Robust Portfolio Optimization with Multi-Factor Stochastic Volatility

Abstract

This paper studies a robust portfolio optimization problem under a multi-factor volatility model. We derive optimal strategies analytically under the worst-case scenario with or without derivative trading in complete and incomplete markets and for assets with jump risk. We extend our study to the case with correlated volatility factors and propose an analytical approximation for the robust optimal strategy. To illustrate the effects of ambiguity, we compare our optimal robust strategy with the strategies that ignore the information of uncertainty, and provide the welfare analysis. We also discuss how derivative trading affects the optimal strategies. Finally, numerical experiments are provided to demonstrate the behavior of the optimal strategy and the utility loss.

Appendices

Appendix A Proof of Proposition 3.1

Solving optimal problem (15) with respect to $e^S_j$ and $e_j^V$, we have

$$\begin{aligned} (e^S_j)^*=\varPsi _j^S(x\beta ^S_jJ_x+\rho _j\sigma _jJ_{v_j})\sqrt{v_j},\quad (e^V_j)^*=\varPsi _j^V(x\beta ^V_jJ_x+\sqrt{1-\rho _j^2}\sigma _jJ_{v_j})\sqrt{v_j}.\nonumber \\ \end{aligned}$$

(54)

Substituting (54) into (15), we obtain the following equation:

$$\begin{aligned} \begin{aligned}&\sup _{\beta ^S_i,\beta ^V_i} \bigg \{J_t+x\big (r+\sum _{j=1}^{2}(\beta ^{S}_j\lambda _jv_j +\beta ^{V}_j\mu _jv_j\big )J_x \\&\quad +\frac{1}{2}x^2\sum _{j=1}^{2}[(\beta ^{S}_j)^2+(\beta ^{V}_j)^2]v_jJ_{xx} +\sum _{j=1}^{2}\big (\kappa _j(\theta _j-v_j)\big )J_{v_j}\\&\quad +\frac{1}{2}\sum _{j=1}^{2}\sigma _j^2v_jJ_{v_jv_j}+\sum _{j=1}^{2}\sigma _jv_jx\big (\beta ^S_j\rho _j+\beta ^V_j\sqrt{1-\rho _j^2}\big )J_{v_jx}\\&\quad -\sum _{j=1}^{2}\frac{1}{2}\varPsi _j^Sv_j\left[ x^2(\beta ^S_j)^2J_x^2\right. \left. +2x\beta ^S_j\rho _j\sigma _jJ_xJ_{V_j}+\rho _j^2\sigma _j^2J^2_{v_j}\right] \\&\quad -\sum _{j=1}^{2}\frac{1}{2}\varPsi _j^Sv_j\left[ x^2(\beta ^V_j)^2J_x^2+2x\beta ^V_j\sqrt{1-\rho _j^2}\sigma _jJ_xJ_{v_j}+(1-\rho _j^2)\sigma _j^2J^2_{v_j}\right] \bigg \}=0. \end{aligned} \end{aligned}$$

(55)

Assuming that the solution J is of the form $J(t,x,v_1,v_2)=\frac{x^{1-\gamma }}{1-\gamma }\exp (H_1(T-t)v_1+H_2(T-t)v_2+h(T-t))$ and choosing

$$\begin{aligned} \varPsi ^S_j=\frac{\phi _j^S}{(1-\gamma )J(t,x,v_1,v_2)}, \quad \varPsi ^V_j=\frac{\phi _j^V}{(1-\gamma )J(t,x,v_1,v_2)}, \quad \phi _j^S, \ \phi _j^V>0, \end{aligned}$$

