Locally Risk-Minimizing Hedging of Counterparty Risk for Portfolio of Credit Derivatives

Abstract

We discuss dynamic hedging of counterparty risk for a portfolio of credit derivatives by the local risk-minimization approach. We study the problem from the perspective of an investor who, trading with credit default swaps (CDS) referencing the counterparty, wants to protect herself/himself against the loss incurred at the default of the counterparty. We propose a credit risk intensity-based model consisting of interacting default intensities by taking into account direct contagion effects. The portfolio of defaultable claims is of generic type, including CDS portfolios, risky bond portfolios and first-to-default claims with payments allowed to depend on the default state of the reference firms and counterparty. Using the martingale representation of the conditional expectation of the counterparty risk price payment stream under the minimal martingale measure, we recover a closed-form representation for the locally risk minimizing strategy in terms of classical solutions to nonlinear recursive systems of Cauchy problems. We also discuss applications of our framework to the most prominent classes of credit derivatives.

A Proofs

Proof of Lemma 2.2

By Definition 2.2, we have the representation of the dividend process of the first-to-default claim given by

$$\begin{aligned} D(t) = -\varepsilon \int _{0}^{t \wedge T} (1-K(u))du + \sum _{i=1}^N\int _{0}^{t \wedge T} L_i(H(u))H_i(u) d K(u). \end{aligned}$$

(89)

The third term of the above dividend process is in fact given by

$$\begin{aligned} \sum _{i=1}^N\int _{0}^{t \wedge T} L_i(H(u))H_i(u) d K(u)&=\sum _{i=1}^NL_i(H({\bar{\tau }}_1))H_i({\bar{\tau }}_1)\mathbf{1}_{{\bar{\tau }}_1\le t\wedge T}\\ {}&=\sum _{i=1}^NL_i(H({\bar{\tau }}_1))\mathbf{1}_{\tau _i\le {\bar{\tau }}_1}\mathbf{1}_{{\bar{\tau }}_1\le t\wedge T}. \end{aligned}$$

Notice that for all $i=1,\ldots ,N$, we have $\tau _i\ge {\bar{\tau }}_1=\tau _1\wedge \cdots \wedge \tau _N$, a.s.. Hence $\mathbf{1}_{\tau _i\le {\bar{\tau }}_1}=\mathbf{1}_{\tau _i={\bar{\tau }}_1}$, a.s.. Thus the above equality becomes that

$$\begin{aligned} \sum _{i=1}^N\int _{0}^{t \wedge T} L_i(H(u))H_i(u) d K(u)&=\sum _{i=1}^NL_i(H({\bar{\tau }}_1))\mathbf{1}_{\tau _i\le {\bar{\tau }}_1}\mathbf{1}_{{\bar{\tau }}_1\le t\wedge T}\\ {}&=\sum _{i=1}^NL_i(H({\bar{\tau }}_1))\mathbf{1}_{\tau _i={\bar{\tau }}_1}\mathbf{1}_{{\bar{\tau }}_1\le t\wedge T}. \end{aligned}$$

This results in the dividend representation given by Eq. (11). $\square$

Proof of Proposition 2.3

Using (7), it holds that, for $t\in [0,T]$,

$$\begin{aligned} D(T)-D(t)&=\xi (H(T))(1-K(T))\mathbf{1}_{t\ne T}+\int _{t}^{T} (1-K(u)) a(u) du \\&\quad \, + \int _{t}^{T} Z(u) d K(u). \end{aligned}$$

Then, it follows from (15) that, for $t\in [0,T]$,

$$\begin{aligned} {S}(t,T)&= {\mathbb {E}} ^{ {\mathbb {Q}} }\Bigg [ \xi (H(T))(1-K(T))\mathbf{1}_{t\ne T}+ \int _{t}^{T} (1-K(H(u))) a(H(u)) du\\&\quad + \, \int _{t}^{T} Z(H(u)) d K(H(u))\Big |{\mathcal {G}}_t\Bigg ]. \end{aligned}$$

Recall that Z(z) and K(z) are deterministic functions on $z\in {{{\mathcal {S}}}}=\{0,1\}^{N+1}$.

