Robust non-zero-sum stochastic differential game of two insurers with common shock and CDS transaction

Abstract

This paper considers the non-zero-sum stochastic differential game problem between two ambiguity-averse insurers (AAIs) with common shock. Each AAI’s surplus process consists of a proportional reinsurance protection and an investment in a money account, a stock and a credit default swap (CDS) with the objective of maximizing the expected utility of her relative terminal surplus with respect to that of her competitors. We consider default contagion risk of CDSs through a Markovian model with interacting default intensities. It is worthwhile to consider the uncertainty of the model on both the insurer herself and her competitors. In our model, we describe the surplus processes of two insurers by two jump-diffusion models with a common shock. Under jump-diffusion models, the robust Nash equilibrium strategies and the value functions for the all-default, one-default and all-alive case are derived under a worst-case scenario, respectively. Finally, through some numerical examples, we found some interesting results about the effects of some model parameters on the robust Nash equilibrium strategies, such as, the common shocks and the individual claims have the opposite effect on reinsurance investment.

Appendices

Appendix: Proofs of theorems, lemmas, and propositions

Proof of Proposition 2.3

Proof

According to the Eq. (2.3) and Itô’s formula, we have

$$\begin{aligned} dC_{i}(t)= & {} (1-H_{i}(t-))d\Phi _{i}(t,{\textbf {H}}(t))-\Phi _{i}(t,{\textbf {H}}(t-))dH_{i}(t)+\Delta (1-H_{i}(t))\Delta \Phi _{i}(t,{\textbf {H}}(t))\\= & {} (1-H_{i}(t-))d\Phi _{i}(t,{\textbf {H}}(t))-\Phi _{i}(t,{\textbf {H}}(t-))dH_{i}(t)-\Delta \Phi _{i}(t,{\textbf {H}}(t))dH_{i}(t)\\= & {} (1-H_{i}(t-))d\Phi _{i}(t,{\textbf {H}}(t))-\Phi _{i}(t,{\textbf {H}}(t-))dH_{i}(t)\\{} & {} -\,[\Phi _{i}(t,{\textbf {H}}^{i}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dH_{i}(t), \end{aligned}$$

where the equality $\Delta \Phi _{i}(t,{\textbf {H}}(t))dH_{i}(t)=[\Phi _{i}(t,{\textbf {H}}^{i}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dH_{i}(t),$ which follows from the fact that our default model excludes the occurrence of simultaneous defaults. Hence, if the ith CDS defaults, any other name $m\ne i$ would not, and $\Delta \Phi _{i}(t,{\textbf {H}}(t))=\Phi _{i}(t,{\textbf {H}}^{i}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))$. If $m\ne i$ CDS defaults, then $dH_{i}(t)=0$ and hence the above equality still holds.

From Eq. (2.6) of Lemma 2.2, we have

$$\begin{aligned} (1-H_{i}(t-))d\Phi _{i}(t,{\textbf {H}}(t))= & {} (1-H_{i}(t))\bigg [r(1-H_{i}(t))\Phi _{i}(t,{\textbf {H}}(t))-r\nu _{i}H_{i}(t)\Phi _{i}(t,{\textbf {H}}^{(1)}(t))\\{} & {} +\,\nu _{i}(1-H_{i}(t))\bigg ]dt\\{} & {} +\,(1-H_{i}(t-))\bigg \{[\Phi _{i}(t,{\textbf {H}}^{i}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dM_{i}(t)\\{} & {} +\,[\Phi _{i}(t,{\textbf {H}}^{m}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dM_{m}(t)\bigg \}\\= & {} \bigg [r(1-H_{i}(t))\Phi _{i}(t,{\textbf {H}}(t))+\nu _{i}(1-H_{i}(t))\bigg ]dt\\{} & {} +\,(1-H_{i}(t-))\bigg \{[\Phi _{i}(t,{\textbf {H}}^{i}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dM_{i}(t)\\{} & {} +\,[\Phi _{i}(t,{\textbf {H}}^{m}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dM_{m}(t)\bigg \}. \end{aligned}$$

Then, we have

$$\begin{aligned} dC_{i}(t)= & {} (1-H_{i}(t-))d\Phi _{i}(t,{\textbf {H}}(t))-\Phi _{i}(t,{\textbf {H}}(t-))dH_{i}(t)\\{} & {} -\,[\Phi _{i}(t,{\textbf {H}}^{i}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dH_{i}(t)\\= & {} (1-H_{i}(t))(r\Phi _{i}(t,{\textbf {H}}(t))+\nu _{i}dt-\Phi _{i}(t,{\textbf {H}}(t))dH_{i}(t)\\{} & {} +\,(1-H_{i}(t-))\bigg \{[\Phi _{i}(t,{\textbf {H}}^{i}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dM_{i}(t)\\{} & {} +\,[\Phi _{i}(t,{\textbf {H}}^{m}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dM_{m}(t)\bigg \}-[\Phi _{i}(t,{\textbf {H}}^{i}(t-))\\{} & {} -\,\Phi _{i}(t,{\textbf {H}}(t-))]dH_{i}(t), \end{aligned}$$