Equation (55) yields

$$\begin{aligned} \begin{aligned}&\sup _{\beta ^S_i,\beta ^V_i} \bigg \{ -h'-\sum _{j=1}^{2}(H_j)'v_j+(1-\gamma )\big (r+\sum _{j=1}^{2}(\beta ^{S}_j\lambda _jv_j +\beta ^{v}_j\mu _jv_j\big )\big )\\&\quad -\frac{\gamma (1-\gamma )}{2}\sum _{j=1}^{2}[(\beta ^{S}_j)^2+(\beta ^{v}_j)^2]v_j\\&\quad +\sum _{j=1}^{2}\big (\kappa _j(\theta _j-v_j))H_j +\frac{1}{2}\sum _{j=1}^{2}\sigma _j^2v_j(H_j)^2+\sum _{j=1}^{2}\sigma _jv_jx\big (\beta ^S_j\rho _j+\beta ^V_j\sqrt{1-\rho _j^2}\big )H_j\\&\quad -\sum _{j=1}^{2}\frac{1}{2}(1-\gamma )\phi _j^Sv_j\left[ (\beta ^S_j)^2+2\beta ^S_j\rho _j\sigma _j\frac{H_j}{1-\gamma }+\rho _j^2\sigma _j^2\frac{(H_j)^2}{(1-\gamma )^2}\right] \\&\quad -\sum _{j=1}^{2}\frac{1}{2}(1-\gamma )\phi _j^Vv_j\left[ (\beta ^V_j)^2+2\beta ^V_j\sqrt{1-\rho _j^2}\sigma _j\frac{H_j}{1-\gamma }+(1-\rho _j^2)\sigma _j^2\frac{(H_j)^2}{(1-\gamma )^2}\right] \bigg \}=0, \end{aligned} \end{aligned}$$

(56)

where $h'$ represents the time derivative of function h(t). From (56), we derive the optimal exposures as

$$\begin{aligned} (\beta ^S_j)^*= & {} \frac{\lambda _j}{\gamma +\phi ^S_j}+\frac{(1-\gamma -\phi ^S_j)\sigma _j\rho _j}{(1-\gamma )(\gamma +\phi ^S_j)}H_j(T-t),\nonumber \\ (\beta ^V_j)^*= & {} \frac{\mu _j}{\gamma +\phi ^V_j}+\frac{(1-\gamma -\phi ^V_j)\sigma _j\sqrt{1-\rho _j^2}}{(1-\gamma )(\gamma +\phi ^V_j)}H_j(T-t). \end{aligned}$$

(57)

To determine $H_1$ and $H_2$ and h, we substitute (57) into (56) to obtain

$$\begin{aligned} \begin{aligned}&-h'-\sum _{j=1}^{2}(H_j)'v_j+(1-\gamma )( r+\sum _{j=1}^{2}\big (\kappa _j(\theta _j-v_j))H_j) \\&+\frac{1}{2}\sum _{j=1}^{2}\sigma _j^2v_j(H_j)^2-\sum _{j=1}^{2}\frac{1}{2}\phi _j^Sv_j\rho _j^2\sigma _j^2\frac{(H_j)^2}{1-\gamma }\\&-\sum _{j=1}^{2}\frac{1}{2}\phi _j^Vv_j(1-\rho _j^2)\sigma _j^2\frac{(H_j)^2}{1-\gamma }+\sum _{j=1}^{2}\left[ \frac{(1-\gamma )\lambda _1^2}{2(\gamma +\phi _j^S)}+\frac{\lambda _1(1-\gamma -\phi _j^S)\sigma _j\rho _iH_j}{\gamma +\phi ^S_j} \right. \\&\left. +\frac{(1-\gamma -\phi _i^S)^2\sigma ^2_i\rho ^2_i(H_j)^2}{2(1-\gamma )(\gamma +\phi ^S_i)} \right] v_j\\&+\sum _{j=1}^{2}\left[ \frac{(1-\gamma )\lambda _2^2}{2(\gamma +\phi _1^V)}+\frac{\lambda _2(1-\gamma -\phi _j^V)\sigma _j\sqrt{1-\rho ^2_i}H_j}{\gamma +\phi ^S_j} \right. \\&\left. +\frac{(1-\gamma -\phi _j^V)^2\sigma ^2_j(1-\rho ^2_j)(H_j)^2}{2(1-\gamma )(\gamma +\phi ^V_j)} \right] v_j=0. \end{aligned} \end{aligned}$$

(58)

