Mean–Variance Asset–Liability Management Problem Under Non-Markovian Regime-Switching Models

Abstract

In this paper, we study an asset–liability management problem under a mean–variance criterion with regime switching. Unlike previous works, the dynamics of assets and liability are described by non-Markovian regime-switching models in the sense that all the model parameters are predictable with respect to the filtration generated jointly by a Markov chain and a Brownian motion. The problem is solved with the aid of backward stochastic differential equations (BSDEs) and bounded mean oscillation martingales. An efficient strategy and an efficient frontier are obtained and represented by unique solutions to several relevant BSDEs. We show that the optimal capital structure can be achieved when the initial asset value is expressed by a linear combination of the initial liability and the expected surplus. It is further found that a mutual fund theorem holds not only for the efficient strategy, but also for the optimal capital structure.

Appendix: Solvability of (4.1) and (4.2)

Before proving Lemmas 4.1 and 4.2, we first consider a more general regime-switching BSDE of quadratic–exponential growth:

$$\begin{aligned} \left\{ \begin{array}{l} d Y (t) = - f (t, Y (t), Z (t), U (t)) d t + Z (t)^\top d W (t) + U (t)^\top d {\widetilde{\Phi }} (t) , \\ Y (T) = \xi , \end{array} \right. \end{aligned}$$

(A.1)

where the terminal value, $\xi : \Omega \rightarrow {{\mathbb {R}}},$ is an ${{\mathcal {F}}}_T$-measurable random variable; the driver f maps $[0, T] \times \Omega \times {{\mathbb {R}}} \times {{\mathbb {R}}}^D \times {{\mathbb {R}}}^N$ onto ${{\mathbb {R}}}$ and is ${{\mathcal {P}}} \otimes {{\mathcal {B}}} ({{\mathbb {R}}}) \otimes {{\mathcal {B}}} ({{\mathbb {R}}}^D) \otimes {{\mathcal {B}}} ({{\mathbb {R}}}^N)$-measurable.

Suppose that the driver f is of quadratic–exponential structure and is locally Lipschitz continuous as follows:

Assumption A.4

(i) For any $(y, z, u) \in {{\mathbb {R}}} \times {{\mathbb {R}}}^D \times {{\mathbb {R}}}^N$ with $u : = (u_1, u_2, \ldots , u_N)^\top ,$ there exist two constants $C > 0$ and $\gamma > 0$ such that

$$\begin{aligned}&-\, C ( 1 + |y| ) - \frac{\gamma }{2} |z|^2 - q (t, \gamma , - u) \le f (t, y, z, u) \le C ( 1 + |y| ) \nonumber \\&\quad + \frac{\gamma }{2} |z|^2 + q (t, \gamma , u) , \quad d t \otimes d {{\mathbb {P}}}\text{-a.e. }, \end{aligned}$$

(A.2)

where

$$\begin{aligned} q (t, \gamma , u) := \sum ^N_{j = 1} \frac{e^{\gamma u_j} - 1 - \gamma u_j}{\gamma } \lambda _j (t) ; \end{aligned}$$

(A.3)

(ii) For any $(y, z, u), (y^\prime , z^\prime , u^\prime ) \in {\mathbb R} \times {{\mathbb {R}}}^D \times {{\mathbb {R}}}^N$ with $u : = (u_1, u_2, \ldots , u_N)^\top$ and $u^\prime : = (u^\prime _1, u^\prime _2, \ldots , u^\prime _N)^\top ,$ satisfying $|y|, |u|, |y^\prime |, |u^\prime | \le K,$ there exists some positive constant $C_K$ depending on K such that

$$\begin{aligned}&|f (t, y, z, u) - f (t, y^\prime , z^\prime , u^\prime )| \le C_K \big ( |y - y^\prime | + (1 + |z| + |z^\prime |) |z - z^\prime | \nonumber \\&\quad + \langle \lambda (t), u - u^\prime \rangle \big ) , \quad d t \otimes d {{\mathbb {P}}}\text{-a.e. } \end{aligned}$$

(A.4)

Lemma A.1

Suppose that the terminal value $\xi$ is essentially bounded, i.e., $\xi \in {{\mathcal {L}}}^\infty _{{\mathbb {P}}} ({{\mathcal {F}}}_T; {{\mathbb {R}}}),$ and the driver f satisfies Assumption A.4. Then, the regime-switching BSDE (A.1) admits a unique solution $(Y, Z, U) \in {{\mathcal {S}}}^\infty _{{\mathbb {P}}} (0, T; {{\mathbb {R}}}) \times {{\mathcal {H}}}^2_{BMO_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^D ) \times \mathcal{J}^2_{BMO_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^N ).$ Furthermore, the last component of the solution is essentially bounded, i.e., $U \in {{\mathcal {J}}}^\infty _{{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^N ).$

Proof

Assumption A.4 states that the driver is of quadratic–exponential growth and satisfies the local Lipschitz condition. It can be shown as in [9] that the BSDE (A.1) admits a unique solution $(Y, Z, U) \in \mathcal{S}^\infty _{{\mathbb {P}}} (0, T; {{\mathbb {R}}}) \times \mathcal{H}^2_{{}_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^D ) \times {{\mathcal {J}}}^2_{{}_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^N ).$ It follows from $\Delta Y (t) = U (t)^\top \Delta \Phi (t)$ that

$$\begin{aligned} \Vert U \Vert _{{{\mathcal {J}}}^\infty _{{\mathbb {P}}} (0, T; {{\mathbb {R}}}^N)} \le 2 N \max _{\begin{array}{c} i, j = 1, 2, \ldots , N \\ i \ne j \end{array}} \{ |a_{ij}| \} \Vert Y \Vert _{{{\mathcal {S}}}^\infty _{{\mathbb {P}}} (0, T; {{\mathbb {R}}})} < \infty . \end{aligned}$$

Thus, $U \in {{\mathcal {J}}}^\infty _{{\mathbb {P}}} (0, T; {{\mathbb {R}}}^N).$ Indeed, replacing the general random measure in [9] by the random measure $\Phi$ related to the chain defined in Sect. 2 of our paper, the proof can be conducted essentially the same as Lemmas 3.1–3.3, Proposition 3.1 and Theorem 4.1 therein. So we do not repeat it here. $\square$