Using $\Phi _{i}(t,{\textbf {z}}^{i})=L_{i}$, we obtain

$$\begin{aligned} dC_{i}(t)= & {} (1-H_{i}(t))[r\Phi _{i}(t,{\textbf {H}}(t))+\upsilon _{i}-h^{Q}_{i}({\textbf {H}}(t))L_{i}]dt-\Phi _{i}(t,{\textbf {H}}(t-))dM_{i}(t)\\{} & {} +\,(1-H_{i}(t-))[\Phi _{i}(t,{\textbf {H}}^{m}(t-))-\Phi _{i}(t,{\textbf {H}}(t-))]dM_{m}(t), \end{aligned}$$

with $C_{i}(0)=(1-H_{i}(0))\Phi _{i}(0,{\textbf {H}}(0))\in [-\upsilon _{i},L_{i}].$ $\square$

Proof of Lemma 3.1

Proof

It is that

$$\begin{aligned}&\frac{\partial W_{i}(t, q_{i}(t))}{\partial q_{i}(t)}\\&\quad =\alpha _{i} e^{r(T-t)}\bigg \{\big [\lambda _{i}\phi _{i3}(t,q_{i}(t))+\lambda \phi _{i5}(t,q_{i}(t))\big ]\cdot M'_{Y_{i}}(\alpha _{i} q_{i}(t)e^{r(T-t)})\\&\qquad -(1+\eta _{i})(\lambda _{i}+\lambda )\mu _{i}\bigg \}\\&\quad =\alpha _{i} e^{r(T-t)}\bigg \{\big [\lambda _{i}\phi _{i3}(t,q_{i}(t))+\lambda \phi _{i5}(t,q_{i}(t))\big ]E[Y_{i}e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}]\\&\qquad -(1+\eta _{i})(\lambda _{i}+\lambda )\mu _{i}\bigg \}, \end{aligned}$$

and

$$\begin{aligned}{} & {} \frac{\partial ^{2}W_{i}(t, q_{i}(t))}{\partial q_{i}^{2}(t)}\\{} & {} \quad =\alpha _{i} e^{r(T-t)}\lambda _{i}\bigg \{\beta _{i3}e^{r(T-t)}\exp \left\{ \frac{\beta _{i3}}{\alpha _{i}}(E[e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}]-1)\right\} E[Y_{i}e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}]^{2}\\{} & {} \qquad +\,\alpha _{i} e^{r(T-t)}\exp \left\{ \frac{\beta _{i3}}{\alpha _{i}}(E[e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}]-1)\right\} E[Y_{i}^{2}e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}] \bigg \}\\{} & {} \qquad +\,\alpha _{i} e^{r(T-t)}\lambda \bigg \{\beta _{i5}e^{r(T-t)}\exp \left\{ \frac{\beta _{i5}}{\alpha _{i}}(E[e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}]-1)\right\} E[Y_{i}e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}]^{2}\\{} & {} \qquad +\,\alpha _{i} e^{r(T-t)}\exp \left\{ \frac{\beta _{i5}}{\alpha _{i}}(E[e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}]-1)\right\} E[Y_{i}^{2}e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}] \bigg \}>0, \end{aligned}$$

then, we have $\frac{\partial W_{i}(t, q_{i}(t))}{\partial q_{i}(t)}$ is increasing in $q_{i}(t)$. On the other hand, $\lim \limits _{q_{i}(t)\rightarrow \infty }\frac{\partial W_{i}(t, q_{i}(t))}{\partial q_{i}(t)}|_{q_{i}(t)=\infty }>0$ and $\lim \limits _{q_{i}(t)\rightarrow 0}\frac{\partial W_{i}(t, q_{i}(t))}{\partial q_{i}(t)}|_{q_{i}(t)=0}=-(\lambda _{i}+\lambda )\alpha _{i} e^{r(T-t)}\eta _{i}\mu _{i}<0$. According the continuously of $q_{i}(t)\in [0,\infty ]$ over time, there exists a solution $q_{i}^{*}(t)\in [0,T]$ such that $\frac{\partial W_{i}(t, q_{i}(t))}{\partial q_{i}(t)}|_{q_{i}(t)=q_{i}^{*}(t)}=0$, that is $W_{i}(t, q_{i}(t))$ has a minimizer $q_{i}^{*}(t)$. $\square$

Proof of Lemma 3.4.

Proof

Since $\tilde{W}'(\phi _{i4})=\frac{\alpha _{i}}{\beta _{i4}}(\ln \phi _{i4}+1)+1$, it is easy to verify $\tilde{W}'(\phi _{i4})$ is a decreasing function on $(0, e^{-\frac{\alpha _{i}+\beta _{i4}}{\alpha _{i}}})$ and increases on $(e^{-\frac{\alpha _{i}+\beta _{i4}}{\alpha _{i}}}, +\infty )$. Moreover, $\lim \limits _{\phi _{i4}\rightarrow 0^{+}}\tilde{W}(\phi _{i4})=-\frac{1}{1+u_{i}}<0$, $\lim \limits _{\phi _{i4}\rightarrow \infty }\tilde{W}(\phi _{i4})=\infty$. Therefore the function $\tilde{W}(\phi _{i4})=0$ admits a unique positive root $\phi _{i4}>0$. $\square$

Proof of Lemma 3.6.