Comparing the terms concerned with $v_j$, we get the following Riccati equations:

$$\begin{aligned} \left\{ \begin{aligned}&H_j'=a_jH_j+b_j(H_j)^2+c_j,\quad H_j(0)=0,\ j=1, 2, \\&h'=\kappa _1\theta _1H_1+\kappa _2\theta _2H_2+(1-\gamma )r,\quad h(0)=0. \end{aligned} \right. \end{aligned}$$

(59)

Let $d_j=\sqrt{a_j^2-4b_jc_j}$  , where

$$\begin{aligned} a_j= & {} -\kappa _j+\frac{\lambda _1(1-\gamma -\phi _j^S)\sigma _j\rho _j}{\gamma +\phi _j^S}+\frac{\lambda _2(1-\gamma -\phi _j^V)\sigma _j}{\gamma +\phi _j^V}\sqrt{1-\rho _j^2},\\ b_j= & {} \frac{\sigma _j^2}{2}-\frac{\phi ^S_j\rho _j^2\sigma _j^2}{2(1-\gamma )}-\frac{\phi ^V_j(1-\rho _j^2)\sigma _j^2}{2(1-\gamma )}+\frac{(1-\gamma -\phi _j^S)\sigma ^2_j\rho ^2_j}{2(1-\gamma )(\gamma +\phi _j^S)}\\&+\frac{(1-\gamma -\phi _j^V)^2\sigma ^2_j(1-\rho _j^2)}{2(1-\gamma )(\gamma +\phi _j^V)},\\ c_j= & {} \frac{(1-\gamma )\lambda _1^2}{2(\gamma +\phi ^S_j)}+\frac{(1-\gamma )\lambda _2^2}{2(\gamma +\phi ^V_j)}. \end{aligned}$$

Functions $H_j$ and h can be obtained from (59), as shown in (17). $\square$

Appendix B Proof of Proposition 3.2

Proof

It is sufficient to show that the Novikov’s condition is satisfied for the worst-case probability measure, that is,

$$\begin{aligned} \mathrm {E}^{{\mathbb {P}}}\left[ \exp \left\{ \frac{1}{2}\int _{0}^{T}((e^S_1(t))^*)^2+((e^S_2(t))^*)^2+((e^V_1(t))^*)^2+((e^V_2(t))^*)^2\mathrm {d}t\right\} \right] <\infty . \end{aligned}$$

Applying (19) in Proposition 3.1, one obtains $\mathrm {E}^{{\mathbb {P}}}\big [\exp \big \{\frac{1}{2}\int _{0}^{T}\big (K_1(T-t)V_1(t)+K_2(T-t)V_2(t)\big )\mathrm {d}t\big \}\big ]<\infty ,$ where

$$\begin{aligned} K_j(T-t)= & {} \left( \frac{\lambda _j}{\gamma +\phi ^S_j}+\frac{\sigma _j\rho _j H_j(T-t)}{(1-\gamma )(\gamma +\phi ^S_j)}\right) ^2(\phi ^S_j)^2\\&+\left( \frac{\mu _j}{\gamma +\phi ^V_j}+\frac{\sigma _j\sqrt{1-\rho _j^2} H_j(T-t)}{(1-\gamma )(\gamma +\phi ^V_j)}\right) ^2(\phi ^V_j)^2,\ j= 1, 2. \end{aligned}$$

Define $k_1=\sup _{t\in [0,T]}K_1(T-t)$ and $k_2=\sup _{t\in [0,T]}K_2(T-t)$, and then,

$$\begin{aligned} \begin{aligned}&\mathrm {E}^{{\mathbb {P}}}\left[ \exp \left\{ \frac{1}{2}\int _{0}^{T}\left( K_1(T-t)V_1(t)+K_2(T-t)V_2(t)\right) \mathrm {d}t\right\} \right] \\&\quad< \sum \limits _{i=1}^2\mathrm {E}^{{\mathbb {P}}}\left[ \exp \left\{ \frac{k_i}{2}\int _{0}^{T}V_i(t)\mathrm {d}t\right\} \right] <\infty . \end{aligned} \end{aligned}$$

(60)