Using integrations by parts, it follows that

$$\begin{aligned} Z(H(T))K(H(T))=\,&Z(H(t))K(H(t))+\int _t^T Z(H(u))dK(H(u))\nonumber \\&\quad +\int _t^T K(H(u^-))dZ(H(u)). \end{aligned}$$

(90)

On the other hand, Itô’s formula gives that for $u\in [t,T]$,

$$\begin{aligned} dZ(H(u))&=\sum _{j=1}^{N+1} [Z(H^j(u^-))-Z(H(u^-))]dH_j(u)\\ {}&=\sum _{j=1}^{N+1} [Z(H^j(u^-))-Z(H(u^-))]dM_j^{ {\mathbb {Q}} }(u) \\&\quad +\sum _{j=1}^{N+1}[Z(H^j(u))-Z(H(u))](1-H_j(u))(1+\vartheta _j(u))X_j(u)du. \end{aligned}$$

For $j=1,\ldots ,N+1$, $M_j^{ {\mathbb {Q}} }=(M_j^{ {\mathbb {Q}} }(t))_{t\in [0,T]}$ is the ${\mathbb {G}}$-martingale given in Proposition 2.1. Hence, Eq. (90) yields that

$$\begin{aligned} \int _t^T Z(H(u))dK(H(u))&=Z(H(T))K(H(T))-Z(H(t))K(H(t))\\&\qquad -\int _t^T K(H(u^-))dZ(H(u))\\&=Z(H(T))K(H(T))-Z(H(t))K(H(t))\\&\qquad -\sum _{j=1}^{N+1}\int _t^T K(H(u^-))[Z(H^j(u^-))-Z(H(u^-))]dM_j^{ {\mathbb {Q}} }(u)\\&\qquad -\sum _{j=1}^{N+1}\int _t^TK(H(u)) [Z(H^j(u))\\&\qquad -Z(H(u))](1-H_j(u))(1+\vartheta _j(u))X_j(u)du. \end{aligned}$$

This results in the price representation given by $S(t,T)=F(t,X(t),H(t))-Z(H(t))K(H(t))$, where

$$\begin{aligned} F(t,x,z)&:= {\mathbb {E}} _{t,x,z}^{ {\mathbb {Q}} }\bigg [ \xi (H(T))(1-K(H(T)))\mathbf{1}_{t\ne T}+ Z(H(T))K(H(T))\nonumber \\&\quad +\int _{t}^{T} (1-K(H(u))) a(H(u)) du\nonumber \\&\quad -\sum _{j=1}^{N+1}\int _t^T K(H(u))[Z(H^j(u))\nonumber \\&\quad -Z(H(u))](1-H_j(u))(1+\vartheta _j(u))X_j(u)du\bigg ], \end{aligned}$$

(91)

using that the pair (X, H) is a ${\mathbb {G}}$-adapted Markov process. Then the price representation (17) follows from the decomposition of F(t, x, z) given by

$$\begin{aligned} F(t,x,z)=\mathbf{1}_{t\ne T} \Lambda _1(t,x,z) + \Lambda _2(t,x,z),\qquad (t,x,z)\in [0,T]\times \mathbb {R}_+^{N+1}\times {{{\mathcal {S}}}}. \end{aligned}$$

(92)

This completes the proof of the lemma. $\square$

Proof of Proposition 2.4

On $(t,x)\in [0,T)\times \mathbb {R}_+^{N+1}$, we rewrite (26) as follows:

$$\begin{aligned}&\left( \frac{\partial }{\partial t}+\tilde{{{\mathcal {A}}}}^{ {\mathbb {Q}} }\right) u(t,x)+h(x)u(t,x)+w(t,x)=0 \end{aligned}$$

(93)

with $u(T,x)=\alpha _1\xi ^{(l)}(1-K^{(l)})+\alpha _2Z^{(l)}K^{(l)}$ for all $x\in \mathbb {R}_+^{N+1}$. The coefficients

$$\begin{aligned} h(x)&:= -\sum _{j\notin \{j_1,\ldots ,j_l\}}(1+\vartheta _j^{(l)}(x))x_j,\\ w(t,x)&:=\sum _{j\notin \{j_1,\ldots ,j_l\}}(1+\vartheta _j^{(l)}(x))x_j\big [F_{\alpha }^{(l+1),j}(t,x+w_j)\\&\quad -\alpha _3K^{(l)} (Z^{(l+1),j}-Z^{(l)})\big ]+\alpha _3(1-K^{(l)})a^{(l)}. \end{aligned}$$

We will apply Theorem 1 of Heath and Schweizer [24] to prove existence and uniqueness of classical solutions to Eq. (93) by verifying that their imposed conditions [A1], [A2], [A3’] and [A3a’]-[A3e’] hold in our case. Consider a sequence of bounded domains $D_n:=(\frac{1}{n},n)^{N+1}$, $n\in \mathbb {N}$, with smoothed corners and satisfying $\bigcup _{n=1}^{\infty }D_n=\mathbb {R}_+^{N+1}$. Thus we verified that the condition [A3’] on the domain of the equation holds. By the assumptions (A1)–(A3), the conditions [A1] and [A2] for the coefficients $\mu (x)+\sigma (x)\widetilde{\theta }(x,z)$ and $\sigma (x)$ can be satisfied. This also implies that [A3a’] holds. Moreover, since $\sigma \sigma ^{\top }(x)$ is continuous and invertible under the assumptions (A1) and (A2), $\sigma \sigma ^{\top }(x)$ is uniformly elliptic on $(t,x)\times {\overline{D}}_n$, i.e. [A3b’] holds. Notice that $F_{\alpha }^{(l+1),j}(t,x+w_j)$ is bounded and $C^{1,2}$ in (t, x) by the induction hypothesis. Additionally, notice that h(x) is linear in x. Then the conditions [A3c’] and [A3d’] on the coefficients h(x) and w(t, x) on $(t,x)\in [0,T]\times {\overline{D}}_n$ are satisfied. Finally we need to verify [A3e’]. For this, it suffices to prove the uniform integrability of the family