We now discuss the solvability of a quadratic–exponential BSDE associated with $(\xi , g),$ that is,

$$\begin{aligned} \left\{ \begin{array}{l} d Y (t) = - g (t, Y (t), Z (t), U (t)) d t + Z (t)^\top d W (t) + U (t)^\top d {\widetilde{\Phi }} (t) , \\ Y (T) = \xi , \end{array} \right. \end{aligned}$$

(A.5)

where the terminal value, $\xi : \Omega \rightarrow {{\mathbb {R}}},$ is an ${{\mathcal {F}}}_T$-measurable random variable; the driver g maps $[0, T] \times \Omega \times {{\mathbb {R}}} \times {{\mathbb {R}}}^D \times {{\mathbb {R}}}^N$ onto ${{\mathbb {R}}}$ and is ${{\mathcal {P}}} \otimes {{\mathcal {B}}} ({{\mathbb {R}}}) \otimes {{\mathcal {B}}} ({{\mathbb {R}}}^D) \otimes {{\mathcal {B}}} ({{\mathbb {R}}}^N)$-measurable. Suppose that the driver g satisfies

Assumption A.5

The dependence of the driver g on different arguments is separable in the sense that for any $(y, z, u) \in {{\mathbb {R}}} \times {\mathbb R}^D \times {{\mathbb {R}}}^N,$ there exists c,  being either 0 or 1,  such that g can be decomposed as

$$\begin{aligned} g (t, y, z, u) = h (t, y, z) + c q (t, 1, u) , \quad d t \otimes d {{\mathbb {P}}}\text{-a.e. }, \end{aligned}$$

(A.6)

where the random mapping h is ${{\mathcal {P}}} \otimes {{\mathcal {B}}} ({\mathbb R}) \otimes {{\mathcal {B}}} ({{\mathbb {R}}}^D)/{{\mathcal {B}}} ({\mathbb R})$-measurable and the random mapping q is defined by (A.3). In addition, h satisfies the following quadratic growth and Lipschtiz conditions: (i) for any $(y, z) \in {{\mathbb {R}}} \times {{\mathbb {R}}}^D,$ there exists $C > 0$ such that

$$\begin{aligned} \big | h (t, y, z) \big | \le C \big ( 1 + |y| + |z|^2 \big ) , \quad d t \otimes d {{\mathbb {P}}}\text{-a.e. } ; \end{aligned}$$

(A.7)

(ii) for any $(y, z), (y^\prime , z^\prime ) \in {{\mathbb {R}}} \times {{\mathbb {R}}}^D$, there exists $C > 0$ such that

$$\begin{aligned}&|h (t, y, z) - h (t, y^\prime , z^\prime )|\nonumber \\&\le C ( |y - y^\prime | + (1 + |z| + |z^\prime |) |z - z^\prime | ) , \quad d t \otimes d {{\mathbb {P}}}\text{-a.e. } \end{aligned}$$

(A.8)

It will turn out that the quadratic–exponential BSDE (A.5) is closely related to the BSRE (4.1). The next lemma validates that (A.5) has a unique solution.

Lemma A.2

Suppose that the terminal value $\xi$ is essentially bounded, i.e., $\xi \in {{\mathcal {L}}}^\infty _{{\mathbb {P}}} ({{\mathcal {F}}}_T; {{\mathbb {R}}}),$ and the driver g satisfies Assumption A.5. Then, the regime-switching BSDE (A.5) admits a unique solution $(Y, Z, U) \in {{\mathcal {S}}}^\infty _{{\mathbb {P}}} (0, T; {{\mathbb {R}}}) \times {{\mathcal {H}}}^2_{BMO_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^D ) \times {{\mathcal {J}}}^2_{BMO_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^N ).$ Furthermore, the last component of the solution is essentially bounded, i.e., $U \in {{\mathcal {J}}}^\infty _{{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^N ).$

Proof

By definition, we know that $\lambda _j (t) \ge 0,$$d t \otimes d {{\mathbb {P}}}$-a.e., for each $j = 1, 2, \ldots , N.$ Thus, $q (t, \gamma , u) \ge 0,$$d t \otimes d {{\mathbb {P}}}$-a.e., for any $\gamma > 0$ and $u \in {{\mathbb {R}}}^N.$ Therefore, when $c = 0,$ the driver g satisfies Assumption A.4 if we take $\gamma = 2 C.$ An application of Lemma A.1 gives the desired results immediately.

It remains to consider the case with $c = 1.$ Differentiating q with respect to $\gamma$ gives

$$\begin{aligned} \frac{\partial }{\partial \gamma } q (t, \gamma , u) = \sum ^N_{j = 1} \frac{(\gamma u_j - 1) e^{\gamma u_j} + 1}{\gamma ^2} \lambda _j (t) . \end{aligned}$$

(A.9)

For each $j = 1, 2, \ldots , N,$ if $\gamma u_j \ge 1,$ then $\frac{(\gamma u_j - 1) e^{\gamma u_j} + 1}{\gamma ^2} > 0;$ otherwise, if $\gamma u_j < 1,$ then $\frac{(\gamma u_j - 1) e^{\gamma u_j} + 1}{\gamma ^2} > \frac{- (1 - \gamma u_j)/(1 - \gamma u_j) + 1}{\gamma ^2} = 0.$ We obtain that $\frac{\partial q}{\partial \gamma } \ge 0,$ thereby q is always an increasing function of $\gamma .$ Hence, when $c = 1,$ the driver g also satisfies the quadratic–exponential growth condition with $\gamma = (2 C) \vee 1,$ i.e., (i) of Assumption  A.4. Indeed, for any $(y, z, u) \in {{\mathbb {R}}} \times {{\mathbb {R}}}^D \times {{\mathbb {R}}}^N,$ there exists $C > 0$ such that $d t \otimes d {{\mathbb {P}}}$-a.e.

$$\begin{aligned}&\quad -\, C (1 + |y| ) - \frac{(2 C) \vee 1}{2} |z|^2 - q (t, (2 C) \vee 1,-u)\nonumber \\\le & {} - \, C (1 + |y| ) - \frac{(2 C) \vee 1}{2} |z|^2 \nonumber \\\le & {} g (t, y, z, u) \\\le & {} C (1 + |y| ) + \frac{(2 C) \vee 1}{2} |z|^2 + q (t, (2 C) \vee 1, u) . \nonumber \end{aligned}$$