The first inequity in (60) holds because $V_1$ and $V_2$ are independent, while the second holds if $k_1\le \frac{\kappa _1^2}{\sigma _1^2}$ and $k_2\le \frac{\kappa _2^2}{\sigma _2^2}$ (see [35]). Recall that $\gamma >1$, $\lambda _j > 0$, $\rho _j<0$ and $\mu _j<0$. Thus, $K_j$ reaches its maximum value at $t=T$ because $H_j$ is maximum at $t= T$. Therefore,

$$\begin{aligned} k_1= & {} (\phi _1^S)^2\frac{\lambda _1^2}{(\gamma +\phi ^S_1)^2}+(\phi _1^V)^2\frac{\mu ^2_1}{(\gamma +\phi ^V_1)^2}, \\ k_2= & {} (\phi _2^S)^2\frac{\lambda ^2_2}{(\gamma +\phi ^S_2)^2}+(\phi _2^V)^2\frac{\mu ^2_2}{(\gamma +\phi ^V_2)^2}, \end{aligned}$$

which yield the desired results immediately. $\square$

Appendix C Proof of Proposition 3.3

Proof

In the incomplete market, solving the optimal problem (20)–(21), we obtain the general optimal wealth invested in stock

$$\begin{aligned} \pi ^S=\frac{\sum _{j=1}^{2}\left[ (1-\gamma )v_j(\lambda _j+\rho _j\sigma _j{\bar{H}}_j)-\phi _j^S\rho _j\sigma _jv_j{\bar{H}}_j\right] }{\sum _{j=1}^{2}(1-\gamma )v_j(\gamma +\phi _j^S)}. \end{aligned}$$

(61)

Substituting (61) into (21), we obtain the following general HJB equation:

$$\begin{aligned}&{\bar{h}}'+\sum _{j=1}^{2}({\bar{H}}_j)'v_j-(1-\gamma )\big (r+\sum _{j=1}^{2}\pi ^S\lambda _jv_j \big )\nonumber \\&\quad +\frac{\gamma (1-\gamma )}{2}\sum _{j=1}^{2}(\pi ^S)^2v_j+\sum _{j=1}^{2}\big (\kappa _j(\theta _j-v_j)){\bar{H}}_j \nonumber \\&\quad -\frac{1}{2}\sum _{j=1}^{2}\sigma _j^2v_j({\bar{H}}_j)^2-\sum _{j=1}^{2}\sigma _jv_jx\rho _j{\bar{H}}_j\pi ^S\nonumber \\&\quad +\sum _{j=1}^{2}\frac{1}{2}(1-\gamma )\phi _j^Sv_j\left[ (\pi ^S)^2+2\pi ^S({\bar{H}}_j)\rho _j\sigma _j\frac{{\bar{H}}_j}{1-\gamma }\right. \nonumber \\&\quad \left. +\rho _j^2\sigma _j^2\frac{({\bar{H}}_j)^2}{(1-\gamma )^2}\right] \nonumber \\&\quad +\sum _{j=1}^{2}\frac{1}{2}(1-\gamma )\phi _j^Vv_j(1-\rho _j^2)\sigma _j^2\frac{({\bar{H}}_j)^2}{(1-\gamma )^2}=0. \end{aligned}$$

(62)

If there only exists one risk factor, say, $W_1$, from (61), we have

$$\begin{aligned} \pi ^S=\frac{\lambda _1}{\gamma +\phi ^S_1}+\frac{(1-\gamma -\phi ^S_1)\sigma _1\rho _1}{(1-\gamma )(\gamma +\phi ^S_1)}{\bar{H}}_1(\tau ). \end{aligned}$$

Alternatively, if the only risk factor is $W_2$, $\pi ^S=\frac{\lambda _2}{\gamma +\phi ^S_2}+\frac{(1-\gamma -\phi ^S_2)\sigma _2\rho _2}{(1-\gamma )(\gamma +\phi ^S_2)}{\bar{H}}_2(\tau ).$

Further, if the risk factors are the same, $W_1=W_2$, then

$$\begin{aligned} \pi ^S=\frac{\lambda _1}{\gamma +\phi ^S_1}+\frac{(1-\gamma -\phi ^S_1)\sigma _1\rho _1}{(1-\gamma )(\gamma +\phi ^S_1)}{\bar{H}}_1(\tau )=\frac{\lambda _2}{\gamma +\phi ^S_2}+\frac{(1-\gamma -\phi ^S_2)\sigma _2\rho _2}{(1-\gamma )(\gamma +\phi ^S_2)}{\bar{H}}_2(\tau ). \end{aligned}$$