$$\begin{aligned} \left\{ \int _t^Tw(s,{\check{X}}^{(t,x)}(s))e^{-\int _t^sh({\check{X}}^{(t,x)}(u))du}ds;\ (t,x)\in [0,T]\times \mathbb {R}_+^{N+1}\right\} . \end{aligned}$$

(94)

Here, the underlying $\mathbb {R}_+^{N+1}$-valued process $({\check{X}}^{(t,x)}(s))_{s\in [t,T]}$ is the unique strong solution of

$$\begin{aligned} d{\check{X}}^{(t,x)}(s)&=(\mu ({\check{X}}^{(t,x)} (s))+\sigma ({\check{X}}^{(t,x)}(s))\widetilde{\theta }(\check{X}^{(t,x)}(s),0^{j_1,\ldots ,j_l}))ds\\&\quad +\sigma ({\check{X}}^{(t,x)}(s))dW(s),\ {\check{X}}^{(t,x)}(t)=x. \end{aligned}$$

By the inductive hypothesis that $F_{\alpha }^{(l+1),j}(t,x)$ is nonnegative and bounded on $[0,T]\times \mathbb {R}_+^{N+1}$ for all $j\notin \{j_1,\ldots ,j_l\}$, there exists a constant $C>0$ independent of (t, x) such that for all $(t,x)\in [0,T]\times \mathbb {R}_+^{N+1}$,

$$\begin{aligned}&{\mathbb {E}} \left[ \left| \int _t^Tw(s,{\check{X}}^{(t,x)}(s))e^{\int _t^sh({ \check{X}}^{(t,x)}(u))du}ds\right| ^2\right] \nonumber \\&\quad \le C {\mathbb {E}} \left[ \left| \int _t^Te^{-\int _t^s(\sum _{k \notin \{j_1,\ldots ,j_l\}}(1+\vartheta _k({\check{X}}^{(t,x)}(u))){ \check{X}}_k^{(t,x)}(u))du} \right. \right. \nonumber \\&\left. \left. \qquad \times \left( 1+\sum _{j\notin \{j_1,\ldots ,j_l\}}(1+\vartheta _j({\check{X}}^{(t,x)}(s))){ \check{X}}_j^{(t,x)}(s)\right) ds\right| ^2 \right] \nonumber \\&\quad \le 2CT^2\nonumber \\&\qquad +2C {\mathbb {E}} \left[ \left| \int _t^Te^{-\int _t^s(\sum _{k\notin \{j_1,\ldots ,j_l\}} (1+\vartheta _k({\check{X}}^{(t,x)}(u))){\check{X}}_k^{(t,x)}(u))du} \right. \right. \nonumber \\&\qquad \left. \left. \times \, d\left( \int _t^s\sum _{j\notin \{j_1,\ldots ,j_l\}}(1+\vartheta _j({ \check{X}}^{(t,x)}(u))){\check{X}}_j^{(t,x)}(u)du\right) \right| ^2\right] \nonumber \\&\quad \le 2CT^2+2C\left\{ 1+\left| {\mathbb {E}} \left[ e^{-\int _t^T(\sum _{k\notin \{j_1,\ldots ,j_l\}} (1+\vartheta _k({\check{X}}^{(t,x)}(u))){\check{X}}_k^{(t,x)}(u))du}\right] \right| ^2\right\} \nonumber \\&\quad \le 2CT^2+4C. \end{aligned}$$

(95)

This yields the existence of a constant $C>0$, independent of (t, x), such that

$$\begin{aligned} \sup _{(t,x)\in [0,T]\times \mathbb {R}_+^{N+1}} {\mathbb {E}} \left[ \left| \int _t^Tw(s, {\check{X}}^{(t,x)}(s))e^{\int _t^sh({\check{X}}^{(t,x)}(u))du}ds\right| ^2\right] \le C<+\infty . \end{aligned}$$

This yields the uniform integrability of the family (94). It implies the condition [A3e’] of Heath and Schweizer [24] is satisfied. Using Theorem 1 of Heath and Schweizer [24], Eq. (93) admits a unique classical solution u(t, x) on $[0,T]\times \mathbb {R}_+^{N+1}$.