(A.10)

On the other hand, we apply the mean-value theorem to verify the local Lipschitz condition, i.e., (ii) of Assumption A.4. For any $(y, z, u), (y^\prime , z^\prime , u^\prime ) \in {{\mathbb {R}}} \times {{\mathbb {R}}}^D \times {{\mathbb {R}}}^N$ satisfying $|y|, |u|, |y^\prime |, |u^\prime | \le K,$ we can find $u^{\prime \prime } : = ( u_1^{\prime \prime }, u_2^{\prime \prime }, \ldots , u_N^{\prime \prime } )^\top ,$ where $u_j^{\prime \prime } \in (u_j \wedge u_j^\prime , u_j \vee u_j^\prime )$ and $|u_j^{\prime \prime }| \le K,$ for each $j = 1, 2, \ldots , N$ (in particular, if $u_j = u_j^\prime ,$ we take $u_j^{\prime \prime } = u_j = u_j^\prime$), such that

$$\begin{aligned}&\quad |g (t, y, z, u) - g (t, y^\prime , z^\prime , u^\prime )| \nonumber \\\le & {} C ( |y - y^\prime | + (1 + |z| + |z^\prime |) |z - z^\prime | ) + | \langle \lambda (t), u - u^\prime \rangle | + \bigg | \sum ^N_{j = 1} ( e^{u_j} - e^{u_j^\prime } ) \lambda _j (t) \bigg | \nonumber \\= & {} C ( |y - y^\prime | + (1 + |z| + |z^\prime |) |z - z^\prime | ) + | \langle \lambda (t), u - u^\prime \rangle | + \bigg | \sum ^N_{j = 1} e^{u_j^{\prime \prime }} ( u_j - u_j^\prime ) \lambda _j (t) \bigg | \nonumber \\= & {} C \vee (e^K + 1) \cdot ( |y - y^\prime | + (1 + |z| + |z^\prime |) |z - z^\prime | + | \langle \lambda (t), u - u^\prime \rangle | ) . \end{aligned}$$

(A.11)

Setting $C_K = C \vee (e^K + 1)$ in (A.11) confirms that the local Lipschitz condition is satisfied. Consequently, when $c = 1,$ the driver also satisfies Assumption A.4. Applying Lemma  A.1 again, we obtain the desired results. $\square$

By a similar truncation technique used in [12], we can relate the BSRE (4.1) to a quadratic BSDE of structure as (A.5), and apply Lemma A.1 to discuss its solvability.

Proof of Lemma 4.1

Let $\delta$ be a positive constant. We consider an auxiliary quadratic BSDE:

$$\begin{aligned} \left\{ \begin{aligned} d Y_1 (t)&= - \bigg \{ [ 2 r (t) - B (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} B (t) ] Y_1 (t)\\&\quad - 2 B (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} \sigma (t) Z_1 (t) \\&\quad - \frac{Z_1 (t)^\top \sigma (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} \sigma (t) Z_1 (t)}{Y_1 (t) \vee \delta } \bigg \} d t\\&\quad + Z_1 (t)^\top d W (t) + U_1 (t)^\top d {\widetilde{\Phi }} (t) , \, t \in [0, T] , \\ Y_1 (T)&= 1 , \end{aligned} \right. \end{aligned}$$

(A.12)

which is in fact a truncated version of the BSRE (4.1). Clearly, (A.12) is a regime-switching BSDE satisfying the quadratic–exponential growth structure in Assumption  A.5 with $c = 0.$ By Lemma A.2, we know that (A.12) admits a unique solution $(Y^\delta _1, Z^\delta _1, U^\delta _1) \in {{\mathcal {S}}}^\infty _{\mathbb P} (0, T; {{\mathbb {R}}}) \times {{\mathcal {H}}}^2_{_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^D ) \times \mathcal{J}^2_{_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^N )$ and, furthermore, $U^\delta _1 \in {{\mathcal {J}}}^\infty _{{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^N ).$ Note that the unique solution $(Y^\delta _1, Z^\delta _1, U^\delta _1)$ is parameterized by $\delta .$

Next we claim that the first component of the solution $Y_1^\delta$ is bounded above zero and the bound does not depend on our choice of $\delta .$ From the boundedness of parameters and $Z^\delta _1 \in {{\mathcal {H}}}^2_{_{{\mathbb {P}}}} ( 0, T; {\mathbb R}^D )$, we have $\big [2 B^\top (\sigma \sigma ^\top )^{-1} \sigma + \frac{(Z_1^\delta )^\top \sigma ^\top (\sigma \sigma ^\top )^{-1} \sigma }{Y_1^\delta \vee \delta } \big ] \in \mathcal{H}^2_{_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^D ).$ Thus, ${{\mathcal {E}}} \left( - \left[ 2 B^\top (\sigma \sigma ^\top )^{-1} \sigma + \frac{(Z_1^\delta )^\top \sigma ^\top (\sigma \sigma ^\top )^{-1} \sigma }{Y_1^\delta \vee \delta } \right] \bullet W\right)$ is a uniformly integrable, $({{\mathbb {F}}}, {{\mathbb {P}}})$-martingale. This allows us to define a probability measure ${{\mathbb {Q}}}$ equivalent to ${{\mathbb {P}}}$ as follows:

$$\begin{aligned} \frac{d {{\mathbb {Q}}}}{d {{\mathbb {P}}}} \bigg |_{{{\mathcal {F}}}_T} := {{\mathcal {E}}} \bigg (- \bigg [ 2 B^\top (\sigma \sigma ^\top )^{-1} \sigma + \frac{(Z_1^\delta )^\top \sigma ^\top (\sigma \sigma ^\top )^{-1} \sigma }{Y_1^\delta \vee \delta } \bigg ] \bullet W \bigg ) (T) .\nonumber \\ \end{aligned}$$

(A.13)

Under ${{\mathbb {Q}}},$ the process

$$\begin{aligned}&W^{{\mathbb {Q}}} (t) := W (t) + \int _0^t \bigg [ 2 \sigma (s)^\top (\sigma (s) \sigma (s)^\top )^{-1} B (s) \nonumber \\&\quad +\, \frac{\sigma (s)^\top (\sigma (s) \sigma (s)^\top )^{-1} \sigma (s) Z_1^\delta (s)}{Y_1^\delta (s) \vee \delta } \bigg ] d s \end{aligned}$$

