2.1 Dynamic contagion process

A definition for the standard DCP with time-homogeneous parameters will be provided by Definition 1, while a definition for the associated CDCP will be provided by Definition 2. We will extend it to a version with time-inhomogeneous parameters later.

We start with a complete probability space $(\Omega , \mathcal{F},\mathbb{P})$ , where $\mathbb{P}$ is a real-world probability measure, which is called a physical probability measure. Suppose that the probability space $(\Omega , \mathcal{F},\mathbb{P})$ is sufficiently rich to allow for the definition of all the random processes or quantities that will be introduced subsequently. A continuous-time model with a finite horizon $\mathcal{T} \, :\!= \, [0, T]$ is considered, where $T \lt \infty$ . Let $\{ N_t \}_{t \in \mathcal{T}}$ denote a point process defined on $(\Omega , \mathcal{F},\mathbb{P})$ . Let $\{ T_{2, j} \}_{j = 1, 2, \ldots }$ be a sequence of random jump times of the point process $\{ N_t \}_{t \in \mathcal{T}}$ . That is, $\{ T_{2, j} \}_{j = 1, 2, \ldots }$ is a sequence of positive-valued random variables on $(\Omega , \mathcal{F},\mathbb{P})$ such that $0 \lt T_{2, 1} \lt T_{2, 2} \lt \cdots$ , $\mathbb{P}$ -a.s. Then, for all $t \in \mathcal{T}$ ,

\begin{align*} N_t = \sum _{j \ge 1} \mathbb{I}_{\{ T_{2, j} \le t \} }, \end{align*}

where $\mathbb{I}_A$ is the indicator function of an event $A$ .

Write $\{ Y_j \}_{j = 1, 2, \ldots }$ for the random positive jump sizes corresponding to the random jump times $\{ T_{2, j} \}_{j = 1, 2, \ldots }$ . Specifically, for each $j = 1, 2, \ldots$ , $Y_j$ is the random positive jump size corresponding to the random jump time $T_{2, j}$ . Assume that $\{ Y_j \}_{j = 1, 2, \ldots }$ is a sequence of independent and identically distributed (i.i.d.) positive random variables on $(\Omega , \mathcal{F}, \mathbb{P})$ with the common distribution $G (y)$ , where $y \gt 0$ .

Let $\{ M_t \}_{t \in \mathcal{T}}$ be a Poisson point process on $(\Omega , \mathcal{F},\mathbb{P})$ such that its intensity process is given by $\rho \gt 0$ , for all $t \in \mathcal{T}$ , $\mathbb{P}$ -a.s. Write $\{ T_{1, i} \}_{i = 1, 2, \ldots }$ and $\{ X_i \}_{i = 1, 2, \ldots }$ for the random jump times and sizes for the Poisson point process $\{ M_t \}_{t \in \mathcal{T}}$ , respectively. Again, suppose that $\{ X_i \}_{i = 1, 2, \ldots }$ is a sequence of i.i.d. positive random variables on $(\Omega , \mathcal{F}, \mathbb{P})$ with the common distribution $H(x)$ , where $x \gt 0$ . Assume that under the probability measure $\mathbb{P}$ , $\{ X_i \}_{i = 1, 2, \ldots }$ , $\{ T_{1,i} \}_{i = 1, 2, \ldots }$ , and $\{ Y_{j} \}_{j = 1, 2, \ldots }$ are independent of each other.

Definition 1 (Dynamic contagion process). Suppose the point process $\{ N_t \}_{t \in \mathcal{T}}$ has a stochastic intensity process $\{ \lambda _t \}_{t \in \mathcal{T}}$ under the probability measure $\mathbb{P}$ . Then $\{ N_t \}_{t \in \mathcal{T}}$ is said to be a DCP if $\{ \lambda _t \}_{t \in \mathcal{T}}$ is given by:

(2.1) \begin{equation} \lambda _{t} = a + \left ( \lambda _{0}-a\right ) \mathrm{e}^{-\delta t}+\sum _{i \ge 1} X_{i} \mathrm{e}^{-\delta \left ( t - T_{1, i} \right )}\mathbb{I}_{\{ T_{1,i} \le t \}} + \sum _{j \ge 1} Y_{j} \mathrm{e}^{-\delta \left (t - T_{2,j} \right )} \mathbb{I}_{ \{ T_{2, j} \le t \}}, \end{equation}

where

• • the constant mean-reverting (baseline) intensity level $a \ge 0$ ;

• • the constant initial intensity at time $t = 0$ , $\lambda _{0} \geq a$ , $\mathbb{P}$ -a.s.;

• • the constant rate of exponential decay $\delta \gt 0$ .

In Definition 1, $\left \{ X_{i}\right \}_{i=1,2,\ldots }$ is interpreted as a sequence of positive externally-excited jumps, while $\left \{ Y_{j}\right \}_{j=1,2,\ldots }$ is interpreted as a sequence of positive self-excited jumps. If there are no externally excited jumps in Eq. (2.1) (i.e., $\rho =0$ for all $t \in \mathcal{T}$ ) $\{ N_{t} \}_{t \in \mathcal{T}}$ becomes a generalized Hawkes process. If $a=0$ and there are no self-excited jumps in Eq. (2.1), $\{ N_{t} \}_{t \in \mathcal{T}}$ becomes the Cox process with shot-noise Poisson intensity. It may be noted that these processes are within the general framework of affine processes. See, for example, Duffie et al. (Reference Duffie, Pan and Singleton2000, Reference Duffie, Filipovic and Schachermayer2003) and Glasserman and Kim (Reference Glasserman and Kim2010). Let $\mathbb{F}^{\lambda }$ and $\mathbb{F}^N$ denote the $\mathbb{P}$ -augmentation of the natural filtration generated by the stochastic intensity process $\{ \lambda _t \}_{t \in \mathcal{T}}$ and the point process $\{ N_t \}_{t \in \mathcal{T}}$ , respectively. That is, $\mathbb{F}^{\lambda } \, :\!= \, \{ \mathcal{F}^{\lambda }_t \}_{t \in \mathcal{T}}$ and $\mathbb{F}^N \, :\!= \, \{ \mathcal{F}^N_t \}_{t \in \mathcal{T}}$ so that for all $t \in \mathcal{T}$ ,

\begin{equation*} \begin{aligned} \mathcal{F}^{\lambda }_t &\, :\!= \, \sigma \{ \lambda _u \,|\, u \in [0, t] \} \vee \mathcal{N}, \\ \mathcal{F}^N_t &\, :\!= \, \sigma \{ N_u \,|\, u \in [0, t] \} \vee \mathcal{N}, \end{aligned} \end{equation*}

where $\mathcal{N}$ is a collection of the $\mathbb{P}$ -null subsets of $\mathcal{F}$ ; $\mathcal{F}_1 \vee \mathcal{F}_2$ is the minimal $\sigma$ -field containing the two $\sigma$ -fields $\mathcal{F}_1$ and $\mathcal{F}_2$ . Note that for all $t \in \mathcal{T}$ , $\mathcal{F}^N_t \subset \mathcal{F}^{\lambda }_t$ .

2.2 Compound dynamic contagion process

This section provides a definition of the CDCP that will be used for the catastrophic component of the liability. We also provide the infinitesimal generator of this process and relevant moments, after which we deal with it separately when the parameters of this process are time-inhomogeneous.

Definition 2 (Dynamic contagion claims). Suppose $\{ T_{2, j} \}_{j = 1, 2, \ldots }$ is a sequence of random jump times of the DCP $\{ N_t \}_{t \in \mathcal{T}}$ having the stochastic intensity process $\{ \lambda _t \}_{t \in \mathcal{T}}$ given by Eq. ( 2.1 ). Let $\{ \Xi _{j} \}_{j = 1, 2, \ldots }$ be a sequence of i.i.d. positive individual loss amounts with the common distribution function $J(\xi )$ , where $\xi \gt 0$ such that for each $j = 1, 2, \ldots$ , $\Xi _j$ is the individual loss amount that occurs at random time $T_{2, j}$ . Assume that under the probability measure $\mathbb{P}$ , $\left \{ X_{i}\right \}_{i=1,2,\ldots }$ , $\left \{ T_{1,i}\right \}_{i=1,2,\ldots }$ , $\left \{ Y_{j}\right \}_{j=1,2,\ldots }$ and $\left \{ \Xi _{j}\right \}_{j=1,2,\ldots }$ are independent of each other. Then a compound point process $\{ C_t \}_{t \in \mathcal{T}}$ is said to be a CDCP if

\begin{align*} C_{t}=\sum \limits _{j\geq 1}\Xi _{j}\mathbb{I}_{\{ T_{2,j} \le t \} }. \\[-20pt] \end{align*}

Let us denote the integrated intensity process

\begin{equation*} {\Lambda _t \,\, :\!= \,\, \int ^t_0 \lambda _u \mathop {}\!\mathrm{d} u}. \end{equation*}

2.2.1 Time-homogeneous compound dynamic contagion process

For the standard DCP with time-homogeneous parameters, the infinitesimal generator $\mathcal{A}$ of the joint process $\{ \left ( \lambda _{t},N_{t},C_{t}, {M_t}, {\Lambda _t}, t\right ) \}_{t \in \mathcal{T}}$ acting on a function $f(\lambda ,n,c,{m},{\Lambda },t)\in \mathcal{D}( \mathcal{A} )$ is given by:

(2.2) \begin{align} &\hspace {-0.5cm}\mathcal{A} f( \lambda , n,c,{m}, {\Lambda }, t) \notag \\ &= \frac {\partial f}{\partial t}+\delta \left ( a-\lambda \right ) \frac {\partial f}{\partial \lambda } + {\lambda \frac {\partial f}{\partial \Lambda }} \notag \\ &\quad +\lambda \left [ \int _{0}^{\infty }\int _{0}^{\infty }f( \lambda +y,n+1,c+\xi ,{m},{\Lambda },t) \mathop {}\!\mathrm{d} G(y)\mathop {}\!\mathrm{d} J(\xi )-f( \lambda ,n,c,{m},{\Lambda },t) \right ] \notag \\ &\quad +\rho \left [ \int _{0}^{\infty }f( \lambda +x,n,c,{m+1},{\Lambda },t) \mathop {}\!\mathrm{d} H(x)-f( \lambda ,n,c,{m},{\Lambda },t) \right ]\!, \end{align}

where $\mathcal{D}( \mathcal{A})$ is the domain of the generator $\mathcal{A}$ such that $f(\lambda ,n,c,{m},{\Lambda },t)$ is differentiable with respect to $\lambda$ and $t$ , and

\begin{equation*} \left \vert \int _{0}^{\infty }\int _{0}^{\infty }f( \lambda +y,n+1,c+\xi ,{m},{\Lambda },t) \mathop {}\!\mathrm{d} G(y)\mathop {}\!\mathrm{d} J(\xi )-f( \lambda ,n,c,{m},{\Lambda },t) \right \vert \lt \infty \text{,} \end{equation*}

\begin{equation*} \left \vert \int _{0}^{\infty }f( \lambda +x,n,c,{m+1},{\Lambda },t) \mathop {}\!\mathrm{d} H(x)-f( \lambda ,n,c,{m},{\Lambda },t) \right \vert \lt \infty \text{.} \end{equation*}

For details on the definition and derivations of the infinitesimal generator in (2.2), please refer to Dassios and Zhao (Reference Dassios and Zhao2011), Dassios and Zhao (Reference Dassios and Zhao2017), Jang and Oh (Reference Jang and Oh2021), and Øksendal (Reference Øksendal2013).

Based on Eq. (2.2), we can easily obtain the moments of $\lambda _t$ , $N_t$ , and $C_{t}$ . To do so, we denote the first-order moments of $X$ , $Y$ , and $\Xi$ , respectively, by

\begin{equation*} {\mu _{H}=\int _{0}^{\infty }x \mathop {}\!\mathrm{d} H(x)},\ \ \mu _{G}=\int _{0}^{\infty }y \mathop {}\!\mathrm{d} G(y),\ \ \mu _{J}=\int _{0}^{\infty }\xi \mathop {}\!\mathrm{d} J(\xi ) . \end{equation*}

and their Laplace transforms by

(2.3) \begin{equation} {\hat {h}(s) = \int _{0}^{\infty }\mathrm{e}^{-s x} \mathop {}\!\mathrm{d} H(x)}, \quad \hat {g}(s) = \int _{0}^{\infty }\mathrm{e}^{-s y} \mathop {}\!\mathrm{d} G(y), \quad \hat {j}(s) = \int _{0}^{\infty }\mathrm{e}^{-s \xi }\mathop {}\!\mathrm{d} J(\xi ), \end{equation}

which are supposed to be finite. The Laplace transforms will be used in Section 3.