Proof

Finding the first and second order partial derivatives of $\hat{p}_{i}$ with respect to $\pi _{i2}(t)$ as follows:

$$\begin{aligned}{} & {} \frac{\partial \hat{p}(\pi _{i2})}{\partial \pi _{i2}}=\alpha _{i} e^{r(T-t)}\frac{g_{i2}(t)}{g_{i3}(t)}\bigg [\phi ^{*}_{i4}(t)h_{i}^{P}[\Phi _{i}(t,{\textbf {0}})-L_{i}]\exp \{\alpha _{i}[\Phi _{i}(t,{\textbf {0}})-L_{i}]\pi _{i2}(t)e^{r(T-t)}\}\nonumber \\{} & {} \quad -\,\phi ^{*}_{m4}(t)h_{m}^{P}[\Phi _{i}(t,{\textbf {z}}^{m})-\Phi _{i}(t,{\textbf {0}})]\exp \{-\alpha _{i}[\Phi _{i}(t,{\textbf {z}}^{m})-\Phi _{i}(t,{\textbf {0}})]\pi _{i2}(t)e^{r(T-t)}\}\bigg ]\nonumber \\{} & {} \quad -\,\alpha _{i} e^{r(T-t)}\bigg [[\Phi _{i}(t,{\textbf {0}})-L_{i}]\frac{h_{i}^{P}}{1+u_{i}}-[\Phi _{i}(t,{\textbf {z}}^{m})-\Phi _{i}(t,{\textbf {0}})]\frac{h_{m}^{P}}{1+u_{m}}\bigg ], \end{aligned}$$

(6.1)

and

$$\begin{aligned}{} & {} \frac{\partial ^{2}\hat{p}(\pi _{i2})}{\partial \pi _{i2}\partial \pi _{i2}}=\alpha ^{2}_{i} e^{2r(T-t)}\frac{g_{i2}(t)}{g_{i3}(t)}\bigg [\phi ^{*}_{i4}(t)h_{i}^{P}[\Phi _{i}(t,{\textbf {0}})-L_{i}]^{2}\exp \{\alpha _{i}[\Phi _{i}(t,{\textbf {0}})-L_{i}]\pi _{i2}(t)e^{r(T-t)}\}\nonumber \\{} & {} \qquad +\,\phi ^{*}_{m4}(t)h_{m}^{P}[\Phi _{i}(t,{\textbf {z}}^{m})-\Phi _{i}(t,{\textbf {0}})]^{2}\exp \{-\alpha _{i}[\Phi _{i}(t,{\textbf {z}}^{m})-\Phi _{i}(t,{\textbf {0}})]\pi _{i2}(t)e^{r(T-t)}\}\bigg ]\nonumber \\{} & {} \quad >0, \end{aligned}$$

(6.2)

then, we have $\lim \limits _{\pi _{i2}(t)\rightarrow \infty }\frac{\partial \hat{p}_{i}(t, \pi _{i2}(t))}{\partial \pi _{i2}(t)}|_{\pi _{i2}(t)=\infty }>0$ and $\lim \limits _{\pi _{i2}(t)\rightarrow 0}\frac{\partial \hat{p}_{i}(t, \pi _{i2}(t))}{\partial \pi _{i2}(t)}|_{\pi _{i2}(t)=0}<0$. we know $\frac{\partial \hat{p}_{i}(t, \pi _{i2}(t))}{\partial \pi _{i2}}$ is increasing in $\pi _{i2}(t)$, then there exists a unique solution $\pi _{i2}^{*}(t)\in [0,T]$ such that $\frac{\partial \hat{p}_{i2}(t, \pi _{i2}(t))}{\partial \pi _{i2}(t)}|_{\pi _{i2}(t)=\pi _{i2}^{*}(t)}=0$, that is $\hat{p}_{i}(t, \pi _{i2}(t))$ has a unique minimizer $\pi _{i2}^{*}(t)$. $\square$

The proof of Lemma 3.8.

Proof

It is that

$$\begin{aligned} \frac{\partial W_{i}(t, q_{i}(t))}{\partial q_{i}(t)}&=\alpha _{i} e^{r(T-t)}\bigg \{(\lambda _{i}+\lambda )\cdot M'_{Y_{i}}(\alpha _{i} q_{i}(t)e^{r(T-t)})-(1+\eta _{i})(\lambda _{i}+\lambda )\mu _{i}\bigg \}\\&=\alpha _{i} e^{r(T-t)}(\lambda _{i}+\lambda )\bigg \{E[Y_{i}e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}] -(1+\eta _{i})\mu _{i}\bigg \}, \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^{2}W_{i}(t, q_{i}(t))}{\partial q_{i}^{2}(t)}= & {} \alpha _{i} e^{r(T-t)}(\lambda _{i}+\lambda )\bigg \{\alpha _{i} e^{r(T-t)}E[Y_{i}^{2}e^{\alpha _{i} q_{i}(t)Y_{i}e^{r(T-t)}}] \bigg \}>0, \end{aligned}$$

then, we have $\lim \limits _{q_{i}(t)\rightarrow 0}\frac{\partial ^{2} W_{i}(t, q_{i}(t))}{\partial q^{2}_{i}(t)}|_{q_{i}(t)=0}>0$ and $\lim \limits _{q_{i}(t)\rightarrow 0}\frac{\partial W_{i}(t, q_{i}(t))}{\partial q_{i}(t)}|_{q_{i}(t)=0}=-(\lambda _{i}+\lambda )\alpha _{i} e^{r(T-t)}\eta _{i}\mu _{i}<0$. we know $\frac{\partial W_{i}(t, q_{i}(t))}{\partial q}$ is increasing in $q_{i}(t)$, then there exists a unique solution $q_{i}^{*}(t)\in [0,T]$ such that $\frac{\partial W_{i}(t, q_{i}(t))}{\partial q_{i}(t)}|_{q_{i}(t)=q_{i}^{*}(t)}=0$, that is $W_{i}(t, q_{i}(t))$ has a unique minimizer $q_{i}^{*}(t)$. $\square$

The proof of Lemma 4.2.