(A.14)

is a D-dimensional, standard Brownian motion; the probability law of the Markov chain remains unchanged, and thus, ${\widetilde{\Phi }}^{{\mathbb {Q}}} := {\widetilde{\Phi }}$ is still a jump martingale. Then under ${{\mathbb {Q}}},$ the truncated BSDE (A.12) can be written as

$$\begin{aligned} \left\{ \begin{aligned} d Y_1 (t)&= - [ 2 r (t) - B (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} B (t) ] Y_1 (t) d t + Z_1 (t)^\top d W^{{\mathbb {Q}}} (t)\\&\quad + U_1 (t)^\top d {\widetilde{\Phi }}^{{\mathbb {Q}}} (t) , \, t \in [0, T] , \\ Y_1 (T)&= 1 . \end{aligned} \right. \end{aligned}$$

(A.15)

Applying the similar argument to prove the Feymann–Kac formula for BSDEs (refer to Proposition 2.2 in [6]), we can express $Y_1$ by the following expectation:

$$\begin{aligned} Y_1 (t) = {{\mathbb {E}}}^{{\mathbb {Q}}} \bigg [ \exp \bigg ( \int ^T_t \big [ 2 r (s) - B (s)^\top (\sigma (s) \sigma (s)^\top )^{-1} B (s) \big ] d s \bigg ) \bigg | {{\mathcal {F}}}_t \bigg ] \ge c ,\nonumber \\ \end{aligned}$$

(A.16)

where c is a positive constant independent of $\delta$ and depending only on T and the bounds of model coefficients.

By setting $\delta = c,$ we see that the parameter $\delta$ can be dropped in the truncated BSDE (A.12) because $\delta$ is the a priori lower bound for $Y_1.$ Therefore, all the results obtained in the above derivations do not depend on our choice of $\delta .$ Particularly, the unique solution $(Y_1, Z_1, U_1)$ to the truncated BSDE (A.12) is exactly the unique solution to the BSRE (4.1). This confirms the existence and uniqueness of the solution and also verifies Assertion (i). Moreover, we know

$$\begin{aligned} \frac{Z_1 (t)}{Y_1 (t)} \le \frac{1}{c} Z_1 (t) , \quad \frac{U_1 (t)}{Y_1 (t)} \le \frac{1}{c} U_1 (t) , \quad d t \otimes d {{\mathbb {P}}}\text{-a.e. } \end{aligned}$$

(A.17)

Hence, Assertion (ii) follows immediately.

We now show Assertion (iii). Let us consider the following BSDE:

$$\begin{aligned} \left\{ \begin{aligned} d Y_1^\prime (t)&= - \bigg \{ [ 2 r (t) - B (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} B (t) ]\\&\quad - 2 B (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} \sigma (t) Z^\prime _1 (t) \\&\quad - (Z_1^\prime (t))^\top \sigma (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} \sigma (t) Z_1^\prime (t) + \frac{1}{2} | Z_1^\prime (t) |^2\\&\quad + \sum \nolimits ^N_{j = 1} ( e^{U^\prime _1 (t)} - 1 - U^\prime _j (t) ) \bigg \} d t \\&\quad + Z_1^\prime (t) d W (t) + U_1^\prime (t) d {\widetilde{\Phi }} (t) , \, t \in [0, T] , \\ Y_1^\prime (T)&= 0 . \end{aligned} \right. \end{aligned}$$

(A.18)

Indeed, this is also a quadratic–exponential BSDE satisfying Assumption A.5, but with $c = 1.$ Then by Lemma  A.2, the BSDE (A.18) admits a unique solution $(Y_1^\prime , Z_1^\prime , U_1^\prime ) \in \mathcal{S}^\infty _{{\mathbb {P}}} (0, T; {{\mathbb {R}}}) \times \mathcal{H}^2_{_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^D ) \times {{\mathcal {J}}}^2_{_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^N ),$ and furthermore, $U_2^\prime \in \mathcal{J}^\infty _{_{{\mathbb {P}}}} ( 0, T; {\mathbb R}^N ).$

By Itô’s formula, we can verify that $(Y_1, Z_1, U_1)$ and $(Y_1^\prime , Z_1^\prime , U_1^\prime )$ satisfy the relationships: $Y_1 (t) = e^{Y_1^\prime (t)}$, $Z_1 (t) = e^{Y_1^\prime (t)} Z_1^\prime (t)$ and $1 + \frac{U_{1j (t)}}{Y_1 (t)} = e^{U_{1j}^\prime (t)},$ for each $j = 1, 2, \ldots , N$ and any $t \in [0, T].$ From the existence and uniqueness of $(Y_1, Z_1, U_1)$ and $(Y_1^\prime , Z_1^\prime , U_1^\prime )$ and the boundedness of $U_{1j}^\prime ,$ we obtain $1 + \frac{U_{1j} (t)}{Y_1 (t)} = e^{U_{1j}^\prime (t)} \ge \epsilon ,$ for any $t \in [0, T ],$ some $\epsilon > 0$ and each $j = 1, 2, \ldots , N.$ This leads to Assertion (iii). $\square$

Remark A.1

We relate the BSRE (4.1) to the quadratic BSDE (A.18) by an exponential transformation. This ingenious technique leads us to obtain $\frac{U_{1j}}{Y_1} \ge - 1 + \epsilon ,$ which seems to be new among the literature of BSDEs with jumps. More importantly, as remarked in the conclusion, this finding plays an indispensable role in various places of the paper.