The following proposition for a standard CDCP with time-homogeneous parameters is directly adapted from Dassios and Zhao (Reference Dassios and Zhao2011); Dassios and Zhao (Reference Dassios and Zhao2017) and Jang and Oh (Reference Jang and Oh2021).

Proposition 1 (Expectations of $\lambda _t$ , $N_t$ , and $C_t$ for time-homogeneous CDCP). The expectation of $\lambda _{t}$ conditional on $\lambda _0$ under the real-world probability measure $\mathbb{P}$ is given by

\begin{equation*} \mathbb{E}\left [ \lambda _{t} \mid \lambda _0 \right ] \,=\, \left \{ \begin{array}{ll} \dfrac {\rho \mu _H + a\delta }{\kappa } + \left ( \lambda _0 - \dfrac {\rho \mu _H + a\delta }{\kappa } \right ) \mathrm{e}^{-\kappa t}, &\quad \kappa \neq 0,\\ \lambda _0 + \left (\rho \mu _H+ a\delta \right )t, &\quad \kappa =0, \end{array} \right . \end{equation*}

where $\kappa \, :\!= \, \delta - \mu _{G}$ . Under the stationary condition $\kappa \gt 0$ , the asymptotic first moment of $\lambda _{t}$ under $\mathbb{P}$ is given by

\begin{equation*} \mu _1\ \, :\!= \,\ \lim _{t\rightarrow \infty }\mathbb{E}\left [ \lambda _{t} \mid \lambda _0 \right ] \ =\ \frac {\rho \mu _H + a\delta }{\kappa }. \end{equation*}

The expectation of $N_{t}$ conditional on $N_0=0$ and $\lambda _0$ under $\mathbb{P}$ is given by

\begin{equation*} \mathbb{E}\left [ N_{t} \mid \lambda _0 \right ] \,=\, \left \{ \begin{array}{ll} \mu _1 t + \left ( \lambda _0 - \mu _1 \right ) \dfrac {1}{ \kappa } ( 1-\mathrm{e}^{- \kappa t} ), &\quad \kappa \neq 0,\\ \lambda _0 t + \dfrac {1}{2}\left (\rho \mu _H+ a\delta \right ){t^2}, &\quad \kappa = 0. \end{array} \right . \end{equation*}

The expectation of $C_{t}$ conditional on $C_0=0$ and $\lambda _0$ under $\mathbb{P}$ is given by

\begin{equation*} \mathbb{E}\left [ C_{t} \mid \lambda _0 \right ] \,=\, \left \{ \begin{array}{ll} \mu _J \left[ \mu _1 t + \left ( \lambda _0 - \mu _1 \right ) \dfrac {1}{ \kappa } ( 1-\mathrm{e}^{- \kappa t} ) \right]\!, &\quad \kappa \neq 0,\\[8pt] \mu _J \left[ \lambda _0 t + \dfrac {1}{2} {\left (\rho \mu _H+ a\delta \right )}{t^2 }\right]\!, &\quad \kappa = 0. \end{array} \right . \end{equation*}

2.2.2 Time-inhomogeneous compound dynamic contagion process

The previously defined DCP $\lambda _t$ from Eq. (2.1) is the result of solving the stochastic differential equation (SDE)

(2.4) \begin{equation} \lambda _{t} = \lambda _0 + \int _{0}^{t} \delta \left ( a - \lambda _s \right ) \mathop {}\!\mathrm{d} s + \sum _{i \ge 1} X_{i} \, \mathbb{I}_{\{ T_{1,i} \le t \}} + \sum _{j \ge 1} Y_{j} \, \mathbb{I}_{ \{ T_{2, j} \le t \}} . \end{equation}

The parameters of this process are all time-homogeneous. More generally, we can extend to the time-inhomogeneous situation by allowing the parameters to vary over time (except the decaying rate $\delta$ ). Specifically, we impose the following assumptions:

1. (A1) The mean-reverting level $a(t)$ and the rate of externally excited jump arrival $\rho (t)$ are deterministic functions of time $t$ ; they are bounded on all intervals $[0, t)$ (no explosions); and $\lambda _0 \geq a(t)$ .

2. (A2) The distribution function of the externally excited jump sizes at any time $t$ is $H(x; \, t)$ for $x \gt 0$ with $\mu _H(t) = \int _0^\infty x \, \mathop {}\!\mathrm{d} H(x; \, t)$ .

3. (A3) The distribution function of the self-excited jump sizes at any time $t$ is $G(y; \, t)$ for $y \gt 0$ with $\mu _G(t) = \int _0^\infty y \, \mathop {}\!\mathrm{d} G(y; \, t)$ .

4. (A4) $a(t)$ , $\rho (t)$ , $G(y; \, t)$ and $H(x; \, t)$ are Riemann-integrable functions of $t$ ; they are all positive.

After replacing the time-homogeneous constants in Eq. (2.4) with time-varying functions and solving the SDE, we arrive at the $\lambda _t$ for this new time-inhomogeneous DCP

(2.5) \begin{equation} \lambda _{t} = \lambda _0 \mathrm{e}^{-\delta t} + \delta \int _{0}^{t} a(s) \mathrm{e}^{-\delta (t-s)} \, \mathop {}\!\mathrm{d} s + \sum _{i \ge 1} X_{i} \mathrm{e}^{-\delta \left ( t - T_{1, i} \right )}\mathbb{I}_{\{ T_{1,i} \le t \}} + \sum _{j \ge 1} Y_{j} \mathrm{e}^{-\delta \left (t - T_{2,j} \right )} \mathbb{I}_{ \{ T_{2, j} \le t \}}. \end{equation}

We allow the model parameters to be time-inhomogeneous or time-varying to reflect evolving exposure and claim dynamics in the insurance and reinsurance portfolios. Specifically, in reality, claim intensity and contagion effects may vary over time due to changes in insured exposure, portfolio composition, seasonal effects, and external risk conditions. Introducing time-inhomogeneity at this stage provides a flexible actuarial framework for modeling claim arrivals, while its role under an equivalent change of measures will become apparent later in Section 3.

For this time-inhomogeneous CDCP, we can obtain the moments of $\lambda _t$ , $N_t$ , and $C_t$ as below, where the relevant parameters are time-dependent. They extend the results in Proposition 1.

Proposition 2 (Expectations of $\lambda _t$ , $N_t$ , and $C_t$ for time-inhomogeneous CDCP). For the time-inhomogeneous CDCP, the expectation of $\lambda _t$ conditional on $\lambda _0$ under the real-world probability measure $\mathbb{P}$ is given by

(2.6) \begin{equation} \mu _{\lambda }(t) \,\, :\!= \,\, \mathbb{E}\left [ \lambda _{t}\mid \lambda _{0}\right ] \,=\, \lambda _{0} \mathrm{e}^{-\int _{0}^{t} \kappa (s) \mathop {}\!\mathrm{d} s} + \mathrm{e}^{-\int _{0}^{t} \kappa (s) \mathop {}\!\mathrm{d} s} \int \limits _{0}^{t} \mathrm{e}^{\int _{0}^{s} \kappa (u) \mathop {}\!\mathrm{d} u} \big \{ \rho (s) \mu _H(s)+a(s) \delta \big \} \mathop {}\!\mathrm{d} s, \end{equation}

where

\begin{equation*} \kappa (t) \, :\!= \, \delta - \mu _G(t). \end{equation*}

The expectation of $N_{t}$ conditional on $N_0=0$ and $\lambda _0$ under $\mathbb{P}$ is given by

(2.7) \begin{equation} \mathbb{E}[ N_{t} \mid \lambda _{0} ] = \int \limits _{0}^{t} \mu _{\lambda }(s) \mathop {}\!\mathrm{d} s. \end{equation}

The expectation of $C_{t}$ conditional on $C_0=0$ and $\lambda _0$ under $\mathbb{P}$ is given by

(2.8) \begin{equation} \mathbb{E}[ C_{t} \mid \lambda _{0} ] = \mu _{J} \int \limits _{0}^{t} \mu _{\lambda }(s) \mathop {}\!\mathrm{d} s. \end{equation}

Proof. The infinitesimal generator $\mathcal{A}$ of the joint process $\{ \left ( \lambda _{t},N_{t},C_{t}, {M_t}, {\Lambda _t}, t\right ) \}_{t \in \mathcal{T}}$ acting on a function $ f(\lambda ,n,c,{m},{\Lambda },t)\in \mathcal{D}\left ( \mathcal{A} \right )$ for the time-inhomogeneous CDCP is given by:

(2.9) \begin{align} &\hspace {-0.5cm}\mathcal{A} f( \lambda , n,c,{m}, {\Lambda }, t) \notag \\ &= \frac {\partial f}{\partial t}+\delta \big ( a(t)-\lambda \big ) \frac {\partial f}{\partial \lambda } + {\lambda \frac {\partial f}{\partial \Lambda }} \notag \\ &\quad +\lambda \left [ \int _{0}^{\infty }\int _{0}^{\infty }f( \lambda +y,n+1,c+\xi ,{m},{\Lambda },t) \mathop {}\!\mathrm{d} G(y; \, t)\mathop {}\!\mathrm{d} J(\xi )-f( \lambda ,n,c,{m},{\Lambda },t) \right ] \notag \\ &\quad +\rho (t) \left [ \int _{0}^{\infty }f( \lambda +x,n,c,{m+1},{\Lambda },t) \mathop {}\!\mathrm{d} H(x; \, t)-f( \lambda ,n,c,{m},{\Lambda },t) \right ]\!, \end{align}

where $\mathcal{D}\left ( \mathcal{A}\right )$ is the domain of the generator $\mathcal{A}$ such that $f(\lambda ,n,c,{m},{\Lambda },t)$ is differentiable with respect to $\lambda$ and $t$ , and

\begin{equation*} \left \vert \int _{0}^{\infty }\int _{0}^{\infty }f( \lambda +y,n+1,c+\xi ,{m},{\Lambda },t) \mathop {}\!\mathrm{d} G(y; \, t)\mathop {}\!\mathrm{d} J(\xi )-f( \lambda ,n,c,{m},{\Lambda },t) \right \vert \lt \infty \text{,} \end{equation*}

\begin{equation*} \left \vert \int _{0}^{\infty }f( \lambda +x,n,c,{m+1},{\Lambda },t) \mathop {}\!\mathrm{d} H(x; \, t)-f( \lambda ,n,c,{m},{\Lambda },t) \right \vert \lt \infty \text{.} \end{equation*}

Setting $f( \lambda ,n , c, m, \Lambda , t) =\lambda$ in Eq. (2.9), we have

\begin{equation*} \mathcal{A}\lambda =- \kappa (t) \lambda +\rho (t) \mu _H(t)+a(t) \delta . \end{equation*}

For each $t \ge 0$ , ${\lambda }^{\dagger }_t \, :\!= \, \lambda _{t}-\lambda _{0}-\int _{0}^{t}\mathcal{A}\lambda _{s}\mathop {}\!\mathrm{d} s$ . Then for any $s \lt t$ , we have:

\begin{eqnarray*} \mathbb{E} [\lambda ^{\dagger }_t - \lambda ^{\dagger }_s \,\mid \, \mathcal{F}_s ] = \mathbb{E} \left [\lambda _t - \lambda _s - \int _{s}^{t}\mathcal{A}\lambda _{u}\mathop {}\!\mathrm{d} u \,\mid \, \mathcal{F}_s \right ], \quad \mbox{$\mathbb{P}$-a.s.} \end{eqnarray*}

By the Dynkin formula, since $t \lt \infty$ ,

\begin{eqnarray*} \mathbb{E} [ \lambda _t \,|\, \mathcal{F}_s ] = \lambda _s + \mathbb{E} \left [ \int _{s}^{t}\mathcal{A}\lambda _{u}\mathop {}\!\mathrm{d} u { \ | \ \mathcal{F}_s } \right ], \quad \mbox{$\mathbb{P}$-a.s.} \end{eqnarray*}

Consequently, by Fubini’s theorem,

\begin{eqnarray*} \mathbb{E} [\lambda ^{\dagger }_t - \lambda ^{\dagger }_s \,|\, \mathcal{F}_s ] = \mathbb{E} \left [\lambda _t - \lambda _s - \int _{s}^{t}\mathcal{A}\lambda _{u}\mathop {}\!\mathrm{d} u \,\mid \, \mathcal{F}_s \right ] = 0, \quad \mbox{$\mathbb{P}$-a.s.} \end{eqnarray*}

This implies that

\begin{eqnarray*} \mathbb{E} [\lambda ^{\dagger }_t \,|\, \mathcal{F}_s ] = \lambda ^{\dagger }_s, \quad \mbox{$\mathbb{P}$-a.s.} \end{eqnarray*}