Proof

Putting $\phi _{i}^{*}$ and $\phi _{m}^{*}$ into (4.2). Because that $\phi _{i}^{*}$ and $\phi _{m}^{*}$ are the deterministic and bound functions on [0, T], we have $I(T)<\infty .$ $\square$

The proof of Lemma 4.3.

Proof

Substituting $(\pi ^{*},\phi ^{*})$ into (2.16), we have the wealth process under $(\pi ^{*},\phi ^{*})$

$$\begin{aligned} \hat{X}_{i}^{\pi _{i}^{*},\pi _{m}^{*}}(t)= & {} e^{rt}\hat{X}_{i}^{\pi _{i}^{*}}(0)+\int _{0}^{t}e^{-r(u-t)}\bigg [(a_{i}-r)\pi ^{*}_{i1}(t)-k_{i}(a_{m}-r)\pi ^{*}_{m1}(t)\nonumber \\{} & {} \quad +\,\pi ^{*}_{i2}(u)\big [[\Phi _{i}(u,0,0)-L_{i}]\frac{h_{i}^{P}}{1+u_{i}}-[\Phi _{i}(u,{\textbf {z}}^{m})-\Phi _{i}(u,0,0)]\frac{h_{m}^{P}}{1+u_{m}}\big ]\nonumber \\{} & {} \quad -\,k_{i}\pi ^{*}_{m2}(u)\big [[\Phi _{m}(u,0,0)-L_{m}]\frac{h_{m}^{P}}{1+u_{m}}-[\Phi _{m}(u,{\textbf {z}}^{i})-\Phi _{m}(u,0,0]\frac{h_{i}^{P}}{1+u_{i}}\big ]+\hat{\theta }_{i}\nonumber \\{} & {} \quad +\,[(\lambda _{i}+\lambda )\mu _{i}\eta _{i}q^{*}_{i}(u)-k_{i}(\lambda _{m}+\lambda )\mu _{m}\eta _{m}q^{*}_{m}(u)]-[\sigma _{0i}\phi ^{*}_{i2}(u)-k_{i}\sigma _{0m}\phi ^{*}_{m2}(u)]\nonumber \\{} & {} \quad -\,b_{i}\phi _{i6}(u)\pi ^{*}_{i1}(u)+k_{i}b_{m}\phi ^{*}_{m6}(u)\pi ^{*}_{m1}(u)-\sigma _{i}\phi _{i1}(u)\pi ^{*}_{i1}(u)+k_{i}\sigma _{m}\phi ^{*}_{m1}(t)\pi ^{*}_{m1}(u)\bigg ]du \nonumber \\{} & {} \quad -\,\int _{0}^{t}e^{-r(u-t)}\bigg [\pi ^{*}_{i2}(u)[\Phi _{i}(u,{\textbf {H}}(u))-L_{i}]-k_{i}\pi ^{*}_{m2}(u)[\Phi _{m}(u,{\textbf {H}}^{i}(u-))\nonumber \\{} & {} \quad -\,\Phi _{m}(u,{\textbf {H}}(u-))]\bigg ]dH_{i}(u)+\int _{0}^{t}e^{-r(u-t)}\bigg [k_{i}\pi ^{*}_{m2}(u)[\Phi _{m}(u,{\textbf {H}}(u))-L_{m}]\nonumber \\{} & {} \quad +\,\pi ^{*}_{i2}(u)[\Phi _{i}(u,{\textbf {H}}^{m}(u-))-\Phi _{i}(u,{\textbf {H}}(u-))]\bigg ]dH_{m}(u)\nonumber \\{} & {} \quad +\,\int _{0}^{t}e^{-r(u-t)}(\pi ^{*}_{i1}(u)\sigma _{i} dW_{i}^{P^{\phi }}(u)-k_{i}\pi ^{*}_{m1}(u)\sigma _{m} dW_{m}^{P^{\phi }}(u))+\int _{0}^{t}e^{-r(u-t)}(\pi ^{*}_{i1}(u)b_{i}\nonumber \\{} & {} \quad -\,k_{i}\pi ^{*}_{m1}(u)b_{m})dB^{P^{\phi }}(t)+\int _{0}^{t}e^{-r(u-t)}(\sigma _{0i}dB^{P^{\phi }}_{i}(u)-k_{i}\sigma _{0m}dB^{P^{\phi }}_{m}(t))\nonumber \\{} & {} \quad -\,\int _{0}^{t}e^{-r(u-t)}\int _{R^{+}}(q^{*}_{i}(u)y_{i}\tilde{N}_{i}(du,dy_{i})-k_{i}q^{*}_{m}(u)y_{m}\tilde{N}_{m}(du,dy_{m}))\nonumber \\{} & {} \quad -\,\int _{0}^{t}e^{-r(u-t)}\int _{D}(q^{*}_{i}(u)y_{i}\tilde{N}(du,dy_{i},dy_{m})-k_{i}q^{*}_{m}(t)y_{m}\tilde{N}(du,dy_{i},dy_{m})). \end{aligned}$$

(6.3)