Proof of Lemma 4.2

Since all the model coefficients are bounded and $\frac{Z_1}{Y_1} \in {{\mathcal {H}}}^2_{_{{\mathbb {P}}}} ( 0, T; {{\mathbb {R}}}^D ),$ we have that the stochastic integral $\left[ \frac{Z_1^\top \sigma _\perp }{Y_1} - B^\top (\sigma \sigma ^\top )^{-1} \sigma \right] \bullet W$ is a $\text{ BMO }_{{\mathbb {P}}}$-martingale. Then we can define a probability measure ${{\mathbb {Q}}}_1$ equivalent to ${{\mathbb {P}}}$ as

$$\begin{aligned} \frac{d {{\mathbb {Q}}}_1}{d {{\mathbb {P}}}} \bigg |_{{{\mathcal {F}}}_T} = \Theta _1 := {{\mathcal {E}}} \bigg ( \bigg [ \frac{Z_1^\top \sigma _\perp }{Y_1} - B^\top (\sigma \sigma ^\top )^{-1} \sigma \bigg ] \bullet W \bigg ) (T) . \end{aligned}$$

(A.19)

Under ${{\mathbb {Q}}}_1,$ the process

$$\begin{aligned} W^{{{\mathbb {Q}}}_1} (t) := W (t) - \int ^t_0 \bigg [ \frac{\sigma _\perp (s) Z_1 (s)}{Y_1 (s)} - \sigma (s)^\top (\sigma (s) \sigma (s)^\top )^{-1} B (s) \bigg ] d s\nonumber \\ \end{aligned}$$

(A.20)

is a D-dimensional standard Brownian motion, and ${\widetilde{\Phi }}^{{{\mathbb {Q}}}_1} (t) := {\widetilde{\Phi }} (t)$ has the same probability law as under ${{\mathbb {P}}}$ and is still an ${\mathbb R}^N$-valued jump martingale associated with the chain.

Since $\frac{U_1}{Y_1} \in {{\mathcal {J}}}^2_{_{{\mathbb {P}}}} (0, T; {\mathbb {R}}^N)$ and the probability measure ${\mathbb {Q}}_1$ is constructed as Eq. (A.19), it can be shown as Theorem 3.3 in [15] that $\frac{U_1}{Y_1} \in \mathcal{J}^2_{_{{\mathbb {Q}}_1}} (0, T; {\mathbb {R}}_N).$ By Lemma A.2 and the equivalence of ${{\mathbb {Q}}}_1$ and ${{\mathbb {P}}},$ we have that $\frac{U_{1j} (t)}{Y_1 (t)} \ge - 1 + \epsilon ,$$d t \otimes d {{\mathbb {Q}}}_1$-a.e. Hence, the techniques of [15] and the jump sizes $\Delta \Phi (t)$ can be used to show that ${{\mathcal {E}}} \left( \frac{U_1^\top }{Y_1} \bullet {\widetilde{\Phi }}^{{{\mathbb {Q}}}_1} \right)$ is a uniformly integrable, $({{\mathbb {F}}}, {{\mathbb {Q}}}_1)$-martingale. Therefore, we can define a probability measure ${{\mathbb {Q}}}_2$ equivalent to ${{\mathbb {Q}}}_1$ as

$$\begin{aligned} \frac{d {{\mathbb {Q}}}_2}{d {{\mathbb {Q}}}_1} \bigg |_{{{\mathcal {F}}}_T} = \Theta _2 := {{\mathcal {E}}} \bigg ( \frac{U_1^\top }{Y_1} \bullet {\widetilde{\Phi }}^{{{\mathbb {Q}}}_1} \bigg ) (T) . \end{aligned}$$

(A.21)

Under ${{\mathbb {Q}}}_2,$ the ${{\mathbb {Q}}}_1$-Brownian motion $W^{{{\mathbb {Q}}}_1}$ is still a D-dimensional standard Brownian motion, which is denoted by $W^{{{\mathbb {Q}}}_2} (t) := W^{{\mathbb Q}_1} (t),$ and the process ${\widetilde{\Phi }}^{{{\mathbb {Q}}}_2} (t) := {\widetilde{\Phi }}^{{{\mathbb {Q}}}_1} (t) - \int ^t_0 \frac{\Lambda (s) U_1 (s)}{Y_1 (s)} d s$ is an ${{\mathbb {R}}}^N$-valued jump martingale associated with the chain. So, the ${\mathbb Q}_2$-intensity of $\Phi$ is given by an ${{\mathbb {R}}}^N$-valued process:

$$\begin{aligned}&\lambda ^{{\mathbb {Q}}_2} (t) := \bigg [ \lambda _1 (t) \bigg ( 1 + \frac{U_{11} (t)}{Y_1 (t)} \bigg ) , \lambda _2 (t) \bigg ( 1 + \frac{U_{12} (t)}{Y_1 (t)} \bigg ) , \ldots , \lambda _N (t) \bigg ( 1 + \frac{U_{1N} (t)}{Y_1 (t)} \bigg ) \bigg ]^\top .\nonumber \\ \end{aligned}$$

(A.22)

Therefore, under ${{\mathbb {Q}}}_2$ the linear BSDE (4.2) becomes

$$\begin{aligned} \left\{ \begin{array}{l} d Y_2 (t) = r (t) Y_2 (t) d t + Z_2 (t)^\top d W^{{{\mathbb {Q}}}_2} (t) + U_2 (t)^\top d {\widetilde{\Phi }}^{{{\mathbb {Q}}}_2} (t) , \, t \in [0, T] , \\ Y_2 (T) = - L (T) - c . \end{array} \right. \quad \end{aligned}$$

(A.23)

Moreover, the ${{\mathbb {Q}}}_2$-dynamics of the liability value process is given by

$$\begin{aligned} d L (t)= & {} L (t) \bigg \{ \bigg [ \rho (t) + \frac{Z_1 (t)^\top \sigma _\perp (t) \beta (t)}{Y_1 (t)} - B (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} \sigma (t) \beta (t) \bigg ] d t \nonumber \\&\quad +\, \beta (t)^\top d W^{{{\mathbb {Q}}}_2} (t) \bigg \} . \end{aligned}$$

(A.24)

Thus,

$$\begin{aligned} L (T)= & {} l_0 \exp \bigg ( \int ^T_0 \bigg [ \rho (t) + \frac{Z_1 (t)^\top \sigma _\perp (t) \beta (t)}{Y_1 (t)} \nonumber \\&\quad - B (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} \sigma (t) \beta (t) \bigg ] d t \bigg ) {{\mathcal {E}}} \left( \beta ^\top \bullet W^{{{\mathbb {Q}}}_2} \right) (T) . \end{aligned}$$

(A.25)