Therefore, under the circumstances that there are only one single risk or two identical risk factors we can write, without loss of generality,

$$\begin{aligned} \begin{aligned} \pi ^S=\pi _j({\bar{H}}_j) = \frac{\lambda _j}{\gamma +\phi ^S_j}+\frac{(1-\gamma -\phi ^S_j)\sigma _j\rho _j}{(1-\gamma )(\gamma +\phi ^S_j)}{\bar{H}}_j(\tau ),\ j = 1, 2. \end{aligned} \end{aligned}$$

Substituting $\pi ^S=\pi _j({\bar{H}}_j)$ into (21), we obtain the following HJB equation:

$$\begin{aligned} \begin{aligned} 0=&{\bar{h}}'+\sum _{j=1}^{2}({\bar{H}}_j)'v_j-(1-\gamma )\big (r+\sum _{j=1}^{2}\pi _j({\bar{H}}_j)\lambda _jv_j \big )\\&+\frac{\gamma (1-\gamma )}{2}\sum _{j=1}^{2}\pi ^2_j({\bar{H}}_j)v_j+\sum _{j=1}^{2}\big (\kappa _j(\theta _j-v_j)){\bar{H}}_j\\&-\frac{1}{2}\sum _{j=1}^{2}\sigma _j^2v_j({\bar{H}}_j)^2-\sum _{j=1}^{2}\sigma _jv_jx\rho _j{\bar{H}}_j\pi _j({\bar{H}}_j)\\&+\sum _{j=1}^{2}\frac{1}{2}(1-\gamma )\phi _j^Vv_j(1-\rho _j^2)\sigma _j^2\frac{({\bar{H}}_j)^2}{(1-\gamma )^2}\\&+\sum _{j=1}^{2}\frac{1}{2}(1-\gamma )\phi _j^Sv_j\left[ (\pi _j({\bar{H}}_j))^2+2\pi _j({\bar{H}}_j)\rho _j\sigma _j\frac{{\bar{H}}_j}{1-\gamma }+\rho _j^2\sigma _j^2\frac{({\bar{H}}_j)^2}{(1-\gamma )^2}\right] . \end{aligned} \end{aligned}$$

(63)

By matching coefficients, we obtain the ordinary differential equations for the solution of ${\bar{H}}_j$ and ${\bar{h}}$ as follows:

$$\begin{aligned} \left\{ \begin{aligned} {\bar{H}}'_j&=(1-\gamma )\pi _j({\bar{H}}_j)\lambda _j -\frac{\gamma (1-\gamma )}{2}\pi ^2_j({\bar{H}}_j)-\kappa _j{\bar{H}}_j +\frac{1}{2}\sigma _j^2({\bar{H}}_j)^2+(1-\gamma )\sigma _j\rho _j{\bar{H}}_j\pi _j({\bar{H}}_j)\\&-\frac{1}{2}(1-\gamma )\phi _j^S\left[ (\pi _j({\bar{H}}_j))^2+2\pi _j({\bar{H}}_j)\rho _j\sigma _j\frac{{\bar{H}}_j}{1-\gamma }+\rho _j^2\sigma _j^2\frac{({\bar{H}}_j)^2}{(1-\gamma )^2}\right] \\&-\frac{1}{2}(1-\gamma )\phi _j^V(1-\rho _j^2)\sigma _j^2\frac{({\bar{H}}_j)^2}{(1-\gamma )^2},\quad&{\bar{H}}_j(0)=0,\\ {\bar{h}}'&=\kappa _1\theta _1{\bar{H}}_1+\kappa _2\theta _2{\bar{H}}_2+(1-\gamma )r,\quad&{\bar{h}}(0)=0. \end{aligned} \right. \end{aligned}$$

(64)

The worst-case measures can be computed directly by (24), and thus, the proof is complete.$\square$