Since $\{ \lambda ^{\dagger }_t \}_{t \ge 0}$ is an $(\mathbb{F}, \mathbb{P})$ -martingale, we must have:

\begin{equation*} \mathbb{E} [ \lambda ^{\dagger }_t \,|\, \mathcal{F}_0 ] = \mathbb{E} [ \lambda ^{\dagger }_t \,|\, \lambda _0 ] = \mathbb{E}\Biggl ( \lambda _{t} {- \lambda _0} -\int _{0}^{t}\mathcal{A}\lambda _{s}\mathop {}\!\mathrm{d} s \,\mid \, \lambda _{0} \Biggr ) = 0, \quad \mbox{$\mathbb{P}$-a.s.} \end{equation*}

Consequently,

\begin{eqnarray*} \mathbb{E} [\lambda _t \,|\, \lambda _0] = \lambda _0 + \int ^{t}_{{0}} \mathbb{E} \left [ \mathcal{A} \lambda _u \,|\, \lambda _0 \right ] \mathop {}\!\mathrm{d} u, \quad \mbox{$\mathbb{P}$-a.s.} \end{eqnarray*}

Then,

\begin{equation*} \mathbb{E}[ \lambda _{t}\mid \lambda _{0} ] = \lambda _{0}-\int _{0}^{t} \kappa (s) \, \mathbb{E}[ \lambda _{s}\mid \lambda _{0} ] \mathop {}\!\mathrm{d} s+\int _{0}^{t}\left \{ \rho (s) \mu _H(s)+a(s) \delta \right \} \mathop {}\!\mathrm{d} s. \end{equation*}

Differentiating it with respect to $t$ , we then have the ODE for $\mu _{\lambda }(t) \, :\!= \,\mathbb{E} \left ( \lambda _{t}\mid \lambda _{0}\right )$ ,

\begin{equation*} \frac {\mathop {}\!\mathrm{d} \mu _{\lambda }(t)}{\mathop {}\!\mathrm{d} t} = - \kappa (t) \mu _{\lambda }(t) +\rho (t) \mu _H(t)+a(t) \delta . \end{equation*}

and solving this ODE, we obtain Eq. (2.6) as its solution. Using Eq. (2.6), given $N_0 = C_0 = 0$ , we can derive

\begin{equation*} \mathbb{E}[ N_t \mid \lambda _0 ] = \int _{0}^{t} \mathbb{E}[ \lambda _s \mid \lambda _0 ] \mathop {}\!\mathrm{d} s \, \end{equation*}

and

\begin{equation*} \mathbb{E}[ C_{t}\mid \lambda _{0} ] = \mu _J \int _0^t \mathbb{E}[ \lambda _s \mid \lambda _0 ] \mathop {}\!\mathrm{d} s. \end{equation*}

The expressions for Eqs. (2.7) and (2.8) would be very long with various simple exponential functions. To save space, we just leave their concise expressions, but we will compare them with corresponding time-homogeneous cases in the context of arbitrage-free catastrophe reinsurance premiums in Section 4. Appendix B contains the algorithm we developed to simulate from this new process.

3.1 Equivalent martingale measures

In this section, we extend the measure change for the Cox process with shot-noise intensity (Dassios & Jang, Reference Dassios and Jang2003) to that for a standard dynamic contagion process (Dassios & Zhao, Reference Dassios and Zhao2011). For further details, we refer to Sondermann (Reference Sondermann1991), where the no-arbitrage approach to pricing reinsurance contracts was first introduced.

Before introducing the formal notion of an equivalent martingale measure, we emphasize that the arbitrage-free pricing method considered in this paper is understood in the actuarial sense, rather than through replication in a liquid financial market. Specifically, we rule out reinsurance strategies that generate non-negative terminal surplus almost surely with strictly positive expected gain, in accordance with Definition 4, and adopt the Esscher transform as a risk-adjusted pricing principle reflecting risk aversion and solvency considerations. In our modeling setup, the market is incomplete, and contingent claims such as catastrophe reinsurance contracts cannot be replicated via dynamic hedging; valuation and hedging therefore “divorce,” as noted in Madan and Schoutens (Reference Madan and Schoutens2022) (Chapter 13, Page 171 therein). This makes financial arbitrage arguments based on dynamic replication, such as those in Artzner et al. (Reference Artzner, Eisele and Schmidt2024), less relevant in our setting. Our approach is particularly appropriate for catastrophe reinsurance contracts, which are written over finite (typically annual or multi-year) horizons and are not dynamically hedgeable, and operates within an actuarial martingale valuation paradigm under which the insurer’s surplus process is a martingale under the Esscher-transformed measure (Definition 5), thereby ensuring actuarial no-arbitrage pricing for non-traded catastrophic risks.

To simplify the discussion, given that we assume a constant interest rate, we take the interest rate to be zero. The framework extends straightforwardly to a non-zero constant interest rate by working with discounted surplus processes, as detailed in Appendix A. However, the incorporation of a stochastic interest rate, while it is more realistic, would complicate the modeling framework. Consequently, to simplify the discussion, we do not consider the situation of a stochastic interest rate here.

Remark 1. It may be noted that Definition 5 (iii) can be expressed as follows: assuming $R_{0}=0$ , (i.e., the surplus at the initial time $0$ is zero), we have:

\begin{equation*} \tilde {\mathbb{E}}\!\left (R_{t}\mid \mathcal{F}_{0}\right ) = \tilde {\mathbb{E}}\!\left (P_{t}-C_{t}\mid \mathcal{F}_{0}\right ) = \tilde {\mathbb{E}}\!\left (P_{t}\mid \mathcal{F}_{0}\right ) - \tilde {\mathbb{E}}\!\left (C_{t}\mid \mathcal{F}_{0}\right ) = 0. \end{equation*}

Consequently,

\begin{equation*} \tilde {\mathbb{E}}\!\left (C_{t}\mid \mathcal{F}_{0}\right ) = \tilde {\mathbb{E}}\!\left (P_{t}\mid \mathcal{F}_{0}\right ) = \text{an arbitrage-free insurance premium.} \end{equation*}

In the sequel, we shall adopt the Esscher transform to select an equivalent martingale measure. To do this, we start with Theorem 1, giving a (local) martingale, which will be considered when specifying the Radon–Nikodym derivative for changing probability measures via the Esscher transform.

Theorem 1. Suppose that $\rho \gt 0$ for all $t \in \mathcal{T}$ . For constants $\theta , \nu , \phi , {\psi }$ , we have an $(\mathbb{F}, \mathbb{P})$ -(local)-martingale

\begin{equation*} \mathrm{e}^{K(t)} \theta ^{N_t} \mathrm{e}^{ B(t) \lambda _t} \, \mathrm{e}^{-\nu C_t} {\mathrm{e}^{\phi \Lambda _t} \psi ^{M_t}}, \quad t \in \mathcal{T}, \end{equation*}

if $\{B(t),K(t)\}_{t \in \mathcal{T}}$ satisfy the non-linear ordinary differential equations (ODE):

\begin{align*} B^{\prime }(t) - \delta B(t) + \theta \, {\hat {j}(\nu )} \, \hat {g}(\!-\!B(t)) \, + {\phi } -1 &= 0,\\ K^{\prime }(t) + a \delta B(t) + \rho \left [{\psi } \hat {h}(\!-\!B(t)) - 1 \right ] &= 0, \end{align*}

where ${\hat h} (\! \cdot \!)$ , ${\hat g} (\! \cdot \!)$ and ${\hat j} (\! \cdot \!)$ are the Laplace transforms defined by Eq. ( 2.3 ).

Proof. Firstly, we consider a function $f( \lambda _{t},N_{t},C_{t}, {M_t}, {\Lambda _t}, t)$ defined by:

\begin{equation*} f( \lambda _{t},N_{t},C_{t}, {M_t}, {\Lambda _t}, t) \,=\, \mathrm{e}^{K(t)} \theta ^{N_t} \mathrm{e}^{ B(t) \lambda _t} \, \mathrm{e}^{-\nu C_t} {\mathrm{e}^{\phi \Lambda _t} \psi ^{M_t}}, \end{equation*}

for constants $\theta ,{\psi },\nu$ .

Applying Itô’s differentiation rule to $f$ gives:

\begin{eqnarray*} f (\lambda _t, N_t, C_t, M_t, \Lambda _t, t) &=& f (\lambda _0, N_0, C_0, M_0, \Lambda _0, 0) + \int ^{t}_{0} \delta (a - \lambda _u) \frac {\partial f}{\partial \lambda } \mathop {}\!\mathrm{d} u + \int ^{t}_{0} \lambda _u \frac {\partial f}{\partial \Lambda } \mathop {}\!\mathrm{d} u \\ && + \int ^{t}_{0} \lambda _{u-} \bigg ( \int ^{\infty }_{0} \int ^{\infty }_{0} f (\lambda _{u-} + y, N_{u-} + 1, C_{u-} + \xi , M_{u-}, \Lambda _{u-}, u) \\ && \quad \times \mathop {}\!\mathrm{d} G(y) \mathop {}\!\mathrm{d} J (\xi ) - f (\lambda _{u-}, N_{u-}, C_{u-}, M_{u-}, \Lambda _{u-}, u) \bigg ) \mathop {}\!\mathrm{d} u \\ && + \int ^{t}_{0} \rho _{u-} \bigg ( \int ^{\infty }_{0} f (\lambda _{u-} + x, N_{u-}, C_{u-}, M_{u-} + 1, \Lambda _{u-}, u) \mathop {}\!\mathrm{d} H (x) \\ && \quad - f(\lambda _{u-}, N_{u-}, C_{u-}, M_{u-}, \Lambda _{u-}, u) \bigg ) \mathop {}\!\mathrm{d} u + \mathcal{M}_t, \end{eqnarray*}

where $\{ \mathcal{M}_t \}_{t \ge 0}$ is an $(\mathbb{F}, \mathbb{P})$ -(local)-martingale.

Using Eq. (2.2), we have:

\begin{align*} &\hspace {-0.5cm} f (\lambda _t, N_t, C_t, M_t, \Lambda _t, t) \\ &= f (\lambda _0, N_0, C_0, M_0, \Lambda _0, 0) + \int ^{t}_{0} \mathcal{A} f (\lambda _{u-}, N_{u-}, C_{u-}, M_{u-}, \Lambda _{u-}, u) \mathop {}\!\mathrm{d} u + \mathcal{M}_t. \end{align*}

If $\mathcal{A} f( \lambda , n,c,{m}, {\Lambda }, t)=0$ then

\begin{equation*} f (\lambda _t, N_t, C_t, M_t, \Lambda _t, t) = f (\lambda _0, N_0, C_0, M_0, \Lambda _0, 0) + \mathcal{M}_t, \end{equation*}

i.e., $\{ f (\lambda _t, N_t, C_t, M_t, \Lambda _t, t) \}_{t \in \mathcal{T}}$ is an $(\mathbb{F}, \mathbb{P})$ -(local)-martingale.

Plug $f(\lambda _t, N_t, C_t, M_t, \Lambda _t, t)$ into the generator in Eq. (2.2) for the standard DCP and set $\mathcal{A} f( \lambda , n,c,{m}, {\Lambda }, t)=0$ , where

\begin{equation*} f( \lambda , n,c,{m}, {\Lambda }, t) = \mathrm{e}^{K(t)} \theta ^{n} \mathrm{e}^{ B(t) \lambda } \, \mathrm{e}^{-\nu c} {\mathrm{e}^{\phi \Lambda } \psi ^{m}} , \end{equation*}

i.e.,

\begin{align*} 0&= K^{\prime }(t) + B^{\prime }(t) \lambda +\delta \left ( a- \lambda \right ) B(t) + \lambda \phi \\ &\quad + \lambda \left [ \theta \int _{0}^{\infty }\int _{0}^{\infty } \mathrm{e}^{ B(t) y} \mathrm{e}^{-\nu \xi } \mathop {}\!\mathrm{d} G(y)\mathop {}\!\mathrm{d} J(\xi )-1 \right ] + \rho \left [ \psi \int _{0}^{\infty } \mathrm{e}^{ B(t) x} \mathop {}\!\mathrm{d} H(x)- 1 \right ], \\ 0 &= \lambda \left \{ B^{\prime }(t) - \delta B(t) + \phi + \theta \int _{0}^{\infty }\int _{0}^{\infty } \mathrm{e}^{ B(t) y} \mathrm{e}^{-\nu \xi } \mathop {}\!\mathrm{d} G(y)\mathop {}\!\mathrm{d} J(\xi )-1 \right \} \\ &\quad + K^{\prime }(t) + \delta a B(t) + \rho \left [ \psi \int _{0}^{\infty } \mathrm{e}^{ B(t) x} \mathop {}\!\mathrm{d} H(x)- 1 \right ] \\ 0 &= \lambda \bigg \{ B^{\prime }(t) - \delta B(t) + \phi + \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) -1 \bigg \} + K^{\prime }(t) + \delta a B(t) + \rho \left [ \psi \hat {h}(\!-\!B(t))- 1 \right ], \end{align*}

where

\begin{equation*} \hat {j}(\nu ) \, :\!= \, \int _{0}^{\infty } \mathrm{e}^{-\nu \xi } \mathop {}\!\mathrm{d} J(\xi ), \qquad \hat {g}(B(t)) \, :\!= \, \int _{0}^{\infty } \mathrm{e}^{ -B(t) y} \mathop {}\!\mathrm{d} G(y), \qquad {\hat {h}}(B(t)) \, :\!= \, \int _{0}^{\infty } \mathrm{e}^{- B(t) x} \mathop {}\!\mathrm{d} {H}(x). \end{equation*}

So, we have the ODEs for $B(t),K(t)$ ,

\begin{align*} B^{\prime }(t) - \delta B(t) + \theta \, {\hat {j}(\nu )} \, \hat {g}(\!-\!B(t)) \, + {\phi } -1 &= 0,\\ K^{\prime }(t) + a \delta B(t) + \rho \left [{\psi } \hat {h}(\!-\!B(t)) - 1 \right ] &= 0. \end{align*}

Hence, the theorem is proved.