By using Theorem 3.3 in [15], we obtain $\frac{Z_1}{Y_1} \in {{\mathcal {H}}}^2_{_{\mathbb Q_2}} ( 0, T; {{\mathbb {R}}}^D ).$ Therefore, we can define

$$\begin{aligned} \frac{d {{\mathbb {Q}}}_3}{d {{\mathbb {Q}}}_2} \bigg |_{{{\mathcal {F}}}_T} := {{\mathcal {E}}} \bigg ( - \frac{Z_1^\top \sigma _\perp }{Y_1} \bullet W^{{{\mathbb {Q}}}_2} \bigg ) (T) . \end{aligned}$$

(A.26)

Under ${{\mathbb {Q}}}_3,$ the process $W^{{{\mathbb {Q}}}_3} (t) := W^{{{\mathbb {Q}}}_2} (t) + \int ^t_0 \frac{\sigma _\perp (s) Z_1 (s)}{Y_1 (s)} d s$ is a D-dimensional standard Brownian motion, and the dynamics of the liability value process is described by

$$\begin{aligned} d L (t) = L (t) \big \{ \big [ \rho (t) - B (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} \sigma (t) \beta (t) \big ] d t + \beta (t)^\top d W^{{{\mathbb {Q}}}_3} (t) \big \} ,\nonumber \\ \end{aligned}$$

(A.27)

which has the following solution

$$\begin{aligned} L (T) = l_0 \exp \bigg ( \int ^T_0 \big [ \rho (t) - B (t)^\top (\sigma (t) \sigma (t)^\top )^{-1} \sigma (t) \beta (t) \big ] d t \bigg ) {{\mathcal {E}}}\left( \beta ^\top \bullet W^{{{\mathbb {Q}}}_3} \right) (T) .\nonumber \\ \end{aligned}$$

(A.28)

Clearly,

$$\begin{aligned} \frac{d {{\mathbb {Q}}}_2}{d {{\mathbb {Q}}}_3} \bigg |_{{{\mathcal {F}}}_T} = \mathcal{E} \bigg ( \frac{Z_1^\top \sigma _\perp }{Y_1} \bullet W^{{\mathbb Q}_3} \bigg ) (T) . \end{aligned}$$

(A.29)

Again by Theorem 3.3 in [15], we have $\frac{Z_1}{Y_1} \in {{\mathcal {H}}}^2_{_{{\mathbb {Q}}_3}} ( 0, T; {{\mathbb {R}}}^D ).$ Moreover, it follows from the reverse Hölder inequality (see [14]) that ${{\mathcal {E}}} \left( \frac{Z_1^\top \sigma _\perp }{Y_1} \bullet W^{{{\mathbb {Q}}}_3} \right) (T) \in \mathcal{L}^{p_1}_{{{\mathbb {Q}}}_3} ({{\mathcal {F}}}_T; {{\mathbb {R}}}),$ for some $p_1 > 1.$ Since $\beta$ is bounded, we have that ${{\mathcal {E}}} ( \beta ^\top \bullet W^{{{\mathbb {Q}}}_3} )$ is in ${{\mathcal {S}}}^p_{{{\mathbb {Q}}}_3} (0, T; {{\mathbb {R}}}),$ for any $p > 1.$ Choosing $q_1 > 1$ such that $1/p_1 + 1/q_1 = 1$ and using Hölder’s inequality, we derive

$$\begin{aligned} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ |L (T)|^p \big ]= & {} {\mathbb E}^{{{\mathbb {Q}}}_3} \bigg [ {{\mathcal {E}}} \bigg ( \frac{Z_1^\top \sigma _\perp }{Y_1} \bullet W^{{{\mathbb {Q}}}_3} \bigg ) (T) \cdot |L (T)|^p \bigg ] \nonumber \\\le & {} K {{\mathbb {E}}}^{{{\mathbb {Q}}}_3} \bigg [ {{\mathcal {E}}} \bigg ( \frac{Z_1^\top \sigma _\perp }{Y_1} \bullet W^{{{\mathbb {Q}}}_3} \bigg )(T) \cdot \big \{ {{\mathcal {E}}} ( \beta ^\top \bullet W^{{{\mathbb {Q}}}_3} ) (T) \big \}^p \bigg ] \nonumber \\\le & {} K \bigg ( {{\mathbb {E}}}^{{{\mathbb {Q}}}_3} \bigg [ \bigg \{ \mathcal{E} \bigg ( \frac{Z_1^\top \sigma _\perp }{Y_1} \bullet W^{{\mathbb Q}_3} \bigg ) (T) \bigg \}^{p_1} \bigg ] \bigg )^\frac{1}{p_1}\nonumber \\&\quad \bigg ( {{\mathbb {E}}}^{{{\mathbb {Q}}}_3} \bigg [ \big \{ {{\mathcal {E}}} ( \beta ^\top \bullet W^{{{\mathbb {Q}}}_3} ) (T) \big \}^{q_1p} \bigg ] \bigg )^\frac{1}{q_1} \nonumber \\\le & {} K \bigg ( {{\mathbb {E}}}^{{{\mathbb {Q}}}_3} \bigg [ \bigg \{ \mathcal{E} \bigg ( \frac{Z_1^\top \sigma _\perp }{Y_1} \bullet W^{{\mathbb Q}_3} \bigg ) (T) \bigg \}^{p_1} \bigg ] \bigg )^\frac{1}{p_1} \nonumber \\&\quad \left\| {{\mathcal {E}}} ( \beta ^\top \bullet W^{{{\mathbb {Q}}}_3} ) \right\| ^p_{{{\mathcal {S}}}^{q_1 p}_{{{\mathbb {Q}}}_3} (0, T; {{\mathbb {R}}})} < \infty . \end{aligned}$$

(A.30)

Thus, $L (T) \in {{\mathcal {L}}}^p_{{{\mathbb {Q}}}_2} ({{\mathcal {F}}}_T; {\mathbb R}),$ for any $p \ge 1.$

Define

$$\begin{aligned} y_2 (t) : = - {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ e^{- \int ^T_0 r (s) d s} ( L (T) + c ) \big | {{\mathcal {F}}}_t \big ] . \end{aligned}$$

(A.31)