Further, the estimate (95) implies that this solution is bounded for all $(t,x)\in [0,T]\times \mathbb {R}_+^{N+1}$. This completes the proof of the proposition. $\square$

Proof of Lemma 2.5

It follows from Eq. (7) that

$$\begin{aligned} D(T) = \xi (H(T))(1-K(T)) +\int _{0}^{T} (1-K(u)) a(u) du + \int _{0}^{T} Z(u) d K(u). \end{aligned}$$

Using integration by parts (90), we have that

$$\begin{aligned} D(T)&= \xi (H(T))(1-K(H(T))) + \int _{0}^{T} (1-K(H(u))) a(u) du + Z(H(T))K(H(T))\\&\quad -Z(H(0))K(H(0))-\int _{0}^{T} K(H(u^-)) d Z(H(u)). \end{aligned}$$

Since $K(0)=0$, it follows from Proposition 2.4 that

$$\begin{aligned} Y(t)&= F_{(1,1,1)}(t,X(t),H(t))+\int _{0}^{t} (1-K(u)) a(u) du -\int _{0}^{t} K(H(u^-)) d Z(H(u)). \end{aligned}$$

(96)

Above, $F_{(1,1,1)}(t,x,z)$ is the unique bounded classical solution to the recursive system of the backward Cauchy problems given by, on $(t,x,z)\in [0,T)\times \mathbb {R}_+^{N+1}\times {{{\mathcal {S}}}}$,

$$\begin{aligned}&\left( \frac{\partial }{\partial t} + {{{\mathcal {A}}}}^{ {\mathbb {Q}} }\right) F_{(1,1,1)}(t,x,z)+(1-K(z))a(z)\nonumber \\&\quad - \sum _{j=1}^{N+1}K(z)[Z(z^j)-Z(z)](1-z_j)(1+\vartheta _j(x))x_j= 0 \end{aligned}$$

(97)

with the terminal condition

$$\begin{aligned} F_{(1,1,1)}(T,x,z)=\xi (z)(1-K(z))+Z(z)K(z),\qquad (x,z)\in \mathbb {R}_+^{N+1}\times {{{\mathcal {S}}}}. \end{aligned}$$

(98)

Applying Itô’s formula and (97), we obtain that

$$\begin{aligned}&F_{(1,1,1)}(t,X(t),H(t)) =F_{(1,1,1)}(0,X(0),H(0))\\&\qquad +\int _0^t\left\{ \sum _{j=1}^{N+1}K(H(u))[Z(H^j(u))-Z(H(u))] (1-H_j(u))\right. \\&\left. \qquad (1 +\,\vartheta _j(u))X_j(u)-(1-K(H(u)))a(H(u))\right\} du\\&\qquad + \int _0^t D_xF_{(1,1,1)}(u,X(u),H(u))^{\top }\sigma (X(u))dW^{ {\mathbb {Q}} }(u)\\&\qquad +\sum _{j=1}^{N+1} \int _0^t [F_{(1,1,1)}(u,X(u^-)+w_j,H^j(u^-))\\&\qquad - F_{(1,1,1)}(u,X(u^-),H(u^-))]dM_j^{ {\mathbb {Q}} }(u). \end{aligned}$$

Using Eq. (96), we deduce

$$\begin{aligned} dY(t)&=D_xF_{(1,1,1)}(t,X(t),H(t))^{\top }\sigma (X(t))dW^{ {\mathbb {Q}} }(t)\\&\quad +\sum _{j=1}^{N+1}[F_{(1,1,1)}(t,X(t^-)+w_j,H^j(t^-))-F_{(1,1,1)} (t,X(t^-),H(t^-))]dM_j^{ {\mathbb {Q}} }(t)\\&\quad -\sum _{j=1}^{N+1}K(H(t^-))[Z(H^j(t^-))-Z(H(t^-))]dM_j^{ {\mathbb {Q}} }(t). \end{aligned}$$

This yields the dynamics (28) of the gain process. $\square$

Proof of Lemma 3.3

We first verify that the density process $\xi$ is strictly positive and square integrable. The assumption of $0<1+{\hat{\lambda }}(t,x,z)\Psi _j(t,x,z)<\nu _j$ implies that $\xi$ is strictly positive using the SDE-representation of the stochastic exponential. We next introduce the so-called mean-variance trade-off process given by

$$\begin{aligned} \Xi (t)&:=\int _0^t {\hat{\lambda }}(s,X(s),H(s))^2 d\left<Q\right>(s) \nonumber \\&=\int _0^t\frac{|\Upsilon (s) \widetilde{\theta }(s)+\sum _{j=1}^{N+1}\Psi _j(s)(1-H_j(s)) \vartheta _j(s)X_j(s)|^2}{|\Upsilon (s)|^2+\sum _{j=1}^{N+1} \Psi _j^2(s)(1-H_j(s))X_j(s)}ds\nonumber \\&\le 2\int _0^t|\widetilde{\theta }(s)|^2ds+2^{N+1}\int _0^t\vartheta _j^2(s)X_j(s)ds. \end{aligned}$$