For the candidate value function (2.20), we get

$$\begin{aligned} |\hat{V}_{i}(t,\hat{X}^{\pi _{i}^{*}}(t),{\textbf {H}}(t))|^{4}= & {} |(1-H_{1}(t))(1-H_{2}(t))V_{i}(t,\hat{x}_{i},0,0)\nonumber \\{} & {} +\,H_{2}(t)(1-H_{1}(t)) V_{i}(t,\hat{x}_{i},0,1)\nonumber \\{} & {} +\,H_{1}(t)(1-H_{2}(t)) V_{i}(t,\hat{x}_{i},1,0)+H_{1}(t)H_{2}(t) V_{i}(t,\hat{x}_{i},1,1)|^{4}\nonumber \\\le & {} 4|\hat{V}_{i}(t,\hat{X}^{\pi _{i}^{*}}(t),0,0)|^{4}+4|\hat{V}_{i}(t,\hat{X}^{\pi _{i}^{*}}(t),0,1)|^{4}\nonumber \\{} & {} +\,4|\hat{V}_{i}(t,\hat{X}^{\pi _{i}^{*}}(t),1,0)|^{4}+4|\hat{V}_{i}(t,\hat{X}^{\pi _{i}^{*}}(t),1,1)|^{4}. \end{aligned}$$

(6.4)

At first, we prove $E^{P^{\phi _{i}*}}(\sup _{t\in [0,T]}|\hat{V}_{i}(t,\hat{X}^{\pi _{i}^{*}}(t),0,0)|^{4})<\infty$. $\pi _{i}^{*},\phi _{i}^{*}$ and $g_{i3}(t)$ are deterministic continuous functions, then are bounded on [0, T]. Substituting (6.3) into (3.14), there are three constants $0<K_{1}<K_{2}<K_{3}$ such that