Clearly, $\{ y_2 (t) | t \in [0, T] \}$ is a square-integrable, $({\mathbb {F}}, {\mathbb {Q}}_2)$-martingale. Applying the martingale representation theorem (refer to [7]), we obtain that there exists a unique pair $(z_2, u_2),$ where $z_2 \in \mathcal{H}^2_{{{\mathbb {Q}}}_2} (0, T; {{\mathbb {R}}}^D)$ and $u_2 \in \mathcal{J}^2_{{{\mathbb {Q}}}_2} (0, T; {{\mathbb {R}}}^N),$ such that

$$\begin{aligned} y_2 (t) = y_2 (0) + \int ^t_0 z_2 (s)^\top d W^{{\mathbb {Q}}_2} (s) + \int ^t_0 u_2 (s)^\top d {\widetilde{\Phi }}^{{\mathbb {Q}}_2} (s) , \quad \forall t \in [0, T] .\nonumber \\ \end{aligned}$$

(A.32)

Then,

$$\begin{aligned} y_2 (t) = y_2 (T) - \int ^T_t z_2 (s)^\top d W^{{\mathbb {Q}}_2} (s) - \int ^T_t u_2 (s)^\top d {\widetilde{\Phi }}^{{\mathbb {Q}}_2} (s) . \end{aligned}$$

(A.33)

Denote by

$$\begin{aligned} (Y_2 (t), Z_2 (t), U_2 (t)) : = e^{\int ^t_0 r (s) d s} (y_2 (t), z_2 (t), u_2 (t)) , \quad \forall t \in [0, T] . \end{aligned}$$

(A.34)

Therefore, an application of the product rule can verify that $(Y_2, Z_2, U_2)$ is the unique solution to (A.23).

Moreover, the first component of the unique solution $(Y_2, Z_2, U_2)$ can be expressed by

$$\begin{aligned} Y_2 (t) = - {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ e^{- \int ^T_t r (s) d s} ( L (T) + c ) \big | {{\mathcal {F}}}_t \big ] . \end{aligned}$$

(A.35)

By Doob’s maximal inequality and Jensen’s inequality, we derive that for any $p \ge 2,$

$$\begin{aligned} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \sup _{t \in [0, T]} |Y_2 (t)|^p \bigg ]\le & {} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \sup _{t \in [0, T]} \bigg | {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ ( L (T) + c ) \big | {{\mathcal {F}}}_t \big ] \bigg |^p \bigg ] \nonumber \\\le & {} \bigg ( \frac{p}{p - 1} \bigg )^p \sup _{t \in [0, T]} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \bigg | {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ ( L (T) + c ) \big | {{\mathcal {F}}}_t \big ] \bigg |^p \bigg ] \nonumber \\\le & {} \bigg ( \frac{p}{p - 1} \bigg )^p \sup _{t \in [0, T]} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ \big | L (T) + c \big |^p \big ] \nonumber \\\le & {} K \big \{ 1 + {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ |L (T)|^p \big ] \big \} < \infty . \end{aligned}$$

(A.36)

That is, $Y_2 \in {{\mathcal {S}}}^p_{{{\mathbb {Q}}}_2} (0, T; {{\mathbb {R}}}),$ for any $p \ge 2.$ From (A.23), we know

$$\begin{aligned} \int ^t_0 Z_2 (s)^\top d W^{{{\mathbb {Q}}}_2} (s) + \int ^t_0 U_2 (s)^\top d {\widetilde{\Phi }}^{{{\mathbb {Q}}}_2} (s) = Y_2 (t) - Y_2 (0) - \int ^t_0 r (s) Y_2 (s) d s .\nonumber \\ \end{aligned}$$

(A.37)

For any $p > 2,$ applying the Burkholder–Davis–Gundy inequality and $Y_2 \in {{\mathcal {S}}}^p_{{{\mathbb {Q}}}_2} (0, T; {{\mathbb {R}}}),$ we can find $c_p > 0$ such that

$$\begin{aligned} c_p {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \bigg ( \int ^T_0 |Z_2 (s)|^2 d s \bigg )^{\frac{p}{2}} \bigg ]\le & {} c_p {{\mathbb {E}}}^{{\mathbb Q}_2} \bigg [ \bigg ( \int ^T_0 |Z_2 (s)|^2 d s + \langle U_2^\top \bullet {\widetilde{\Phi }}^{{{\mathbb {Q}}}_2} \rangle (T) \bigg )^{\frac{p}{2}} \bigg ] \nonumber \\\le & {} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \sup _{t \in [0, T]} \bigg | \int ^t_0 Z_2 (s)^\top d W^{{{\mathbb {Q}}}_2} (s) + \int ^t_0 U_2 (s)^\top d {\widetilde{\Phi }}^{{{\mathbb {Q}}}_2} (s) \bigg |^p \bigg ] \nonumber \\\le & {} K {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \sup _{t \in [0, T]} | Y_2 (t) |^p \bigg ] < \infty \end{aligned}$$

(A.38)

and

$$\begin{aligned} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \bigg ( \sum ^N_{j = 1} \int ^T_0 |{{\mathscr {U}}}_{2j} (s)|^2 d s \bigg )^{\frac{p}{2}} \bigg ]= & {} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \bigg ( \langle U_2^\top \bullet {\widetilde{\Phi }}^{{{\mathbb {Q}}}_2} \rangle (T) \bigg )^{\frac{p}{2}} \bigg ] \\\le & {} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \bigg ( \int ^T_0 |Z_2 (s)|^2 d s + \langle U_2^\top \bullet {\widetilde{\Phi }}^{{\mathbb Q}_2} \rangle (T) \bigg )^{\frac{p}{2}} \bigg ] < \infty . \nonumber \end{aligned}$$

(A.39)

Thus, we obtain that $Z_2 \in {{\mathcal {H}}}^p_{{{\mathbb {Q}}}_2} (0, T; {{\mathbb {R}}}^D)$ and ${{\mathscr {U}}}_2 \in {{\mathcal {J}}}^p_{{{\mathbb {Q}}}_2} (0, T; {{\mathbb {R}}}^N).$

Recalling (A.19) and (A.21), we have

$$\begin{aligned} \frac{d {{\mathbb {P}}}}{d {{\mathbb {Q}}}_1} \bigg |_{{{\mathcal {F}}}_T} = \Gamma _1 := {{\mathcal {E}}} \bigg ( - \bigg [ \frac{Z_1^\top \sigma _\perp }{Y_1} - B^\top (\sigma \sigma ^\top )^{-1} \sigma \bigg ] \bullet W^{{{\mathbb {Q}}}_1} \bigg ) (T) \end{aligned}$$