(99)

Then Assumption (A3) yields that $\Xi =(\Xi (t))_{t\in [0,T]}$ is uniformly bounded. Using Proposition 3.7 of Choulli et al. [17], the process $\xi$ satisfies the reverse Hölder inequality, see also Assumption 3.2 in Arai [2]. On the other hand, the structural condition given by $B=-\int _0^{\cdot }{\hat{\lambda }}(s,X(s^-),H(s^-))d\left<Q\right>(s)$ implies that $Y_{N+1}\xi$ is a local ${\mathbb {P}}$-martingale (see Ansel and Stricker [1]). Using the arguments in Sect. 3 of Arai [2], we have that $\xi$ is the density process of the MMM ${\hat{ {\mathbb {P}} }}$ w.r.t. ${\mathbb {P}}$. $\square$

Proof of Theorem 3.5

Without any loss of generality, we set $L_{N+1}(z)=1$ for all $z\in {{{\mathcal {S}}}}$. Then, in the default state $z=0^{j_1,\ldots ,j_l}$, we rewrite Eq. (62) in the following abstract form: on $(t,x)\in [0,T)\times \mathbb {R}_+^{N+1}$,

$$\begin{aligned} \left( \frac{\partial }{\partial t}+\bar{{{\mathcal {A}}}}\right) u(t,x)+h(t,x)u(t,x)+w(t,x)=0 \end{aligned}$$

(100)

with $u(T,x)=0$ for all $x\in \mathbb {R}_+^{N+1}$. The coefficients are given by

$$\begin{aligned} h(t,x)&:=-\sum _{j\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_j^{(l)}(t,x),\\ w(t,x)&:=\left\{ \sum _{i=1}^{{\bar{N}}} b_i(1-K_i^{(l+1),N+1})\big [F_{(1,1,1)i}(t,x\right. \\&\left. \quad +\,w_{N+1},0^{j_1, \ldots ,j_l,N+1})-Z_i^{(l+1),N+1}K_i^{(l+1),N+1}\big ]\right\} _+\\&\qquad \times {\hat{F}}_{N+1}^{(l)}(t,x)\mathbf{1}_{j_1,\ldots ,j_l\ne N+1}+\sum _{j\notin \{j_1,\ldots ,j_l\}}g^{(l+1),j}(t,x+w_j){\hat{F}}_j^{(l)}(t,x). \end{aligned}$$

We next apply Theorem 1 of Heath and Schweizer [24] to prove existence and uniqueness of classical solutions of Eq. (100) by verifying that their series of conditions [A1], [A2], [A3’] and [A3a’]-[A3e’] hold in our case. We first consider bounded domains $D_n:=(\frac{1}{n},n)^{N+1}$, $n\in \mathbb {N}$, with smoothed corners such that $\bigcup _{n=1}^{\infty }D_n=\mathbb {R}_+^{N+1}$. We can then verify that the condition [A3’] holds in the domain of the equation. Using assumptions (A1)–(A3), the conditions [A1] and [A2] hold. The same assumption also implies that [A3a’] holds. Moreover $\sigma \sigma ^{\top }(x)$ is uniformly elliptic on $(t,x)\times {\overline{D}}_n$, i.e. [A3b’] holds. Notice that the solution $g^{(l+1),j}(t,x+w_j)$ is bounded and $C^{1,2}$ in (t, x) by the induction hypothesis for $j\notin \{j_1,\ldots ,j_l\}$. The function $F_{(1,1,1)i}(t,x)$ is also bounded and $C^{1,2}$ in (t, x) for $i=1,\ldots ,{\bar{N}}$ by Proposition 2.4. Note that the positive ${\hat{F}}_j^{(l)}(t,x)$ is $C^1$ in (t, x). Then the conditions [A3c’] and [A3d’] on the coefficients h(t, x) and w(t, x), $(t,x)\in [0,T]\times {\overline{D}}_n$, are satisfied. It is left to verify [A3e’]. For this, it suffices to prove the uniform integrability of the family

$$\begin{aligned} \left\{ \int _t^Tw(s,{\hat{X}}^{(t,x)}(s))e^{-\int _t^sh(u,{\hat{X}}^{(t,x)}(u))du}ds;\ (t,x)\in [0,T]\times \mathbb {R}_+^{N+1}\right\} . \end{aligned}$$

(101)

Here, for $t\in [0,T]$, the $N+1$-dimensional Markov process $({\hat{X}}^{(t,x)}(s))_{s\in [t,T]}$ satisfies a SDE with ${\hat{X}}^{(t,x)}(t)=x$ such that its infinitesimal generator is given by $\bar{{{\mathcal {A}}}}$ in (60).