$$\begin{aligned}{} & {} |\hat{V}_{i}(t,\hat{X}^{\pi ^{*}}_{i}(t),0,0)|^{4}=\frac{g_{i3}^{4}(t)}{\alpha _{i}^{4}}\exp \bigg \{-4\alpha _{i}\bigg [e^{rt}\hat{X}^{\pi ^{*}}_{i}(0)+\int _{0}^{t}e^{-r(u-T)}\big [(a_{i}-r)\pi ^{*}_{i1}(t)\nonumber \\{} & {} \quad -\,k_{i}(a_{m}-r)\pi ^{*}_{m1}(t)+\pi ^{*}_{i2}(u)\bigg [[\Phi _{i}(u,0,0)-L_{i}]\frac{h_{i}^{P}}{1+u_{i}}\nonumber \\{} & {} \quad -\,[\Phi _{i}(u,{\textbf {z}}^{m})-\Phi _{i}(u,0,0)]\frac{h_{m}^{P}}{1+u_{m}}\bigg ]\nonumber \\{} & {} \quad -\,k_{i}\pi ^{*}_{m2}(u)\big [[\Phi _{m}(u,0,0)-L_{m}]\frac{h_{m}^{P}}{1+u_{m}}-[\Phi _{m}(u,{\textbf {z}}^{i})-\Phi _{m}(u,0,0]\frac{h_{i}^{P}}{1+u_{i}}\big ]+\hat{\theta }_{i}\nonumber \\{} & {} \quad +\,[(\lambda _{i}+\lambda )\mu _{i}\eta _{i}q^{*}_{i}(u)-k_{i}(\lambda _{m}+\lambda )\mu _{m}\eta _{m}q^{*}_{m}(u)]-b_{i}\phi _{i6}(u)\pi ^{*}_{i1}(u)+k_{i}b_{m}\phi ^{*}_{m6}(u)\pi ^{*}_{m1}(u) \nonumber \\{} & {} \quad -\,\sigma _{i}\phi _{i1}(u)\pi ^{*}_{i1}(u)+k_{i}\sigma _{m}\phi ^{*}_{m1}(t)\pi ^{*}_{m1}(u)-[\sigma _{0i}\phi ^{*}_{i2}(u)-k_{i}\sigma _{0m}\phi ^{*}_{m2}(u)]\big ]du\nonumber \\{} & {} \quad +\,\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{i1}(u)\sigma _{i} dW_{i}^{P^{\phi }}(u)-k_{i}\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{m1}(u)\sigma _{m} dW_{m}^{P^{\phi }}(u))dW_{m}^{P^{\phi }}(u) \nonumber \\{} & {} \quad +\,\int _{0}^{t}e^{-r(u-T)}(b_{i}\pi ^{*}_{i1}(u)-k_{i}b_{m}\pi ^{*}_{m1}(u)dB^{P^{\phi }}(u)\nonumber \\{} & {} \quad +\,\int _{0}^{t}e^{-r(u-T)}\sigma _{0i}dB^{P^{\phi _{i}}}_{i}(u)-k_{i}\int _{0}^{t}e^{-r(u-T)}\sigma _{0m}dB^{P^{\phi _{m}}}_{m}(u)\nonumber \\{} & {} \quad -\,\int _{0}^{t}e^{-r(u-t)}\bigg [\pi ^{*}_{i2}(u)[\Phi _{i}(u,{\textbf {H}}(u))-L_{i}]-k_{i}\pi ^{*}_{m2}(u)[\Phi _{m}(u,{\textbf {H}}^{i}(u-))\nonumber \\{} & {} \quad -\,\Phi _{m}(u,{\textbf {H}}(u-))]\bigg ]dH_{i}(u)+\int _{0}^{t}e^{-r(u-t)}\bigg [k_{i}\pi ^{*}_{m2}(u)[\Phi _{m}(u,{\textbf {H}}(u))-L_{m}]\nonumber \\{} & {} \quad +\,\pi ^{*}_{i2}(u)[\Phi _{i}(u,{\textbf {H}}^{m}(u-))-\Phi _{i}(u,{\textbf {H}}(u-))]\bigg ]dH_{m}(u)\nonumber \\{} & {} \quad -\,\int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{i}(u)y_{i}\tilde{N}_{i}(du,dy_{i})+k_{i}\int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{m}(u)y_{m}\tilde{N}_{m}(du,dy_{m})\nonumber \\{} & {} \quad -\,\int _{0}^{t}e^{-r(u-T)}\int _{D}q^{*}_{i}(u)y_{i}\tilde{N}(du,dy_{i},dy_{m})+k_{i}\int _{0}^{t}e^{-r(u-T)}\nonumber \\{} & {} \quad \int _{D}q^{*}_{m}(u)y_{m}\tilde{N}(du,dy_{i},dy_{m})\bigg ]\bigg \}\nonumber \\{} & {} \le K_{1} \exp \bigg \{-4\alpha _{i}\bigg [\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{i1}(u)\sigma _{i} dW_{i}^{P^{\phi }}(u)-k_{i}\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{m1}(u)\sigma _{m} dW_{m}^{P^{\phi }}(u)\nonumber \\{} & {} \quad +\,\int _{0}^{t}e^{-r(u-T)}(b_{i}\pi ^{*}_{i1}(u)-k_{i}b_{m}\pi ^{*}_{m1}(u))dB^{P^{\phi }}(u)+\int _{0}^{t}e^{-r(u-T)}\sigma _{0i}dB^{P^{\phi }}_{i}(u)\nonumber \\{} & {} \quad -\,k_{i}\int _{0}^{t}e^{-r(u-T)}\sigma _{0m}dB^{P^{\phi }}_{m}(u)-\int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{i}(u)y_{i}\tilde{N}_{i}(du,dy_{i})\nonumber \\{} & {} \quad +\,k_{i}\int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{m}(u)y_{m}\tilde{N}_{m}(du,dy_{m})\nonumber \\{} & {} \quad -\,\int _{0}^{t}e^{-r(u-T)}\int _{D}\big (q^{*}_{i}(u)y_{i}-k_{i}q^{*}_{m}(u)y_{m}\big )\tilde{N}(du,dy_{i},dy_{m})\bigg ]\bigg \}\nonumber \\{} & {} \le K_{2} \exp \bigg \{32\alpha ^{2}_{i}\int _{0}^{t}e^{-2r(u-T)}\big (-2\pi ^{*2}_{i1}(u)\sigma _{i}^{2}+2k^{2}_{i}\pi ^{*2}_{m1}(u)\sigma _{m}^{2}+(b_{i}\pi ^{*}_{i1}(u)\nonumber \\{} & {} \quad -\,k_{i}b_{m}\pi ^{*}_{m1}(u))^{2}\nonumber \\{} & {} \quad +\,2\sigma _{0i}^{2}-2k^{2}_{i}\sigma _{0m}^{2}\big ) du+64\alpha ^{2}_{i}\int _{0}^{t}e^{-2r(u-T)}\pi ^{*2}_{i1}(u)\sigma _{i}^{2} du\nonumber \\{} & {} \quad -\,4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{i1}(u)\sigma _{i} dW_{i}^{P^{\phi }}(u)\nonumber \\{} & {} \quad -\,64k^{2}_{i}\alpha ^{2}_{i}\int _{0}^{t}e^{-2r(u-T)}\pi ^{*2}_{m1}(u)\sigma _{m}^{2} du+4k_{i}\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{m1}(u)\sigma _{m} dW_{m}^{P^{\phi }}(u)\nonumber \\{} & {} \quad -\,32\alpha _{i}^{2}\int _{0}^{t}e^{-2r(u-T)}(b_{i}\pi ^{*}_{i1}(u)-k_{i}b_{m}\pi ^{*}_{m1}(u))^{2} du\nonumber \\{} & {} \quad +\,4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}(b_{i}\pi ^{*}_{i1}(u)-k_{i}b_{m}\pi ^{*}_{m1}(u))dB^{P^{\phi }}(u)\nonumber \\{} & {} \quad -\,64\alpha _{i}^{2}\int _{0}^{t}e^{-2r(u-T)}\sigma _{0i}^{2} du+4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\sigma _{0i}dB_{i}^{P^{\phi }}(u)\nonumber \\{} & {} \quad +\,64k^{2}_{i}\alpha _{i}^{2}\int _{0}^{t}e^{-2r(u-T)}\sigma _{0m}^{2} du-4k_{i}\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\sigma _{0m}dB_{m}^{P^{\phi _{m}}}(u)\nonumber \\{} & {} \quad +\,4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{i}(u)y_{i}\tilde{N}_{i}(du,dy_{i})-4\alpha _{i}k_{i}\int _{0}^{t}e^{-r(u-T)}\nonumber \\{} & {} \quad \int _{R^{+}}q^{*}_{m}(u)y_{m}\tilde{N}_{m}(du,dy_{m})\nonumber \\{} & {} \quad +\,4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\int _{D}\big (q^{*}_{i}(u)y_{i}-k_{i}q^{*}_{m}(u)y_{m}\big )\tilde{N}(du,dy_{i},dy_{m})\bigg \}\nonumber \\{} & {} \le K_{3} \exp \bigg \{64\alpha ^{2}_{i}\int _{0}^{t}e^{-2r(u-T)}\pi ^{*2}_{i1}(u)\sigma _{i}^{2} du-4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{i1}(u)\sigma _{i} dW_{i}^{P^{\phi }}(u)\nonumber \\{} & {} \quad -\,64k^{2}_{i}\alpha ^{2}_{i}\int _{0}^{t}e^{-2r(u-T)}\pi ^{*2}_{m1}(u)\sigma _{m}^{2} du+4k_{i}\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{m1}(u)\sigma _{m} dW_{m}^{P^{\phi }}(u)\nonumber \\{} & {} \quad -\,32\alpha _{i}^{2}\int _{0}^{t}e^{-2r(u-T)}(b_{i}\pi ^{*}_{i1}(u)-k_{i}b_{m}\pi ^{*}_{m1}(u))^{2} du\nonumber \\{} & {} \quad +\,4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}(b_{i}\pi ^{*}_{i1}(u)-k_{i}b_{m}\pi ^{*}_{m1}(u))dB^{P^{\phi }}(u)\nonumber \\{} & {} \quad -\,64\alpha _{i}^{2}\int _{0}^{t}e^{-2r(u-T)}\sigma _{0i}^{2} du+4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\sigma _{0i}dB_{i}^{P^{\phi }}(u)\nonumber \\{} & {} \quad +\,64k^{2}_{i}\alpha _{i}^{2}\int _{0}^{t}e^{-2r(u-T)}\sigma _{0m}^{2} du-4k_{i}\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\sigma _{0m}dB_{m}^{P^{\phi _{m}}}(u)\nonumber \\{} & {} \quad +\,4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{i}(u)y_{i}\tilde{N}_{i}(du,dy_{i})-4\alpha _{i}k_{i}\int _{0}^{t}e^{-r(u-T)}\nonumber \\{} & {} \quad \int _{R^{+}}q^{*}_{m}(u)y_{m}\tilde{N}_{m}(du,dy_{m})\nonumber \\{} & {} \quad +\,4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\int _{D}\big (q^{*}_{i}(u)y_{i}-k_{i}q^{*}_{m}(u)y_{m}\big )\tilde{N}(du,dy_{i},dy_{m})\bigg \}\nonumber \\{} & {} \triangleq K_{3}\exp \{D_{1}(t)+D_{2}(t)+D_{3}(t)+D_{4}(t)+D_{5}(t)+D_{6}(t)\}, \end{aligned}$$