The following theorem provides the existence and uniqueness results for the solutions to the nonlinear ODEs in Theorem 1.

Theorem 2. Assume that the following conditions hold:

\begin{equation*} \theta ,\psi \geq 1, \quad \nu \lt 0 \quad (\hat {j}(\nu )\gt 1), \qquad \phi = -(\theta \, \hat {j}(\nu ) -1) \lt 0, \qquad \delta \gt \theta \, \hat {j}(\nu ) \mu _G. \end{equation*}

Then the following statements hold:

1. 1. There exists a unique solution for $B(t)$ to the nonlinear ODE in Theorem  1 with the initial condition $B(0)=b\gt 0$ and the stationary condition $\delta \gt \mu _G$ such that

   \begin{equation*} B(t)\ =\ \mathcal{G}^{-1}(t), \qquad t \geq 0, \end{equation*}
   where
   \begin{equation*} \mathcal{G}(B) \, :\!= \, \int ^{B}_b \frac {\mathop {}\!\mathrm{d} u }{f_1( u )}, \qquad B\in (b, B^+ ), \end{equation*}
   \begin{equation*} f_1 (B) \, :\!= \, \delta B -\theta \, \hat {j}(\nu ) \, ( \hat {g}(\!-\!B ) -1 ). \end{equation*}
   and $B^+\gt 0$ is the smallest positive solution to $f_1 (B)=0$ .

2. 2. Then, there exists a unique solution for $K(t)$ to the nonlinear ODE in Theorem  1 with the boundary condition $K(0) = 0$ such that

   \begin{equation*} K (t) = -a \delta \int _0^t B(s)\mathop {}\!\mathrm{d} s + \rho \int _0^t \big [ 1 - {\psi } \hat {h}(-B(s)) \big ] \mathop {}\!\mathrm{d} s. \end{equation*}

Proof. Assume that

\begin{equation*} \theta ,\psi \geq 1, \quad \nu \lt 0 \quad (\hat {\,j}(\nu )\gt 1), \qquad \phi = -(\theta \, \hat {j}(\nu ) -1) \lt 0, \qquad \delta \gt \theta \, \hat {j}(\nu ) \mu _G. \end{equation*}

Then the nonlinear ODE in Theorem 1 can be written as:

(3.2) \begin{equation} \frac {\mathop {}\!\mathrm{d} B(t)}{\mathop {}\!\mathrm{d} t} = \delta B - \theta \, \hat {j}(\nu ) \, \left ( \hat {g}(\!-\!B ) -1 \right ), \end{equation}

with the initial condition $B(0)=b\gt 0$ and the stationary condition $\delta \gt \mu _G$ . Define

(3.3) \begin{equation} f_1(B) \, :\!= \, \delta B -\theta \, \hat {j}(\nu ) \, \left ( \hat {g}(\!-\!B ) -1 \right )\!. \end{equation}

Then

(3.4) \begin{equation} f_1(0)= 0. \end{equation}

Also,

(3.5) \begin{equation} \left . \frac {\mathop {}\!\mathrm{d} f_1 (B)}{\mathop {}\!\mathrm{d} B} \right |_{B=0} =\ \delta - \theta \hat {j} (\nu ) \mu _G\ \gt \ 0. \end{equation}

Taking the second-order derivative of $f_1 (B)$ with respect to $B$ gives:

\begin{equation*} \frac {\mathop {}\!\mathrm{d}^2 f_1 (B) }{\mathop {}\!\mathrm{d} B^2} = - \theta \hat {j} (\nu ) \int ^{\infty }_{0} {y^2 \mathrm{e}^{B y} \mathop {}\!\mathrm{d} G(y)} \lt 0. \end{equation*}

Then $f_1 (B)$ is strictly concave. This, together with Eqs. (3.4) and (3.5), imply that $f_1(B) \gt 0$ , for $B\in \left (0, B^+\right )$ , where $B^+\gt 0$ is the smallest positive solution to $f_1 (B)=0$ .

Taking $B = u$ and $t = \tau$ , Eqs. (3.2) and (3.3) lead to:

(3.6) \begin{equation} \frac {\mathop {}\!\mathrm{d} u}{f_1 (u)} = \mathop {}\!\mathrm{d} \tau , \quad u \in (0, B^+). \end{equation}

Recall that $B (0) = b$ and $B (t) = B$ . Then integrating both sides of Eq. (3.6) gives:

(3.7) \begin{equation} \int ^{B}_{b} \frac {\mathop {}\!\mathrm{d} u}{f_1 (u)} = \int ^{t}_{0} \mathop {}\!\mathrm{d} \tau = t, \quad B \in (0, B^+). \end{equation}

Define the left-hand side of Eq. (3.7) as:

(3.8) \begin{equation} \mathcal{G}(B) \, :\!= \, \int ^{B}_b \frac {\mathop {}\!\mathrm{d} u}{f_1( u )}, \qquad B\in \left (b, B^+ \right )\!. \end{equation}

Note that $\mathcal{G}(B)$ is a continuous and strictly increasing function for $B \in (b, B^+)$ . Consequently, its inverse $\mathcal{G}^{-1}$ exists and is uniquely determined. The function $\mathcal{G} \, : \, (b, B^+) \to (0, \infty )$ is defined in Eq. (3.8) is an isomorphism. The inverse $\mathcal{G}^{-1} (t)$ then gives the solution $B(t) = B$ to the nonlinear ODE in Theorem 1 with the initial condition $B (0) = b \gt 0$ as follows:

\begin{equation*} B(t) \,\,=\,\, \mathcal{G}^{-1}(t) \,\, \in \,\, (b, B^+), \qquad t \gt 0. \end{equation*}

Once $B(t)$ is uniquely determined as above, $K(t)$ can be obtained from the second nonlinear ODE in Theorem 1 with the boundary condition $K(0) = 0$ simply as

\begin{equation*} K (t) = -a \delta \int _0^t B(s)\mathop {}\!\mathrm{d} s + \rho \int _0^t \left [ 1 - {\psi } \hat {h}(\!-\!B(s)) \right ] \mathop {}\!\mathrm{d} s. \end{equation*}

Recall that $\{\mathrm{e}^{K(t)} \theta ^{N_t} \mathrm{e}^{ B(t) \lambda _t} \, \mathrm{e}^{-\nu C_t} {\mathrm{e}^{\phi \Lambda _t} \psi ^{M_t}} \}_{t \in \mathcal{T}}$ is a (local)-martingale. We suppose that some (suitable) square integrability conditions hold so that $\{\mathrm{e}^{K(t)} \theta ^{N_t} \mathrm{e}^{ B(t) \lambda _t} \, \mathrm{e}^{-\nu C_t} {\mathrm{e}^{\phi \Lambda _t} \psi ^{M_t}} \}_{t \in \mathcal{T}}$ is a square-integrable martingale. We now describe how the probability laws of $N_{t}$ and $C_t$ change after changing the probability measures by the Esscher transform. Firstly, from Theorem 1, $\{ \mathrm{e}^{K(t)} \theta ^{N_t} \mathrm{e}^{ B(t) \lambda _t} \, \mathrm{e}^{-\nu C_t} {\mathrm{e}^{\phi \Lambda _t} \psi ^{M_t}} \}_{t \in \mathcal{T}}$ is an $(\mathbb{F}, \mathbb{P})$ -martingale. Then, for all $t \in \mathcal{T}$ ,

\begin{equation*} \mathbb{E}_0 [\mathrm{e}^{K(t)} \theta ^{N_t} \mathrm{e}^{ B(t) \lambda _t} \, \mathrm{e}^{-\nu C_t} {\mathrm{e}^{\phi \Lambda _t} \psi ^{M_t}} ] = \mathrm{e}^{K(0)} \theta ^{N_0} \mathrm{e}^{ B(0) \lambda _0} \, \mathrm{e}^{-\nu C_0} {\mathrm{e}^{\phi \Lambda _0} \psi ^{M_0}}. \end{equation*}

Consider the following (normalized) $\mathbb{F}$ -adapted process $\{\zeta _t \}_{t \in \mathcal{T}}$ :

(3.9) \begin{equation} \zeta _t \, :\!= \, \frac { \mathrm{e}^{K(t)} \theta ^{N_t} \mathrm{e}^{ B(t) \lambda _t} \, \mathrm{e}^{-\nu C_t} {\mathrm{e}^{\phi \Lambda _t} \psi ^{M_t}} }{\mathbb{E}_0 [\mathrm{e}^{K(t)} \theta ^{N_t} \mathrm{e}^{ B(t) \lambda _t} \, \mathrm{e}^{-\nu C_t} {\mathrm{e}^{\phi \Lambda _t} \psi ^{M_t}} ]} = \frac {\mathrm{e}^{K(t)} \theta ^{N_t} \mathrm{e}^{ B(t) \lambda _t} \, \mathrm{e}^{-\nu C_t} {\mathrm{e}^{\phi \Lambda _t} \psi ^{M_t}} }{\mathrm{e}^{K(0)} \theta ^{N_0} \mathrm{e}^{ B(0) \lambda _0} \, \mathrm{e}^{-\nu C_0} {\mathrm{e}^{\phi \Lambda _0} \psi ^{M_0}}}. \end{equation}

It is clear that by its definition in Eq. (3.9) and Theorem 1, $\{ \zeta _t \}_{t \in \mathcal{T}}$ is an $(\mathbb{F}, \mathbb{P})$ -martingale. For each $t \in \mathcal{T}$ , $\zeta _t \gt 0$ , $\mathbb{P}$ -a.s. Furthermore, $\mathbb{E}_0 [\zeta _t] = \zeta _0 = 1$ . Consequently, a new probability measure $\tilde {\mathbb{P}}$ , which is equivalent to the original probability measure $\mathbb{P}$ on $\mathcal{F}_T$ , can be defined by putting:

\begin{equation*} \frac {d \tilde {\mathbb{P}}}{d \mathbb{P}} \bigg |_{\mathcal{F}_T} \, :\!= \, \zeta _T. \end{equation*}

Given that $\{ \zeta _t \}_{t \in \mathcal{T}}$ is a square-integrable $(\mathbb{F}, \mathbb{P})$ –martingale, then

\begin{eqnarray*} \mathbb{E} \left [ \bigg | \frac {d \tilde {\mathbb{P}}}{d \mathbb{P}} \bigg |^2 \right ] = \mathbb{E} [ | \zeta _T |^2] \lt \infty . \end{eqnarray*}

Consequently, Definition 5 (ii) is satisfied.

We refer to related works employing the exponential change of measure: Palmowski and Rolski (Reference Palmowski and Rolski2002) for Markov processes, Kallsen and Shiryaev (Reference Kallsen and Shiryaev2002) via the cumulant process and Esscher transform, and Palmowski et al. (Reference Palmowski, Pojer and Thonhauser2025) for ruin probabilities with Hawkes arrivals.