(A.40)

and

$$\begin{aligned} \frac{d {{\mathbb {Q}}}_1}{d {{\mathbb {Q}}}_2} \bigg |_{{{\mathcal {F}}}_T} = \Gamma _2 := {{\mathcal {E}}} \bigg ( - \frac{U_1^\top }{Y_1} \bullet {\widetilde{\Phi }}^{{{\mathbb {Q}}}_2} \bigg ) (T) . \end{aligned}$$

(A.41)

Since $\left[ \frac{Z_1^\top \sigma _\perp }{Y_1} - B^\top (\sigma \sigma ^\top )^{-1} \sigma \right] \bullet W^{{{\mathbb {Q}}}_1}$ is a $\text{ BMO }_{{\mathbb {Q}}_1}$-martingale, it follows from [14] that the following reverse Hölder inequality holds:

$$\begin{aligned} {{\mathbb {E}}}^{{{\mathbb {Q}}}_1} \big [ \Gamma _1^{p_2} \big ] < \infty , \quad \text{ for } \text{ some } \ p_2 > 1 . \end{aligned}$$

(A.42)

On the other hand, since $\frac{U_1}{Y_1}$ is essentially bounded under both ${{\mathbb {Q}}}_1$ and ${{\mathbb {Q}}}_2,$ we have that the following inequalities hold:

$$\begin{aligned} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ \Gamma _2^p \big ]< \infty , \quad {{\mathbb {E}}}^{{{\mathbb {Q}}}_1} \big [ \Theta _2^p \big ] < \infty , \quad \text{ for } \text{ any } \ p > 1 . \end{aligned}$$

(A.43)

Using Young’s inequality and taking $p_2, q_2 > 1$ such that $1 / \sqrt{p_2} + 1 / \sqrt{q_2} = 1,$ we derive that for any $p \ge 1,$

$$\begin{aligned} {{\mathbb {E}}} \bigg [ \sup _{t \in [0, T]} |Y_2 (t)|^p \bigg ]= & {} {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \Gamma _1 \Gamma _2 \sup _{t \in [0, T]} |Y_2|^p \bigg ] \nonumber \\\le & {} K \bigg \{ {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ \Gamma ^{\sqrt{p_2}}_1 \big ] + {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \Gamma ^{\sqrt{q_2}}_2 \sup _{t \in [0, T]} |Y_2|^{\sqrt{q_2}p} \bigg ] \bigg \} \nonumber \\\le & {} K \bigg \{ {{\mathbb {E}}}^{{{\mathbb {Q}}}_1} \big [ \Theta _2 \Gamma ^{\sqrt{p_2}}_1 \big ] + {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ \Gamma ^{2\sqrt{q_2}}_2 \big ] + {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \sup _{t \in [0, T]} |Y_2|^{2\sqrt{q_2}p} \bigg ] \bigg \} \nonumber \\\le & {} K \bigg \{ {{\mathbb {E}}}^{{{\mathbb {Q}}}_1} \big [ \Gamma ^{p_2}_1 \big ] + {{\mathbb {E}}}^{{{\mathbb {Q}}}_1} \big [ \Theta ^{\sqrt{q_2}}_2 \big ] + {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \big [ \Gamma ^{2\sqrt{q_2}}_2 \big ]\nonumber \\&+\, {{\mathbb {E}}}^{{{\mathbb {Q}}}_2} \bigg [ \sup _{t \in [0, T]} |Y_2|^{2\sqrt{q_2}p} \bigg ] \bigg \} < \infty . \end{aligned}$$

(A.44)

This validates that $Y_2 \in {{\mathcal {S}}}^p_{{{\mathbb {P}}}} (0, T; {{\mathbb {R}}}).$ Similarly, we can show that $Z_2 \in \mathcal{H}^p_{{{\mathbb {P}}}} (0, T; {{\mathbb {R}}}^D)$ and ${{\mathscr {U}}}_2 \in {{\mathcal {J}}}^p_{{{\mathbb {P}}}} (0, T; {{\mathbb {R}}}^N).$ This confirms that the unique solution of the linear BSDE (4.2) is p-integrable under the original measure ${{\mathbb {P}}}$ in the sense of Assertions (i) and (ii). $\square$

Remark A.2

Though similar linear BSDEs with unbounded coefficients were studied by [23, 24], the proof of Lemma 4.2 has some merits to be mentioned. The solvability result of the linear BSDE in [23] was incomplete, as the integrability of the solution to the linear BSDE therein was not proved under the original measure. Let us compare the result in [24] with ours. Firstly, [24] only obtained the square-integrability of the solution, while our paper shows that the solution to (4.2) is p-integrable. Secondly, [24]’s result relied on the boundedness of terminal value; in our paper the terminal value of the linear BSDE is integrable, but in general is unbounded. Thirdly, [24] imposed some restrictive assumptions on the model structure so that a density process is strictly positive and the variance optimal martingale measure (VMM) can be well-defined by this density process. In general, it is difficult to verify these restrictive assumptions. Although [24] provided more explicit assumptions in some special cases, they anyhow undermined the generality of the result in [24]. In our paper, those restrictive assumptions are not needed. In fact, the ${\mathbb Q}_2$-measure used in the proof of our Lemma 4.2 is the so-called VMM, which is defined by

$$\begin{aligned} \frac{d {{\mathbb {Q}}}_2}{d {{\mathbb {P}}}} \bigg |_{{{\mathcal {F}}}_T} = \Theta _1 \cdot \Theta _2 = {{\mathcal {E}}} \bigg ( \bigg [ \frac{Z_1^\top \sigma _\perp }{Y_1} - B^\top (\sigma \sigma ^\top )^{-1} \sigma \bigg ] \bullet W + \frac{U_1^\top }{Y_1} \bullet {\widetilde{\Phi }} \bigg ) (T) .\nonumber \\ \end{aligned}$$

(A.45)

The property of BMO martingales guarantees that the density $\Theta _1 \cdot \Theta _2$ is strictly positive and ${{\mathbb {Q}}}_2$ is a well-defined probability measure. This last point may be due to the absence of jumps in the stock price models in our paper. Refer to Remark 3.2 for the comparison of the different model structures of [24] and our paper.