(6.5)

where

$$\begin{aligned} D_{1}(t)= & {} 64\alpha ^{2}_{i}\int _{0}^{t}e^{-2r(u-T)}\pi ^{*2}_{i1}(u)\sigma _{i}^{2} du-4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{i1}(u)\sigma _{i} dW_{i}^{P^{\phi }}(u),\\ D_{2}(t)= & {} -\,64k^{2}_{i}\alpha ^{2}_{i}\int _{0}^{t}e^{-2r(u-T)}\pi ^{*2}_{m1}(u)\sigma _{m}^{2} du+4k_{i}\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\pi ^{*}_{m1}(u)\sigma _{m} dW_{m}^{P^{\phi }}(u),\\ D_{3}(t)= & {} -\,32\alpha _{i}^{2}\int _{0}^{t}e^{-2r(u-T)}(b_{i}\pi ^{*}_{i1}(u)-k_{i}b_{m}\pi ^{*}_{m1}(u))^{2} du\\{} & {} +\,4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}(b_{i}\pi ^{*}_{i1}(u)-k_{i}b_{m}\pi ^{*}_{m1}(u))dB^{P^{\phi }}(u),\\ D_{4}(t)= & {} -\,64\alpha _{i}^{2}\int _{0}^{t}e^{-2r(u-T)}\sigma _{0i}^{2} du+4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\sigma _{0i}dB_{i}^{P^{\phi }}(u),\\ D_{5}(t)= & {} 64k^{2}_{i}\alpha _{i}^{2}\int _{0}^{t}e^{-2r(u-T)}\sigma _{0m}^{2} du-4k_{i}\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\sigma _{0m}dB_{m}^{P^{\phi _{m}}}(u),\\ D_{6}(t)= & {} 4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{i}(u)y_{i}\tilde{N}_{i}(du,dy_{i})-4\alpha _{i}k_{i}\\{} & {} \int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{m}(u)y_{m}\tilde{N}_{m}(du,dy_{m})\\{} & {} +\,4\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\int _{D}\big (q^{*}_{i}(u)y_{i}-k_{i}q^{*}_{m}(u)y_{m}\big )\tilde{N}(du,dy_{i},dy_{m}), \end{aligned}$$

and the last unequal sign is due to the deterministic continuous functions $\pi _{i}^{*},\phi _{i}^{*}$ are bounded on [0, T].

Firstly, it easy to check that

$$\begin{aligned}{} & {} E^{P^{\phi _{i}*}}\left( \exp \{4D_{6}(t)\}\right) =16\alpha _{i}(\int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{i}(u)y_{i}\tilde{N}_{i}(du,dy_{i})\nonumber \\{} & {} \quad -\,16\alpha _{i}k_{i}\int _{0}^{t}e^{-r(u-T)}\int _{R^{+}}q^{*}_{m}(u)y_{m}\tilde{N}_{m}(du,dy_{m})\nonumber \\{} & {} \quad +\,16\alpha _{i}\int _{0}^{t}e^{-r(u-T)}\int _{D}\big (q^{*}_{i}(u)y_{i}-k_{i}q^{*}_{m}(u)y_{m}\big )\tilde{N}(du,dy_{i},dy_{m})<\infty .~~~~~~~ \end{aligned}$$

(6.6)