3.2 Infinitesimal generator under the equivalent martingale measure

Let $\tilde {\mathcal{A}}$ denote the infinitesimal generator of the joint process $\{ ( \lambda _{t},N_{t},C_{t}, M_t, \Lambda _t, t) \}_{t \in \mathcal{T}}$ under the new probability measure $\tilde {\mathbb{P}}$ . The key to study how the probability laws of $N_{t}$ and $C_t$ change after changing probability measures by the Esscher transform is to obtain the infinitesimal generator $\tilde {\mathcal{A}}$ . This is to be achieved in Theorem 3 and Corollary 1 to be presented in the sequel. To do so, we start with a technical lemma.

Lemma 1. Assume that $\tilde {f}( \lambda , n,c,{m}, {\Lambda }, t) = \tilde {f}( \lambda , t)$ for all $n$ , $c$ , $m$ , and $\Lambda$ , and that $\mathrm{e}^{-B(t)\lambda _t}$ is a martingale. Consider $B(t)\gt 0$ for some $t \gt 0$ . Then

(3.10) \begin{align} \tilde {\mathcal{A}} \tilde {f} (\lambda , 0) = \frac { \mathcal{A} \bigl (\, \tilde {f}(\lambda , 0)\, \mathrm{e}^{{-}B(t)\lambda _t}\bigr ) } { \mathrm{e}^{-B(t)\lambda _t} }. \\[-23pt] \nonumber \end{align}

Proof. The generator of the process $(\lambda _t, t)$ acting on a function $\tilde {f}(\lambda , t)$ with respect to the equivalent martingale probability measure is

(3.11) \begin{equation} \tilde {\mathcal{A}} \tilde {f} (\lambda , 0) = \lim _{t \downarrow 0} \frac {\tilde {\mathbb{E}}\!\left [\tilde {f}(\lambda _t, t) \mid \lambda _0 = \lambda \right ] - \tilde {f}(\lambda , 0)}{t}. \end{equation}

We will use

\begin{equation*} \frac {\mathrm{e}^{-B(t)\lambda _t}}{\mathbb{E}\!\left (\mathrm{e}^{-B(t)\lambda _t}\right )} \end{equation*}

as the Radon–Nikodym derivative to define the equivalent martingale probability measure, where $\mathbb{E}\!\left (\mathrm{e}^{-B(t)\lambda _t}\right ) \lt \infty$ . Hence, the expected value of $\tilde {f}(\lambda _t, t)$ given $\lambda$ with respect to the equivalent martingale probability measure is

(3.12) \begin{equation} \tilde {\mathbb{E}}\!\left [\tilde {f}(\lambda _t, t) \mid \lambda _0 = \lambda \right ] = \frac {\mathbb{E}\!\left [\tilde {f}(\lambda _t, t)\, \mathrm{e}^{-B(t)\lambda _t} \mid \lambda _0 = \lambda \right ]} {\mathbb{E}\!\left (\mathrm{e}^{-B(t)\lambda _t} \mid \lambda _0 = \lambda \right )}. \end{equation}

Since the denominator in (3.12) is a martingale, it becomes

(3.13) \begin{equation} \tilde {\mathbb{E}}\!\left [\tilde {f}(\lambda _t, t) \mid \lambda _0 = \lambda \right ] = \frac {\tilde {f}(\lambda , 0)\, \mathrm{e}^{-B(t)\lambda } + \int _0^t \mathbb{E}\!\left [ \mathcal{A} \tilde {f}(\lambda _s, s)\, \mathrm{e}^{-B(t)\lambda _s} \mid \lambda _0 = \lambda \right ] \mathop {}\!\mathrm{d} s} {\mathrm{e}^{-B(t)\lambda }}. \end{equation}

Substituting (3.13) into (3.11) gives

(3.14) \begin{equation} \tilde {\mathcal{A}} \tilde {f}(\lambda , 0) = \frac {1}{\mathrm{e}^{-B(t)\lambda }} \lim _{t \downarrow 0} \frac {1}{t} \int _0^t \mathbb{E}\!\left [ \mathcal{A} \tilde {f}(\lambda _s, s)\, \mathrm{e}^{-B(t)\lambda _s} \mid \lambda _0 = \lambda \right ] \mathop {}\!\mathrm{d} s. \end{equation}

Therefore, by Dynkin’s formula (see Øksendal, Reference Øksendal2013), (3.10) follows immediately.

Theorem 3. Suppose that the following conditions hold:

\begin{equation*} \theta ,\psi \geq 1, \quad \nu \lt 0 \quad (\hat {j}(\nu )\gt 1), \qquad \phi = -(\theta \, \hat {j}(\nu ) -1) \lt 0, \qquad \delta \gt \theta \, \hat {j}(\nu ) \mu _G. \end{equation*}

Write, for all $t \in \mathcal{T}$ ,

\begin{align*} \mathop {}\!\mathrm{d} \tilde {G}(y; \, t) &\, :\!= \, \frac {\mathrm{e}^{B(t) y}}{\hat {g}(\!-\!B(t))} \mathop {}\!\mathrm{d} G(y), & \mathop {}\!\mathrm{d} {\tilde {H}}(x; \, t) &\, :\!= \, \frac {\mathrm{e}^{ B(t) x}}{\hat {h}(\!-\!B(t))} \mathop {}\!\mathrm{d} {H}(x), \\ \mathop {}\!\mathrm{d} \tilde {J}(\xi ) &\, :\!= \, \frac {\mathrm{e}^{-\nu \xi }}{\hat {j}(\nu )} \mathop {}\!\mathrm{d} J(\xi ), & \tilde {\rho }(t) &\, :\!= \, \psi \hat {h}(\!-\!B(t))\, \rho . \end{align*}

Let $\mathcal{D} (\mathcal{A})$ and $\mathcal{D} (\tilde {\mathcal{A}})$ denote the domains of the infinitesimal generators $\mathcal{A}$ and $\tilde {\mathcal{A}}$ , respectively. Then, for any function $\tilde {f} \in \mathcal{D} (\mathcal{A}) \cap \mathcal{D} (\tilde {\mathcal{A}})$ ,

(3.15) \begin{align} &\hspace {-0.5cm}\tilde {\mathcal{A}} \tilde {f}( \lambda , n,c,{m}, {\Lambda }, t) \notag \\ &= \frac {\partial \tilde {f}}{\partial t} + \lambda \frac {\partial \tilde {f}}{\partial \Lambda } + \delta \left ( a-\lambda \right )\frac {\partial \tilde {f}}{\partial \lambda } \notag \\ &\quad + \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t))\, \lambda \left [ \int _{0}^{\infty }\int _{0}^{\infty } \tilde {f}(\lambda +y,n+1,c+\xi ,{m},{\Lambda },t) \mathop {}\!\mathrm{d} \tilde {G}(y; \, t) \mathop {}\!\mathrm{d} \tilde {J}(\xi ) - \tilde {f} \right ] \notag \\ &\quad + \tilde {\rho }(t) \left [ \int _{0}^{\infty }\tilde {f}( \lambda +x,n,c,{m+1},{\Lambda },t) \mathop {}\!\mathrm{d} \tilde {H}(x; \, t) - \tilde {f} \right ]. \end{align}

Proof. Set

\begin{equation*} \mathrm{e}^{K(t)} \theta ^{n} \mathrm{e}^{ B(t) \lambda } \, \mathrm{e}^{-\nu c} {\mathrm{e}^{\phi \Lambda } \psi ^{m}} \tilde {f}( \lambda , n,c,{m}, {\Lambda }, t), \end{equation*}

into the generator in Eq. (2.2) and from Lemma 1, we have

\begin{align*} &\hspace {-0.1cm} \tilde {\mathcal{A}} \tilde {f}( \lambda , n,c,{m}, {\Lambda }, t) \\ &= \frac { \mathcal{A} \bigg \{ \mathrm{e}^{K(t)} \theta ^{n} \mathrm{e}^{ B(t) \lambda } \, \mathrm{e}^{-\nu c} {\mathrm{e}^{\phi \Lambda } \psi ^{m}} \tilde {f}( \lambda , n,c,{m}, {\Lambda }, t) \bigg \} }{\mathrm{e}^{K(t)} \theta ^{n} \mathrm{e}^{ B(t) \lambda } \, \mathrm{e}^{-\nu c} {\mathrm{e}^{\phi \Lambda } \psi ^{m}}} + + \\ &= \frac {\partial \tilde {f}}{\partial t} + \bigg ( K^{\prime }(t) + B^{\prime }(t) \lambda \bigg ) \tilde {f} + \delta \left ( a-\lambda \right ) \left ( \frac {\partial \tilde {f}}{\partial \lambda } + B(t) \tilde {f} \right ) + {\lambda \left ( \frac {\partial \tilde {f}}{\partial \Lambda } + \phi \tilde {f} \right )} \\ &\quad + \lambda \left [ \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \int _{0}^{\infty }\int _{0}^{\infty } \tilde {f}(\lambda +y,n+1,c+\xi ,{m},{\Lambda },t) \frac {\mathrm{e}^{ B(t) y}}{\hat {g}(\!-\!B(t))} \mathop {}\!\mathrm{d} G(y) \frac {\mathrm{e}^{-\nu \xi }}{\hat {j}(\nu )} \mathop {}\!\mathrm{d} J(\xi ) -\tilde {f} \right ] \\ &\quad +\rho \left [ \psi \hat {h}(\!-\!B(t)) \int _{0}^{\infty }\tilde {f}( \lambda +x,n,c,{m+1},{\Lambda },t) \frac {\mathrm{e}^{ B(t) x}}{\hat {h}(\!-\!B(t))} \mathop {}\!\mathrm{d} H(x) - \tilde {f} \right ]\!. \end{align*}

Define

\begin{equation*} \mathop {}\!\mathrm{d} \tilde {G}(y; \, t)\, :\!= \, \frac {\mathrm{e}^{ B(t) y}}{\hat {g}(\!-\!B(t))} \mathop {}\!\mathrm{d} G(y), \qquad \mathop {}\!\mathrm{d} {\tilde {H}}(x; \, t) \, :\!= \, \frac {\mathrm{e}^{ B(t) x}}{\hat {h}(\!-\!B(t))} \mathop {}\!\mathrm{d} {H}(x), \qquad \mathop {}\!\mathrm{d} \tilde {J}(\xi ) \, :\!= \, \frac {\mathrm{e}^{-\nu \xi }}{\hat {j}(\nu )} \mathop {}\!\mathrm{d} J(\xi ), \end{equation*}

then,

\begin{align*} &\hspace {-0.1cm} \tilde {\mathcal{A}} \tilde {f}( \lambda , n,c,{m}, {\Lambda }, t) \\ &= \frac {\partial \tilde {f}}{\partial t} + \bigg ( K^{\prime }(t) + B^{\prime }(t) \lambda \bigg ) \tilde {f} + \delta \left ( a-\lambda \right ) \left ( \frac {\partial \tilde {f}}{\partial \lambda } + B(t) \tilde {f} \right ) + {\lambda \left ( \frac {\partial \tilde {f}}{\partial \Lambda } + \phi \tilde {f} \right )} \\ &\quad + \lambda \left [ \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \int _{0}^{\infty }\int _{0}^{\infty } \tilde {f}(\lambda +y,n+1,c+\xi ,{m},{\Lambda },t) \mathop {}\!\mathrm{d} \tilde {G}(y; \, t) \mathop {}\!\mathrm{d} \tilde {J}(\xi ) -\tilde {f} \right ] \\ &\quad +\rho \left [ \psi \hat {h}(\!-\!B(t)) \int _{0}^{\infty }\tilde {f}( \lambda +x,n,c,{m+1},{\Lambda },t) \mathop {}\!\mathrm{d} \tilde {H}(x; \, t) - \tilde {f} \right ] \\ &= \bigg [ K^{\prime }(t) + a \delta B(t) + \bigg ( B^{\prime }(t) - \delta B(t) +\phi \bigg ) \lambda \bigg] \,\tilde {f} + \frac {\partial \tilde {f}}{\partial t} + \lambda \frac {\partial \tilde {f}}{\partial \Lambda } + \delta \left ( a-\lambda \right )\frac {\partial \tilde {f}}{\partial \lambda } \\ &\quad + \lambda \left [ \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \int _{0}^{\infty }\int _{0}^{\infty } \tilde {f}(\lambda +y,n+1,c+\xi ,{m},{\Lambda },t) \mathop {}\!\mathrm{d} \tilde {G}(y; \, t) \mathop {}\!\mathrm{d} \tilde {J}(\xi ) -\tilde {f} \right ] \\ &\quad +\rho \left [ \psi \hat {h}(\!-\!B(t)) \int _{0}^{\infty }\tilde {f}( \lambda +x,n,c,{m+1},{\Lambda },t) \mathop {}\!\mathrm{d} \tilde {H}(x; \, t) - \tilde {f} \right ] \\ &= \frac {\partial \tilde {f}}{\partial t} + \lambda \frac {\partial \tilde {f}}{\partial \Lambda } + \delta \left ( a-\lambda \right )\frac {\partial \tilde {f}}{\partial \lambda } \\ &\quad + \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \lambda \left [ \int _{0}^{\infty }\int _{0}^{\infty } \tilde {f}(\lambda +y,n+1,c+\xi ,{m},{\Lambda },t) \mathop {}\!\mathrm{d} \tilde {G}(y; \, t) \mathop {}\!\mathrm{d} \tilde {J}(\xi ) - \tilde {f} \right ] \\ &\quad +\psi \hat {h}(\!-\!B(t)) \,\rho \left [ \int _{0}^{\infty }\tilde {f}( \lambda +x,n,c,{m+1},{\Lambda },t) \mathop {}\!\mathrm{d} \tilde {H}(x; \, t) - \tilde {f} \right ]\!. \end{align*}

Define

\begin{equation*} \tilde {\rho }(t)\, :\!= \,\psi \hat {h}(\!-\!B(t))\, \rho , \end{equation*}

and consequently, Eq. (3.15) is obtained.