Consider first the case $N+1\in \{j_1,\ldots ,j_l\}$. Because $g^{(l+1),j}(t,x)$ is bounded on $[0,T]\times \mathbb {R}_+^{N+1}$ by the induction hypothesis, there exists a constant $C>0$ such that

$$\begin{aligned}&{\mathbb {E}} \left[ \left| \int _t^Tw(s,{\hat{X}}^{(t,x)}(s))e^{\int _t^sh(u,{ \hat{X}}^{(t,x)}(u))du}ds\right| ^2\right] \\&\quad \le C {\mathbb {E}} \left[ \left| \int _t^T\left( \sum _{j\notin \{j_1,\ldots ,j_l\}}{ \hat{F}}_j^{(l)}(s,{\hat{X}}^{(t,x)}(s))\right) e^{-\int _t^s\sum _{k \notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}ds \right| ^2\right] \\&\quad =C {\mathbb {E}} \left[ \left| \int _t^Te^{-\int _t^s\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}d\left( \int _t^s\sum _{j \notin \{j_1,\ldots ,j_l\}}{\hat{F}}_j^{(l)}(u,{\hat{X}}^{(t,x)}(u))du\right) \right| ^2\right] \\&\quad \le C\left\{ 1+\left| {\mathbb {E}} \left[ e^{-\int _t^T\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}\right] \right| ^2\right\} \\&\quad \le C, \end{aligned}$$

where $C>0$ is independent of (t, x). Next, consider the case $N+1\notin \{j_1,\ldots ,j_l\}$. Also notice that $F_{(1,1,1)i}(t,x)$ is bounded and $C^{1,2}$ in (t, x) for $i=1,\ldots ,{\bar{N}}$ by Proposition 2.4. Then there exists a constant $C>0$ such that

$$\begin{aligned}&{\mathbb {E}} \left[ \left| \int _t^Tw(s,{\hat{X}}^{(t,x)}(s))e^{\int _t^sh(u,{\hat{X}}^{(t,x)} (u))du}ds\right| ^2\right] \\&\quad \le C {\mathbb {E}} \left[ \left| \int _t^Te^{-\int _t^s\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}\right. \right. \\&\left. \left. \qquad \times \left( {\hat{F}}_{N+1}^{(l)} (s,{\hat{X}}^{(t,x)}(s))+\sum _{j\notin \{j_1,\ldots ,j_l\}}{\hat{F}}_{j}^{(l)} (s,{\hat{X}}^{(t,x)}(s))\right) ds\right| ^2 \right] . \end{aligned}$$

Since $N+1\in \{j_1,\ldots ,j_l\}^c$, ${\hat{F}}_{N+1}^{(l)}(s,{\hat{X}}^{(t,x)}(s))\le \sum _{k\notin \{j_1,\ldots ,j_l\}}{\hat{F}}_{k}^{(l)}(s,{\hat{X}}^{(t,x)}(s))$, a.s.. This implies that

$$\begin{aligned}&{\mathbb {E}} \left[ \left| \int _t^Tw(s,{\hat{X}}^{(t,x)}(s))e^{\int _t^sh(u, {\hat{X}}^{(t,x)}(u))du}ds\right| ^2\right] \\&\quad \le 4C {\mathbb {E}} \left[ \left| \int _t^Te^{-\int _t^s\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}d\left( \int _t^s\sum _{j \notin \{j_1,\ldots ,j_l\}}{\hat{F}}_{j}^{(l)}(u,{\hat{X}}^{(t,x)}(u))du\right) \right| ^2\right] \\&\quad \le 4C\left\{ 1+\left| {\mathbb {E}} \left[ e^{-\int _t^T\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}\right] \right| ^2\right\} \\&\quad \le 4C, \end{aligned}$$

where $C>0$ is independent of (t, x). Thus we have verified the existence of a constant $C>0$, independent of (t, x), such that

$$\begin{aligned} \sup _{(t,x)\in [0,T]\times \mathbb {R}_+^{N+1}} {\mathbb {E}} \left[ \left| \int _t^Tw(s,{ \hat{X}}^{(t,x)}(s))e^{\int _t^sh(u,{\hat{X}}^{(t,x)}(u))du}ds\right| ^2\right] \le C<+\infty . \end{aligned}$$

This yields the uniform integrability of the family (101). It implies that the condition [A3e’] of Heath and Schweizer [24] holds. Using Theorem 1 of Heath and Schweizer [24], we conclude that Eq. (100) admits a unique classical solution u(t, x) on $[0,T]\times \mathbb {R}_+^{N+1}$.