Applying Lemma 4.3 in Zeng and Taksar [47], we know that $\exp \{8D_{1}(t)\}$, $\exp \{8D_{2}(t)\}$, $\exp \{4D_{3}(t)\}$, $\exp \{8D_{4}(t)\}$, $\exp \{8D_{5}(t)\}$ and $\exp \{4D_{6}(t)\}$ are martingales, then

$$\begin{aligned} E^{P^{\phi _{i}*}}(\exp \{4D_{3}(t)\})<\infty ~~and~~E^{P^{\phi _{i}*}}(\exp \{8D_{i}(t)\})<\infty , i=1, 2, 4, 5. \end{aligned}$$

(6.7)

Applying (6.6), (6.7), according to Cauchy-Schwarz inequality, we have

$$\begin{aligned}{} & {} E^{P^{\phi _{i}*}}(|\hat{V}_{i}(t,\hat{X}^{\pi ^{*}}_{i}(t),0)|^{4})\nonumber \\{} & {} \quad \le K_{3}E^{P^{\phi _{i}*}}[\exp \{D_{1}(t)+D_{2}(t)+D_{3}(t)+D_{4}(t)+D_{5}(t)+D_{6}(t)\}]\nonumber \\{} & {} \quad \le K_{3}E^{P^{\phi _{i}*}}[\exp \{2D_{1}(t)+2D_{2}(t)+2D_{3}(t)\}]^{\frac{1}{2}}\nonumber \\{} & {} \qquad \times \, E^{P^{\phi _{i}*}}[\exp \{2D_{4}(t)+2D_{5}(t)+2D_{6}(t)\}]^{\frac{1}{2}}\nonumber \\{} & {} \quad \le K_{3}E^{P^{\phi _{i}*}}[\exp \{4D_{1}(t)+4D_{2}(t)]^{\frac{1}{4}}\cdot E^{P^{\phi _{i}*}}[\exp \{4D_{3}(t)]^{\frac{1}{4}}\nonumber \\{} & {} \qquad \times \, E^{P^{\phi _{i}*}}[\exp \{4D_{4}(t)+4D_{5}(t)\}])^{\frac{1}{4}}\cdot E^{P^{\phi _{i}*}}[\exp \{4D_{6}(t)]^{\frac{1}{4}}\nonumber \\{} & {} \quad \le K_{3}(E^{P^{\phi _{i}*}}[\exp \{8D_{1}(t)\}]^{\frac{1}{8}}E^{P^{\phi _{i}*}}[\exp \{8D_{2}(t)\}]^{\frac{1}{8}}E^{P^{\phi _{i}*}}[\exp \{4D_{3}(t)\}]^{\frac{1}{4}}\nonumber \\{} & {} \qquad \times \, E^{P^{\phi _{i}*}}[\exp \{8D_{4}(t)\}]^{\frac{1}{8}}E^{P^{\phi _{i}*}}[\exp \{8D_{5}(t)\}]^{\frac{1}{8}}E^{P^{\phi _{i}*}}[\exp \{4D_{6}(t)\}])^{\frac{1}{4}}\nonumber \\{} & {} \quad \le \infty . \end{aligned}$$

(6.8)

Consequently, $E^{P^{\phi _{i}*}}(\sup _{t\in [0,T]}|\hat{V}_{i}(t,\hat{X}_{i}^{\pi _{i}^{*}}(t),0,0)|^{4})<\infty$. Similarity, we can also prove

$$\begin{aligned}{} & {} E^{P^{\phi _{i}*}}(\sup _{t\in [0,T]}|\hat{V}_{i}(t,\hat{X}_{i}^{\pi _{i}^{*}}(t),0,1)|^{4})<\infty ,~~~E^{P^{\phi _{i}*}}(\sup _{t\in [0,T]}|\hat{V}_{i}(t,\hat{X}_{i}^{\pi _{i}^{*}}(t),1,0)|^{4})<\infty ,\\{} & {} E^{P^{\phi _{i}*}}(\sup _{t\in [0,T]}|\hat{V}_{i}(t,\hat{X}_{i}^{\pi _{i}^{*}}(t),1,1)|^{4})<\infty , \end{aligned}$$

then the formula $E^{P^{\phi _{i}*}}(\sup _{t\in [0,T]}|\hat{V}_{i}(t,\hat{X}_{i}^{\pi ^{*}}(t),{\textbf {H}}(t))|^{4})<\infty$ holds. The first part of Lemma 4.3 is proved.

Since f(t) is obviously bounded, we get

$$\begin{aligned}{} & {} E^{P^{\phi _{i}*}}(\sup _{t\in [0,T]}|G_{i}(t, \hat{X}_{i}^{\pi _{i}^{*}}(t),\phi _{i}^{*}(t))|^{2})=E^{P^{\phi _{i}*}}(\sup _{t\in [0,T]}|f(t)|^{2}|\hat{V}_{i}(t,\hat{X}_{i}^{\pi _{i}^{*}}(t),{\textbf {z}})|^{2}) \nonumber \\{} & {} \quad \le (E^{P^{\phi _{i}*}}\sup _{t\in [0,T]}|f(t)|^{4})^{\frac{1}{2}}(E^{P^{\phi _{i}*}}\sup _{t\in [0,T]}|\hat{V}_{i}(t,\hat{X}_{i}^{\pi _{i}^{*}}(t),{\textbf {z}})|^{4})^{\frac{1}{2}}\nonumber \\{} & {} \quad <\infty . \end{aligned}$$

(6.9)

The first inequality follows from Cauchy-Schwarz inequality. The last inequality follows from (4.2). Lemma 4.3 is proved. $\square$