Theorem 3 yields the following:

1. (i) The intensity process $\lambda _t$ has changed to $\tilde {\lambda }_t \, :\!= \, \theta \hat {j}(\nu )\hat {g}(\!-\!B(t))\,\lambda _t$ , which is time-dependent;

2. (ii) The self-exciting jump size measure $\mathop {}\!\mathrm{d} G(y)$ has changed to $\mathop {}\!\mathrm{d} \tilde {G}(y; \, t)\, :\!= \, \frac {\mathrm{e}^{ B(t) y}}{\hat {g}(\!-\!B(t))} \mathop {}\!\mathrm{d} G(y)$ , which now depends on time;

3. (iii) The rate of external-exciting jump arrival $\rho$ has changed to $\tilde {\rho }(t)\, :\!= \,\psi \hat {h}(\!-\!B(t))\, \rho$ , which now depends on time;

4. (iv) The external-exciting jump size measure $\mathop {}\!\mathrm{d} H(x)$ has changed to $\mathop {}\!\mathrm{d} {\tilde {H}}(x; \, t) \, :\!= \, \frac {\mathrm{e}^{ B(t) x}}{\hat {h}(\!-\!B(t))} \mathop {}\!\mathrm{d} {H}(x)$ , which now depends on time;

5. (v) The claim/loss size measure $\mathop {}\!\mathrm{d} J\left ( \xi \right )$ has changed to $\mathop {}\!\mathrm{d} \tilde {J}(\xi ) \, :\!= \, \frac {\mathrm{e}^{-\nu \xi }}{\hat {j}(\nu )} \mathop {}\!\mathrm{d} J(\xi )$ .

The Esscher transform provides a transparent and actuarially consistent mechanism for risk-adjusted pricing of catastrophe reinsurance. Under the transformed probability measure based on the Esscher transformation, the arrival intensity $\lambda _t$ , jump size measures $\tilde {G}(\! \cdot; \, t)$ , $\tilde {H}(\! \cdot; \, t)$ , $\tilde {J}(\! \cdot \!)$ , and external excitation factor $\tilde {\rho }(t)$ are adjusted to reflect aversion to extreme losses and catastrophic risk, jointly determining an arbitrage-free premium consistent with actuarial principles. In particular, (i) scaling of the intensity process represents a frequency loading; (ii) tilted self-exciting jump sizes place greater weights on contagion-driven shocks; (iii) time-dependent adjustment of external-exciting arrivals loads for systemic shocks; (iv) transformed external-exciting jump sizes increase the likelihood of large external shocks; and (v) the Esscher tilting on claim sizes acts as a severity-based loading. Together, these adjustments allow reinsurers to incorporate risk aversion in a granular and interpretable manner, distinguishing between frequency, contagion, external shocks, and severity risks, while preserving a consistent arbitrage-free valuation framework.

The result in Theorem 3 can be rewritten by defining

\begin{equation*} \tilde {\lambda }_t \, :\!= \, \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \lambda _t, \qquad \tilde {\Lambda }_t \, :\!= \, \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \Lambda _t, \end{equation*}

for any time $t \in \mathcal{T}$ , which is presented in Corollary 1 below.

Corollary 1. The generator in Eq. (3.15) can be alternatively expressed by

(3.16) \begin{align} &\hspace {-0.5cm} \tilde {\mathcal{A}} \tilde {f}( \tilde {\lambda }, n,c,{m}, {\tilde {\Lambda }}, t) \notag \\ &= \frac {\partial \tilde {f}}{\partial t} + \tilde {\lambda } \frac {\partial \tilde {f}}{\partial \tilde {\Lambda } } + \delta \bigg ( \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \, a - \tilde {\lambda } \bigg )\frac {\partial \tilde {f}}{\partial \tilde {\lambda } } \notag \\ &\quad + \tilde {\lambda } \left [ \int _{0}^{\infty }\int _{0}^{\infty } \tilde {f}(\tilde {\lambda } +u,n+1,c+\xi ,{m},{\tilde {\Lambda }},t) \mathop {}\!\mathrm{d} \tilde {G}\left ( \frac {u}{\theta \hat {j}(\nu ) \hat {g}(\!-\!B(t))} ; t \right ) \mathop {}\!\mathrm{d} \tilde {J}(\xi ) - \tilde {f} \right ] \notag \\ &\quad + \tilde {\rho }(t) \left [ \int _{0}^{\infty }\tilde {f}( \tilde {\lambda } + v,n,c,{m+1},{\tilde {\Lambda }},t) \mathop {}\!\mathrm{d} \tilde {H}\left ( \frac {v}{\theta \hat {j}(\nu ) \hat {g}(\!-\!B(t))} ; t\right ) - \tilde {f} \right ]. \end{align}

Proof. Define

\begin{equation*} \tilde {\lambda } \, :\!= \, \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \lambda , \qquad \tilde {\Lambda } \, :\!= \, \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \Lambda , \end{equation*}

and we have

\begin{align*} &\hspace {-0.5cm}\tilde {\mathcal{A}} \tilde {f}( \tilde {\lambda }, n,c,{m}, {\tilde {\Lambda }}, t) \\ &= \frac {\partial \tilde {f}}{\partial t} + \tilde {\lambda } \frac {\partial \tilde {f}}{\partial \tilde {\Lambda } } + \delta \bigg ( \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \, a - \tilde {\lambda } \bigg )\frac {\partial \tilde {f}}{\partial \tilde {\lambda } } \\ &\quad + \tilde {\lambda } \left [ \int _{0}^{\infty }\int _{0}^{\infty } \tilde {f}(\tilde {\lambda } +\theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) y,n+1,c+\xi ,{m},{\tilde {\Lambda }},t) \mathop {}\!\mathrm{d} \tilde {G}(y; \, t) \mathop {}\!\mathrm{d} \tilde {J}(\xi ) - \tilde {f} \right ] \\ &\quad + \tilde {\rho }(t) \left [ \int _{0}^{\infty }\tilde {f}( \tilde {\lambda } + \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) x,n,c,{m+1},{\tilde {\Lambda }},t) \mathop {}\!\mathrm{d} \tilde {H}(x; \, t) - \tilde {f} \right ]\!. \end{align*}

Changing variables $u=\theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) y, v=\theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) x$ , we have Eq. (3.16).

Corollary 2. More explicitly, for

\begin{equation*} \mathop {}\!\mathrm{d} \tilde {G}(y; \, t) = \tilde {g}(y; \, t) \mathop {}\!\mathrm{d} y, \qquad \mathop {}\!\mathrm{d} {\tilde {H}}(x; \, t) = {\tilde {h}}(x; \, t) \mathop {}\!\mathrm{d} x, \end{equation*}

we have the transforms

\begin{align*} a \ &\rightarrow \ \theta \hat {j}(\nu ) \hat {g}(\!-\!{B(t)}) \, a; \\ \rho \ &\rightarrow \ \psi \hat {h}(\!-\!{B(t)})\, \rho ;\\ h(v) \ &\rightarrow \ \frac {\tilde {h}\left ( \frac {v}{\theta \hat {j}(\nu ) \hat {g}(\!-\!{B(t)})}; \, t \right )}{\theta \hat {j}(\nu ) \hat {g}(\!-\!{B(t)})} ; \\ g(u) \ &\rightarrow \ \frac { \tilde {g}\left ( \frac {u}{\theta \hat {j}(\nu ) \hat {g}(\!-\!{B(t)})}; \, t\right ) }{\theta \hat {j}(\nu ) \hat {g}(\!-\!{B(t)})} ; \\ j(\xi ) \ &\rightarrow \ \tilde {j}(\xi ) , \end{align*}

where

\begin{equation*} \tilde {g}(y; \, t) \, :\!= \, \frac {\mathrm{e}^{B(t) y}}{\hat {g}(\!-\!B(t))} g (y), \qquad \tilde {h}(x; \, t) \, :\!= \, \frac {\mathrm{e}^{B(t) x}}{\hat {h}(\!-\!B(t))} h(x), \qquad \tilde {j}(\xi ) \, :\!= \, \frac {\mathrm{e}^{-\nu \xi }}{\hat {j}(\nu )} j (\xi ). \end{equation*}

Proof. Based on Corollary 1, in particular for

\begin{equation*} \mathop {}\!\mathrm{d} \tilde {G}(y; \, t) = \tilde {g}(y; \, t) \mathop {}\!\mathrm{d} y, \qquad \mathop {}\!\mathrm{d} {\tilde {H}}(x; \, t) = {\tilde {h}}(x; \, t) \mathop {}\!\mathrm{d} x, \end{equation*}

we have

\begin{align*} &\hspace {-0.5cm}\tilde {\mathcal{A}} \tilde {f}( \tilde {\lambda }, n,c,{m}, {\tilde {\Lambda }}, t) \\ &= \frac {\partial \tilde {f}}{\partial t} + \tilde {\lambda } \frac {\partial \tilde {f}}{\partial \tilde {\Lambda } } + \delta \bigg ( \theta \hat {j}(\nu ) \hat {g}(\!-\!B(t)) \, a - \tilde {\lambda } \bigg )\frac {\partial \tilde {f}}{\partial \tilde {\lambda } } \\ &\quad + \tilde {\lambda } \left [ \int _{0}^{\infty }\int _{0}^{\infty } \tilde {f}(\tilde {\lambda } +u,n+1,c+\xi ,{m},{\tilde {\Lambda }},t) \mathop {}\!\mathrm{d} \tilde {J}(\xi ) \, \frac { \tilde {g}( \frac {u}{\theta \hat {j}(\nu ) \hat {g}(\!-\!B(t))}\, ;t) }{\theta \hat {j}(\nu ) \hat {g}(\!-\!B(t))} \mathop {}\!\mathrm{d} u - \tilde {f} \right ] \\ &\quad + \tilde {\rho }(t) \left [ \int _{0}^{\infty }\tilde {f}( \tilde {\lambda } + v,n,c,{m+1},{\tilde {\Lambda }},t) \frac {\tilde {h}( \frac {v}{\theta \hat {j}(\nu ) \hat {g}(\!-\!B(t))} ;t )}{\theta \hat {j}(\nu ) \hat {g}(\!-\!B(t))} \mathop {}\!\mathrm{d} v - \tilde {f} \right ]\!. \end{align*}

Then, we have the transforms.

Note that the parameters of a CDCP under the original measure $\mathbb{P}$ are all constants, whereas the parameters of a CDCP under the new measure $\tilde {\mathbb{P}}$ become time-varying (except for the intensity-decay rate $\delta$ ), i.e., a time-inhomogeneous CDCP.

In reality, a “hard” market in a (re)insurance industry occurs when the supply of coverage available is lower than its demand. This leads to a higher (re)insurance premium, reduced (or stopped) coverage, and less competition among (re)insurance providers. The Southern California wildfires of January 2025 further exacerbated this, as rising losses from catastrophic events, coupled with the increasing costs of reinsurance and construction, have prompted insurers like State Farm to limit or withdraw coverage in high-risk areas, intensifying the strain on the market.

There are several reasons why a hardening reinsurance market prevails. Firstly, there has been increasing frequency and severity of losses attributed to catastrophic events due to El Niño and La Niña. Secondly, a high inflation regime has been in force, which may be partly due to turbulence in macroeconomic conditions. Thirdly, there has been a growing number of emerging events such as cyber and pandemic events.