We next prove the solution is nonnegative and bounded on $[0,T]\times \mathbb {R}_+^{N+1}$. Using the Feymann-Kac’s representation of the classical solution u(t, x), for $(t,x)\in [0,T]\times \mathbb {R}_+^{N+1}$,

$$\begin{aligned} u(t,x)&= {\mathbb {E}} \Bigg [\int _t^Te^{-\int _t^s\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}\nonumber \\&\quad \times \Bigg (\sum _{j\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_j^{(l)}(s,{\hat{X}}^{(t,x)}(s))g^{(l+1),j}(t,{\hat{X}}^{(t,x)} (s)+w_j)\nonumber \\&\qquad +\left\{ \sum _{i=1}^{{\bar{N}}} b_i(1-K_i^{(l+1),N+1})\big [F_{(1,1,1)i}(s,{\hat{X}}^{(t,x)}(s) \right. \nonumber \\&\left. \quad +\,w_{N+1},0^{j_1,\ldots ,j_l,N+1})-Z_i^{(l+1),N+1}K_i^{(l+1),N+1}\big ] \right\} _+\nonumber \\&\qquad \times {\hat{F}}_{N+1}^{(l)}(s,{\hat{X}}^{(t,x)}(s))\mathbf{1}_{j_1,\ldots ,j_l\ne N+1}\Bigg )ds \Bigg ]. \end{aligned}$$

(102)

If $N+1\in \{j_1,\ldots ,j_l\}$, then Eq. (102) reduces to

$$\begin{aligned} u(t,x)&= {\mathbb {E}} \left[ \int _t^Te^{-\int _t^s\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}\right. \\&\quad \left. \times \left( \sum _{j \notin \{j_1,\ldots ,j_l\}}{\hat{F}}_j^{(l)}(s, {\hat{X}}^{(t,x)}(s))g^{(l+1),j}(t,{\hat{X}}^{(t,x)}(s)+w_j)\right) ds\right] . \end{aligned}$$

Since the nonnegative function $g^{(l+1),j}(t,x)$ is bounded on $[0,T]\times \mathbb {R}_+^{N+1}$ by the inductive hypothesis, there exists a constant $C>0$ such that

$$\begin{aligned} 0\le u(t,x)&\le C\sum _{j\notin \{j_1,\ldots ,j_l\}} {\mathbb {E}} \left[ \int _t^T{\hat{F}}_j^{(l)} (s,{\hat{X}}^{(t,x)}(s))e^{-\int _t^s\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}ds\right] \\&=C\left\{ 1- {\mathbb {E}} \left[ e^{-\int _t^T\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}\right] \right\} . \end{aligned}$$

Obviously, the above inequality yields the existence of a constant $C>0$ such that $0\le u(t,x)\le C$ for all $(t,x)\in [0,T]\times \mathbb {R}_+^{N+1}$. Next, consider the case $N+1\notin \{j_1,\ldots ,j_l\}$. It follows from (102) that

$$\begin{aligned} u(t,x)&= {\mathbb {E}} \Bigg [\int _t^Te^{-\int _t^s\sum _{k\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}\\&\qquad \quad \times \Bigg (\sum _{j\notin \{j_1,\ldots ,j_l\}} {\hat{F}}_j^{(l)}(s,{\hat{X}}^{(t,x)}(s))g^{(l+1),j}(t, {\hat{X}}^{(t,x)}(s)+w_j)\\&\qquad +\left\{ \sum _{i=1}^{{\bar{N}}} b_i(1-K_i^{(l+1),N+1})\big [F_{(1,1,1)i}(t,{\hat{X}}^{(t,x)} (s) \right. \\&\left. \qquad +\,w_{N+1},0^{j_1,\ldots ,j_l,N+1})-Z_i^{(l+1),N+1}K_i^{ (l+1),N+1}\big ]\right\} _+ \\&\qquad \times {\hat{F}}_{N+1}^{(l)}(s,{\hat{X}}^{(t,x)}(s))\Bigg )ds \Bigg ]. \end{aligned}$$

Then there exists a constant $C>0$ such that

$$\begin{aligned} 0\le u(t,x)&\le C {\mathbb {E}} \left[ \int _t^Te^{-\int _t^s\sum _{k \notin \{j_1,\ldots ,j_l\}}{\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du} \right. \\&\left. \quad \times \left( \sum _{j\notin \{j_1,\ldots ,j_l\}}{\hat{F}}_j^{(l)}(s,{\hat{X}}^{(t,x)}(s)) +{\hat{F}}_{N+1}^{(l)}(s,{\hat{X}}^{(t,x)}(s))\right) ds \right] . \end{aligned}$$

Since $N+1\in \{j_1,\ldots ,j_l\}^c$, we have that

$$\begin{aligned} 0\le u(t,x)&\le 2C\left\{ 1- {\mathbb {E}} \left[ e^{-\int _t^T\sum _{k\notin \{j_1, \ldots ,j_l\}} {\hat{F}}_k^{(l)}(u,{\hat{X}}^{(t,x)}(u))du}\right] \right\} . \end{aligned}$$

The above inequality gives a constant $C>0$ such that $0\le u(t,x)\le C$ for all $(t,x)\in [0,T]\times \mathbb {R}_+^{N+1}$. This completes the proof of the theorem. $\square$

