurance and investment rates admit closed form expressions and are independent from stochastic clocks. The optimal dividend is a linear function of the wealth and of intensities of chronometers. Finally, the optimal policy depends on parameters defining asset and liability dynamics on the market time scale and not on the chronological time scale.