In a hardening reinsurance market, it may be reasonable for a reinsurer to calculate the premium of a catastrophe reinsurance contract in a more conservative and prudent way. To illustrate this situation in our current modeling setup, certain assumptions are imposed on $\theta ,{\psi }, \nu , B(t)$ such that there would be more external-exciting jump arrivals in a given period of time, a higher value of self-exciting intensity, higher values of their jump sizes, as well as a larger size of claim or loss. To take on these additional risks in the hardening market, a reinsurer requires higher compensation. Furthermore, it may be reasonable to postulate that the reinsurer aims to maximize their shareholders’ wealth via earning profits. To this end, the reinsurer may charge a higher premium than the “break-even” premium. The latter is given by the expected claims evaluated under the original (real-world) probability measure. In the hardening market, $\theta ,{\psi }, \nu , B(t)$ may be thought of as security loading factors by which positive gross premium should be finally charged. In fact, based on Corollary 2, if the gross premium is always positive, we shall have the conditions

\begin{equation*} \theta ,{\psi }\geq 1, \qquad \nu \lt 0, \qquad {\mbox{$B(t)\gt 0$ for some $t \gt 0$}}. \end{equation*}

From an economic perspective, the Esscher parameters $(\theta ,\psi ,\nu )$ provide a parsimonious representation of reinsurer risk preferences under hard market conditions. In particular, the transformed intensity process

\begin{equation*} \tilde {\lambda }_t \, :\!= \, \theta \hat {j}(\nu )\hat {g}(\!-\!B(t))\,\lambda _t \end{equation*}

introduces a time-dependent loading on claim arrivals, reflecting heightened sensitivity to both claim frequency and loss severity during periods of elevated risk. The parameter $\theta$ acts as a frequency loading on the baseline arrival intensity, while the factor $\hat {j}(\nu )\hat {g}(\!-\!B(t))$ captures the interaction between severity aversion and contagion-driven dynamics. Similarly, the parameter $\psi$ scales the arrival rate of externally driven shocks, representing increased concern about systemic events. The Esscher parameter $\nu \lt 0$ induces a severity-based loading by tilting the claim size distribution toward larger losses, thereby increasing the contribution of catastrophic events to the premium.

In a hard reinsurance market, these choices of the parameters are consistent with observed outcomes such as higher premiums and tighter underwriting, resulting in a conservative valuation. On the other hand, reinsurers may also limit coverage in high-risk regions by transferring part of the risk whenever alternative capacity is available. In this sense, the Esscher parameters $(\theta ,\psi ,\nu )$ may serve as reduced-form measures incorporating market risk aversion and capital scarcity into catastrophe reinsurance pricing.

3.3 Special case: exponential/gamma distributions for jump sizes

Before closing this section, we show the explicit transformation for the computation of arbitrage-free catastrophe stop-loss reinsurance premiums in Section 4.

Corollary 3 (Special case: exponential/gamma distributions for jump sizes). In particular, we assume that external-exciting and self-exciting jump sizes in the intensity process follow exponential distributions $H \sim \mathsf{Exp}(\alpha ), G \sim \mathsf{Exp}(\beta )$ , respectively, i.e.,

\begin{equation*} h(x) =\alpha \mathrm{e}^{-\alpha x}, \qquad g(y) =\beta \mathrm{e}^{-\beta y},\ \qquad \alpha ,\beta \gt 0, \end{equation*}

and claim/loss sizes follow the gamma distribution $J \sim \mathsf{Gamma}(\gamma ,\eta )$ with the rate parameter $\gamma$ and shape parameter $\eta$ , i.e.,

\begin{equation*} j\left ( \xi \right ) = \frac {\gamma ^{\eta }\xi ^{\eta -1}\mathrm{e}^{-\gamma \xi }}{\left ( \eta -1\right ) !}, \qquad \gamma \gt 0, \eta \geq 1. \end{equation*}

Then, we have the transforms

\begin{align*} a \ &\rightarrow \ \theta \Bigl (1 + \frac {\nu }{\gamma }\Bigr )^{-\eta } \frac {\beta }{\beta - B(t)} a; \\ \rho \ &\rightarrow \ \psi \frac {\alpha }{\alpha - B(t) } \, \rho ;\\ J \sim \mathsf{Gamma}(\gamma ,\eta )\ &\rightarrow \ \mathsf{Gamma}(\gamma +\nu , \eta ); \\ H \sim \mathsf{Exp}(\alpha ) \ &\rightarrow \ \mathsf{Exp}(\lambda _h); \\ G \sim \mathsf{Exp}(\beta ) \ &\rightarrow \ \mathsf{Exp}(\lambda _g); \end{align*}

where

\begin{equation*} \lambda _h \, :\!= \, \frac {\alpha - B(t)}{\theta \left (1 + \frac {\nu }{\gamma }\right )^{-\eta } \frac {\beta }{\beta -B(t)}}, \qquad \lambda _g \, :\!= \, \frac {\beta - B(t)}{\theta \left (1 + \frac {\nu }{\gamma }\right )^{-\eta } \frac {\beta }{\beta -B(t)}}, \qquad \nu \in (\!-{\gamma} ,0). \end{equation*}

Proof. In particular for exponential distributions, $H \sim \mathsf{Exp}(\alpha ), G \sim \mathsf{Exp}(\beta )$ , i.e.,

\begin{equation*} h(x) =\alpha \mathrm{e}^{-\alpha x}, \qquad g(y) =\beta \mathrm{e}^{-\beta y}, \qquad \alpha ,\beta \gt 0, \end{equation*}

we have

\begin{equation*} \hat {g}(\!-\!B(t)) = \frac {\beta }{\beta -B(t)}, \qquad \hat {h}(\!-\!B(t)) = \frac {\alpha }{\alpha -B(t)}, \end{equation*}

\begin{equation*} \tilde {g}(y) = (\beta -B(t)) \mathrm{e}^{-(\beta -B(t)) y}, \qquad \tilde {h}(x) = (\alpha -B(t)) \mathrm{e}^{-(\alpha -B(t)) x}. \end{equation*}

Similarly, for gamma distribution $J \sim \mathsf{Gamma}(\gamma ,\eta )$ , we have the transform to $\mathsf{Gamma}({\gamma +\nu ,\eta })$ , since

\begin{align*} j\left ( \xi \right ) &= \frac {\gamma ^{\eta }\xi ^{\eta -1}\mathrm{e}^{-\gamma \xi }}{\left ( \eta -1\right ) !}, \qquad \gamma \gt 0, \eta \geq 1, \qquad \hat {j}(\nu ) = \left (\frac {\gamma }{\gamma + \nu } \right )^{\eta }, \\ \tilde {j}(\xi ) &= \frac {\mathrm{e}^{-\nu \xi }}{\hat {j}(\nu )} j (\xi ) \,\,=\,\, \frac {\mathrm{e}^{-\nu \xi }}{\hat {j}(\nu )} \frac {\gamma ^{\eta }\xi ^{\eta -1}\mathrm{e}^{-\gamma \xi }}{\left ( \eta -1\right ) !} \,\,=\,\, \frac { (\gamma + \nu )^{\eta } \xi ^{\eta -1} \mathrm{e}^{-(\gamma +\nu ) \xi } }{\left ( \eta -1\right ) !}. \end{align*}

Finally, we obtain the results by applying the transforms in Corollary 2.

In fact, the original distributional structure of the underlying risk process via this change of measure is preserved. We may also need the stationary condition for a CDCP under the new measure, i.e.,

\begin{equation*} \delta \,\,\gt \,\, \frac {1}{\lambda _g} \,\,=\,\, \theta \left ( \frac {\gamma }{\gamma +\nu }\right )^{\eta } \frac {\beta }{(\beta - B(t))^2}, \end{equation*}

where $\delta$ and $\frac {1}{\lambda _g}$ are the decay rate and the mean of exponentially distributed self-excited jump sizes under the new measure for a CDCP, respectively, according to Corollary 3. So, for implementations, we have to set the decay rate $\delta$ relatively large together with the condition $\delta \gt \theta \, \hat {j}(\nu ) \mu _G =\theta \, \hat {j}(\nu ) \frac {1}{\beta }$ required in Theorem 2 for the existence and uniqueness of the solutions of the ODEs to hold. This constraint on $\delta$ is required in our model. In other words, the model may not be applicable if this constraint on $\delta$ does not hold. This may represent a limitation of the model.

Note that, for the ODE of $B(t)$ , we have

\begin{equation*} f_1 (B) = \delta B \left ( \frac {B - \left ( \beta - \frac {1}{\delta } \theta \, \hat {j}(\nu ) \right ) }{B-\beta } \right)\!, \end{equation*}

where $B^+\, :\!= \,\beta - \frac {1}{\delta } \theta \, \hat {j}(\nu ) \gt 0$ or $\delta \beta \gt \theta \, \hat {j}(\nu )$ . Define

(3.17) \begin{equation} \mathcal{G}(B) \, :\!= \, \int ^{B}_b \frac {\mathop {}\!\mathrm{d} u}{f_1(u)}, \qquad B\in \left (b, B^+ \right )\!. \end{equation}

Note that $\mathcal{G}(B)$ is a strictly increasing function and

\begin{equation*} \lim _{B \downarrow b}\mathcal{G}(B) = 0, \qquad \lim _{B \uparrow B^+ }\mathcal{G}(B) = \infty . \end{equation*}

We have its inverse as the solution to $B(t)$ as

\begin{equation*} B(t) \ = \ \mathcal{G}^{-1}(t), \qquad t \geq 0. \end{equation*}

Finally, we have the following three types of parameter settings:

1. 1. If $\alpha \geq B^+$ , then $B(t) \in (0, \alpha )$ for any time $t \geq 0$ ;

2. 2. If $\alpha \in (b,B^+)$ , then $B(t) \in (0, \alpha )$ for $t \in [0, t^* )$ , where $t^* \, :\!= \,\mathcal{G}(\alpha ) \gt 0$ from Eq. (3.17);

3. 3. If $\alpha \in (0,b]$ , then $B(t) \in (0, \alpha )$ ; there is no solution for $B(t) \in (0, \alpha ),t\geq 0$ ;

where $b\gt 0,{B^+=\beta - \frac {1}{\delta } \theta \, {\hat {j}}(\nu ) \gt 0}$ . Note that the function $\mathcal{G}(\! \cdot \!)$ in Eq. (3.17) shall be derived fully analytically. However, its inverse $\mathcal{G}^{-1}(\! \cdot \!)$ has to be solved numerically.

The first type of parameter setting (i.e., $\alpha \geq B^+$ ) is the most ideal. For numerical examples, it is suggested that $b$ is chosen to be small and close to zero, and $\alpha$ is relatively large.

In particular, if $\beta \rightarrow \infty$ , then all self-exciting jump sizes approach to zero, which degenerates to the pure shot-noise intensity model of Dassios and Jang (Reference Dassios and Jang2003).

Corollary 4 (Expectations for exponential/gamma distributions). When $H \sim \mathsf{Exp}(\alpha )$ , $G \sim \mathsf{Exp}(\beta )$ , and $J \sim \mathsf{Gamma}(\gamma ,\eta )$ , the expectations are as follows:

\begin{equation*} \begin{aligned} \tilde {\mathbb{E}}[ \lambda _t \mid \lambda _0]\ &= \ \lambda _0 \mathrm{e}^{-\int _0^t I(s) \mathop {}\!\mathrm{d} s} + \mathrm{e}^{-\int _0^t I(s) \mathop {}\!\mathrm{d} s} \theta \left ( \frac {\gamma }{\gamma +\nu }\right ) ^{\eta } \\ &\quad \times \int _0^t \mathrm{e}^{\int _0^s I(u) \mathop {}\!\mathrm{d} u} \left ( \frac {\beta }{\beta -B(s)}\right ) \left \{ \psi \alpha \rho \left ( \frac {1}{\alpha -B(s)}\right )^{2} + a \delta \right \} \mathop {}\!\mathrm{d} s. \end{aligned} \end{equation*}

\begin{equation*} \tilde {\mathbb{E}}[ N_t \mid \lambda _0 ] = \int _{0}^{t} \tilde {\mathbb{E}}[ \lambda _{s} \mid \lambda _{0} ] \mathop {}\!\mathrm{d} s \end{equation*}

\begin{equation*} \tilde {\mathbb{E}}[ C_t \mid \lambda _0 ] = \mu _{\widetilde {J}} \int _0^t \tilde {\mathbb{E}}[ \lambda _s \mid \lambda _0 ] \mathop {}\!\mathrm{d} s \end{equation*}

where $ I(s) = \delta - \left [\theta \left ( \frac {\gamma }{\gamma +\nu }\right )^{\eta } \left ( \frac {\beta }{\beta -B(s) }\right ) \right ]/[\beta -B(s)]$ and $\mu _{\widetilde {J}} = \eta /(\gamma + \nu )$ .