Proof of Lemma 3.8

It follows from (69) that

$$\begin{aligned}&\int _0^tD_{x}g(s,X(s),H(s))^{\top }\sigma (X(s))d{W}(s) \\&\quad =\int _0^tD_xg(s,X(s), H(s))^{\top }\sigma (X(s))({\hat{\lambda }}\Upsilon )(s,X(s),H(s))^{\top }ds \\&\quad \quad +\int _0^tL_{N+1}(H^{N+1}(s))\left\{ \sum _{i=1}^{{\bar{N}}}b_i (1-K_i(H^{N+1}(s)))F_{i}(t,X(s)+w_{N+1},H^{N+1}(s))\right\} _+ \\&\quad \quad \times (1-H_{N+1}(s)){\hat{F}}_{N+1}(s,X(s),H(s))ds+g(t,X(t), H(t))-g(0,X(0),H(0)) \\&\quad \quad +\sum _{j=1}^{N+1}\int _0^t\left[ g(s,{X}(s)+w_j,H^j(s))-g(s,{X}(s),H(s)) \right] \\&\qquad \times (1-H_j(s))\vartheta _j(X(s),H(s))X_j(s)ds \\&\quad \quad -\sum _{j=1}^{N+1}\left[ g(s,{X}(s^-)+w_j,H^j(s^-))-g(s,{X}(s^-),H(s^-)) \right] d{M}_j(s) \\&\quad =:\int _0^tD_xg(s,X(s),H(s))^{\top }\sigma (X(s))({\hat{\lambda }}\Upsilon ) (s,X(s),H(s))^{\top }ds+E(t). \end{aligned}$$

Then, for any $\varepsilon >0$,

$$\begin{aligned}&{\mathbb {E}} \left[ \left| \int _0^tD_{x}g(s,X(s),H(s))^{\top }\sigma (X(s))d{W}(s)\right| ^2\right] \\&\qquad = {\mathbb {E}} \left[ \left| \int _0^tD_xg(s,X(s),H(s))^{\top }\sigma (X(s)) ({\hat{\lambda }}\Upsilon )(s,X(s),H(s))^{\top }ds+E(t)\right| ^2\right] \\&\qquad \le (1+\varepsilon ) {\mathbb {E}} \left[ \left| \int _0^tD_xg(s,X(s),H(s))^{\top }\sigma (X(s)) ({\hat{\lambda }}\Upsilon )(s,X(s),H(s))^{\top }ds\right| ^2\right] \\&\qquad \quad +\left( 1+\frac{1}{\varepsilon }\right) {\mathbb {E}} [|E(t)|^2]\\&\qquad \le (1+\varepsilon ) {\mathbb {E}} \left[ \left( \int _0^t\left| ({\hat{\lambda }}\Upsilon ) (s,X(s),H(s))\right| ^2ds\right) \right. \\&\left. \qquad \quad \times \left( \int _0^t|D_xg(s,X(s),H(s))^{\top } \sigma (X(s))|^2ds\right) \right] \\&\qquad \quad +\left( 1+\frac{1}{\varepsilon }\right) {\mathbb {E}} [|E(t)|^2]. \end{aligned}$$

By the assumption of the lemma, we have that ${\mathbb {E}} [\int _0^T|({\hat{\lambda }}\Upsilon )(s,X(s),H(s))|^2ds]\le |{\hat{\lambda }}\Upsilon |_{\infty }^2T$. Since g(t, x, z) is the unique bounded classical solution of Eq. (59) by Theorem 3.5. Further, by Proposition 2.4 and Assumption (A3), there exists a constant $C=C(T)>0$ such that ${\mathbb {E}} [|E(T)|^2]\le C(T)+C(T) {\mathbb {E}} [\int _0^T\sum _{j=1}^{N+1}X_j^2(s)ds]$. This gives that

$$\begin{aligned}&{\mathbb {E}} \left[ \int _0^T\left| D_{x}g(s,X(s),H(s))^{\top }\sigma (X(s))\right| ^2ds\right] \le \left( 1+\frac{1}{\varepsilon }\right) C(T) \\&\qquad + (1+\varepsilon )|{\hat{\lambda }}\Upsilon |_{\infty }^2T {\mathbb {E}} \left[ \int _0^T \left| D_{x}g(s,X(s),H(s))^{\top }\sigma (X(s))\right| ^2ds\right] \\&\qquad +\left( 1+\frac{1}{\varepsilon }\right) C(T) {\mathbb {E}} \left[ \int _0^T \sum _{j=1}^{N+1}X_j^2(s)ds\right] . \end{aligned}$$

Using the condition $(1+\varepsilon )|{\hat{\lambda }}\Upsilon |_{\infty }^2T<1$ for some $\varepsilon >0$, we get the estimate (70).

$\square$