For illustrative purposes, we have used a Gamma distribution for claim sizes, which yields a closed-form expression under the Esscher transform, $\tilde {J} \sim \mathsf{Gamma}(\gamma +\nu ,\eta )$ . The Gamma family is a standard choice in actuarial modeling of claim severities (Bühlmann, Reference Bühlmann1970, Page 7) and offers analytical tractability that facilitates clear interpretation of the pricing mechanism. While heavier-tailed alternatives, such as the Pareto distribution,

\begin{equation*} j(\xi ) = \frac {\Gamma (\omega + k)\, \zeta ^\omega \, \xi ^{k-1}} {\Gamma (\omega )\, \Gamma (k)\, (\zeta + \xi )^{\omega + k}}, \qquad \omega \gt 0,\ \zeta \gt 0,\ k\gt 0, \end{equation*}

are commonly employed to model large insurance losses (Albrecher et al., Reference Albrecher, Beirlant and Teugels2017, Chapters 3–4), their form is not preserved under the Esscher transform and instead acquires an exponential tilt. Nevertheless, the corresponding Laplace transform $\hat {j}(\nu )$ exists under suitable parameter conditions and can be evaluated numerically. Heavier-tailed claim size distributions typically lead to higher premium levels, reflecting increased tail risk. In this sense, the Gamma specification, including the exponential distribution as a special case, serves as a benchmark for assessing the impact of tail heaviness on catastrophe reinsurance pricing. The results presented here therefore illustrate the qualitative behavior of the model, while extensions to Pareto or other heavy-tailed distributions may be explored in future work via numerical sensitivity analysis.

4.1 Pricing of a stop-loss reinsurance contract

The total loss excess over $L$ , which is a retention limit, up to time $t$ is $(C_{t}-L)^+$ . Then the stop-loss reinsurance gross premium at time 0 (i.e., under $\tilde {\mathbb{P}}$ ) is

(4.1) \begin{equation} \tilde {\mathbb{E}}[(C_t - L)^+]. \end{equation}

Its Monte Carlo estimate will be denoted by $\widehat {\tilde {\mathbb{E}}}[(C_t - L)^+]$ , and the corresponding estimate under $\mathbb{P}$ will be denoted by $\widehat {\mathbb{E}}[(C_t - L)^+]$ . For this section, we consider

\begin{equation*} \alpha = 2, \beta = 1, \eta = 3, \gamma = 0.4 , \end{equation*}

\begin{equation*} \delta = 3, \rho = 4, a = 1, \lambda _0 = 1, t = 1 , \end{equation*}

\begin{equation*} \nu = -0.05, \theta = 1.25, \psi = 1.25, \phi = - (\theta \hat {j}(\nu ) - 1). \end{equation*}

The parameter values adopted in the numerical examples are chosen for illustrative purposes. They are used to demonstrate the qualitative behavior of the model, as well as the impacts of contagion and risk loading on reinsurance premiums, rather than to provide an empirical calibration to a specific portfolio. Similar parameter ranges are commonly adopted in the dynamic contagion and self-exciting point process literature to study clustering effects and catastrophic risk (Dassios and Zhao (Reference Dassios and Zhao2011); Jang and Oh (Reference Jang and Oh2021, Reference Jang and Oh2025)).

In practice, model parameters may be estimated from historical catastrophe or large-loss data using standard inference techniques for point processes with latent or stochastic intensities, including likelihood-based methods, moment-based estimators, and filtering approaches. General statistical frameworks for point process inference are discussed in Karr (Reference Karr1991) and Daley and Vere-Jones (Reference Daley and Vere-Jones2008), while modern likelihood-based methodologies for self-exciting and Hawkes-type processes are presented in Laub et al. (Reference Laub, Lee and Taimre2022). Related filtering approaches in insurance risk models are studied in Dassios and Jang (Reference Dassios and Jang2005) for shot-noise Cox processes, and similar ideas may be adapted to the dynamic contagion setting considered here.

While maximum likelihood estimators for point process parameters and some of their statistical properties have been studied, the estimation of shot-noise Cox processes, and in particular the analysis of their asymptotic properties, has received comparatively less attention. These challenges are further compounded when time-varying parameters are present, suggesting promising opportunities for future research, particularly in the context of dynamic contagion models for insurance risk.

For each combination of parameters, $10^5$ crude Monte Carlo (CMC) samples of $C_t$ under $\tilde {\mathbb{P}}$ have been used to estimate Eq. (4.1) in Table 1. The corresponding values under $\mathbb{P}$ are also provided in the same table.

Table 1.

The stop-loss CMC estimates $\widehat {\mathbb{E}}[(C_t - L)^+]$ under the original measure and $\widehat {\tilde {\mathbb{E}}}[(C_t - L)^+]$ under the Esscher measure for different $L$ retention levels

Figure 1

A collection of 25 sample paths of $C_t$ and the corresponding $\lambda _t$ for the compound dynamic contagion process under the original measure $\mathbb{P}$ (left column) and the tilted measure $\tilde {\mathbb{P}}$ (right column) given the constants outlined above. It indicates that paths under $\tilde {\mathbb{P}}$ exhibit higher intensities and larger claims, reflecting risk loading and increased emphasis on extreme losses.

Figure 1 Long description

The image contains four line graphs arranged in a two-by-two grid. The left column represents the compound dynamic contagion process under the original measure, while the right column represents the process under the tilted measure. The top row shows the cumulative claims over time, and the bottom row displays the intensity of claim arrivals. Each graph contains multiple colored lines representing different sample paths. The graphs on the right indicate higher intensities and larger claims compared to those on the left, reflecting risk loading and increased emphasis on extreme losses. All values are approximated.

Table 1 shows that arbitrage-free catastrophe stop-loss reinsurance (i.e., gross) premiums are significantly higher than the corresponding net premiums (see Figure 1). Depending on the choice of parameters, the gross premiums could be either lowered or elevated. It is not the purpose of this paper to determine which premium is the most appropriate; rather, the insurance companies’ risk preferences dictate which equivalent martingale probability measure should be adopted.

The simplest way to obtain a gross premium is through the use of the parameter $\theta$ , a well-known concept in insurance where a security loading is added to the expected value under the original probability measure $\mathbb{P}$ . An appealing feature of the Esscher transform is that it guarantees the existence of at least one equivalent martingale probability measure in incomplete markets.

4.2 Sensitivity analysis for the Esscher parameters

To examine the sensitivity analyses for the Esscher parameters, we focus on the mean, $\widehat {\tilde {\mathbb{E}}}[C_t]$ , and the stop-loss retention $L = 25$ for the above combination of parameters. Hence, we have:

\begin{equation*} \widehat {\tilde {\mathbb{E}}}[C_t] = 38.152252 \quad \text{ and } \quad \widehat {\tilde {\mathbb{E}}}[(C_t - 25)^+] = 19.153988, \end{equation*}

where $\widehat {\tilde {\mathbb{E}}}[C_t]$ itself can be used for arbitrage-free catastrophe insurance premiums.

The following tables show the way in which these values change when the Esscher parameters are adjusted individually, with all other parameters held constant (except $\phi$ , which is updated so that it always satisfies $\phi = - (\theta \hat {j}(\nu ) - 1)$ ).

4.2.1 Changing $\theta$

Table 2.

The expected loss $\tilde {\mathbb{E}}[C_t]$ , the CMC loss $\widehat {\tilde {\mathbb{E}}}[C_t]$ , and the stop-loss $\widehat {\tilde {\mathbb{E}}}[(C_t - 25)^+]$ estimates (with 95% confidence intervals) for different values of the tilting parameter $\theta$

4.2.2 Changing $\psi$

Table 3.

The expected loss $\tilde {\mathbb{E}}[C_t]$ , the CMC loss $\widehat {\tilde {\mathbb{E}}}[C_t]$ , and stop-loss $\widehat {\tilde {\mathbb{E}}}[(C_t - 25)^+]$ estimates (with 95% confidence intervals) for different values of the tilting parameter $\psi$

4.2.3 Changing $\nu$

Table 4.

The expected loss $\tilde {\mathbb{E}}[C_t]$ , the CMC loss $\widehat {\tilde {\mathbb{E}}}[C_t]$ , and stop-loss $\widehat {\tilde {\mathbb{E}}}[(C_t - 25)^+]$ estimates (with 95% confidence intervals) for different values of the tilting parameter $\nu$

Comparing the numerical values in Table 2 with those in Table 3, the increases in gross insurance and reinsurance premium estimates due to changes in $\theta$ are greater than those resulting from changes in $\psi$ . This is because $\hat {j}(\nu )$ is involved in the self-exciting intensity function $\lambda _{t}$ . Table 4 illustrates the increase in gross insurance and reinsurance premium estimates resulting from changes in $\nu$ , which is due to both a higher self-exciting intensity and larger claim/loss sizes.

4.3 Sensitivity analysis for the intensity parameters

We next examine sensitivity with respect to the intensity parameters governing the dynamics of the claim arrival process. Using the same setup (i.e., the same baseline parameter values) as in the previous subsection, we focus on the mean $\widehat {\tilde {\mathbb{E}}}[C_t]$ and the stop-loss premium with retention $L = 25$ .

The following tables illustrate how these quantities change when key intensity parameters are varied individually, with all other parameters held constant (except $\phi$ , which is updated to satisfy $\phi = -(\theta \hat {j}(\nu ) - 1)$ ).

4.3.1 Changing $\delta$

Table 5.

The expected loss $\tilde {\mathbb{E}}[C_t]$ , the CMC loss $\widehat {\tilde {\mathbb{E}}}[C_t]$ , and stop-loss $\widehat {\tilde {\mathbb{E}}}[(C_t - 25)^+]$ estimates (with 95% confidence intervals) for different values of the tilting parameter $\delta$

4.3.2 Changing $\alpha$

Table 6.

The expected loss $\tilde {\mathbb{E}}[C_t]$ , the CMC loss $\widehat {\tilde {\mathbb{E}}}[C_t]$ , and stop-loss $\widehat {\tilde {\mathbb{E}}}[(C_t - 25)^+]$ estimates (with 95% confidence intervals) for different values of the tilting parameter $\alpha$

4.3.3 Changing $\beta$

Table 7.

The expected loss $\tilde {\mathbb{E}}[C_t]$ , the CMC loss $\widehat {\tilde {\mathbb{E}}}[C_t]$ , and stop-loss $\widehat {\tilde {\mathbb{E}}}[(C_t - 25)^+]$ estimates (with 95% confidence intervals) for different values of the tilting parameter $\beta$

Table 5 shows the complicated relationship that $\delta$ has on the expected losses. Changing $\delta$ changes the solution $B(t)$ , and that changes nearly all of the Esscher-transformed parameters (except for claim sizes), making the impacts non-monotonic. Tables 6 and 7 present the reduction in them as the external-exciting and self-exciting jump size parameters $\alpha$ and $\beta$ increase. An increase in these jump size parameters results in a decrease in the exponential average.

4.4 Comparison with generalized Hawkes and shot-noise Cox processes

If there are no externally excited jumps in Eq. (2.2) (i.e., $\rho =0$ ) $C_{t}$ becomes a generalized compound Hawkes process. If $a=0$ and there are no self-excited jumps in Eq. (2.2), $C_{t}$ becomes a compound Cox process with shot-noise Poisson intensity.

The following two tables show the corresponding stop-loss CMC estimates in Table 1 (DCP case) for the generalized Hawkes case and the Cox case, respectively.

4.4.1 Generalized Hawkes process

Table 8.

The stop-loss CMC estimates $\widehat {\mathbb{E}}[(C_t - L)^+]$ under the original measure and $\widehat {\tilde {\mathbb{E}}}[(C_t - L)^+]$ under the Esscher measure for different $L$ retention levels. The point process driving $C_t$ is a generalized Hawkes process

If we simulate using the same parameters as above, except setting $\rho = 0$ , then the underlying point process becomes a generalized Hawkes process.

4.4.2 Shot-noise Cox process

Table 9.

The stop-loss CMC estimates $\widehat {\mathbb{E}}[(C_t - L)^+]$ under the original measure and $\widehat {\tilde {\mathbb{E}}}[(C_t - L)^+]$ under the Esscher measure for different $L$ retention levels. The point process driving $C_t$ is a shot-noise cox process

If we simulate using the same parameters as above, except setting $a = 0$ and $\beta = \infty$ (i.e., $Y_j \equiv 0$ for all $j$ ), then the underlying point process becomes a shot-noise Cox process.

Compared to Tables 8 and 9, Table 1 shows that gross insurance and reinsurance premium estimates significantly increase when changing $C_{t}$ from the Cox process with shot-noise Poisson intensity to a generalized Hawkes process due to self-excited jumps and to the dynamic contagion process due to both externally excited jumps and self-excited jumps.