Appendix D Proof of Proposition 3.4

Proof

Without loss of generality, we assume the indirect utility function of all suboptimal strategy $\varPi$ as

$$\begin{aligned} J^{\varPi }=\frac{x^{1-\gamma }}{1-\gamma }\exp \left( H^{\varPi }_1(\tau )v_1+H^{\varPi }_2(\tau )v_2+h^{\varPi }(\tau )\right) . \end{aligned}$$

(65)

For mathematical tractability, we choose

$$\begin{aligned} \displaystyle \varPsi ^S_j=\frac{{\widetilde{\phi }}_j^S}{(1-\gamma )J(t,x,v_1,v_2)} \quad \text{ and } \quad \varPsi ^V_j=\frac{{\widetilde{\phi }}_j^V}{(1-\gamma )J(t,x,v_1,v_2)}, \end{aligned}$$

where the ambiguity aversion parameters ${\widetilde{\phi }}_j^S$, ${\widetilde{\phi }}_j^V>0$.

The functions $H^{\varPi }_1$, $H^{\varPi }_2$ and $h^{\varPi }$ in (65) should satisfy the HJB PDE (26), which leads to

$$\begin{aligned} \begin{aligned} 0=&-(h^{\varPi })'-\sum _{j=1}^{2}(H^{\varPi }_j)'v_j+(1-\gamma )\big (r+\sum _{j=1}^{2}(\beta ^{S}_j\lambda _jv_j \\&+\beta ^{v}_j\mu _jv_j\big )\big ) -\frac{\gamma (1-\gamma )}{2}\sum _{j=1}^{2}[(\beta ^{S}_j)^2+(\beta ^{v}_j)^2]v_j+\sum _{j=1}^{2}\big (\kappa _j(\theta _j-v_j))H^{\varPi }_j\\&+\frac{1}{2}\sum _{j=1}^{2}\sigma _j^2v_j(H^{\varPi }_j)^2+\sum _{j=1}^{2}\sigma _jv_jx\big (\beta ^S_j\rho _j+\beta ^V_j\sqrt{1-\rho _j^2}\big )H^{\varPi }_j\\&-\sum _{j=1}^{2}\frac{1}{2}(1-\gamma ){\widetilde{\phi }}_j^Sv_j\left[ (\beta ^S_j)^2+2\beta ^S_j\rho _j\sigma _j\frac{H^{\varPi }_j}{1-\gamma }+\rho _j^2\sigma _j^2\frac{(H^{\varPi }_j)^2}{(1-\gamma )^2}\right] \\&-\sum _{j=1}^{2}\frac{1}{2}(1-\gamma ){\widetilde{\phi }}_j^Vv_j\left[ (\beta ^V_j)^2+2\beta ^V_j\sqrt{1-\rho _j^2}\sigma _j\frac{H^{\varPi }_j}{1-\gamma }+(1-\rho _j^2)\sigma _j^2\frac{(H^{\varPi }_j)^2}{(1-\gamma )^2}\right] . \end{aligned} \end{aligned}$$

(66)

Comparing the terms with and without multiplier $v_j$, one can certainly obtain the following Riccati equations:

$$\begin{aligned} \left\{ \begin{aligned}&(H^{\varPi }_j)'=a_jH^{\varPi }_j+b_j(H^{\varPi }_j)^2+c_j,\quad&H^{\varPi }_j(0)=0,\ j=1, 2, \\&(h^{\varPi })'=\kappa _1\theta _1H^{\varPi }_1+\kappa _2\theta _2H^{\varPi }_2+(1-\gamma )r,\quad&h^{\varPi }(0)=0, \end{aligned} \right. \end{aligned}$$

(67)

where

$$\begin{aligned} \begin{aligned} a_j=&-\kappa _j+\sigma _j(1-\gamma )(\beta ^S_j\rho +\beta ^V_j\sqrt{1-\rho ^2})-{\widetilde{\phi }}^S_j\beta ^S_j\rho \sigma _j-{\widetilde{\phi }}^V_j\beta ^V_j\sqrt{1-\rho ^2}\sigma _j,\\ b_j=&\frac{\sigma _j^2}{2}-\frac{{\widetilde{\phi }}^S_j\rho _j^2\sigma _j^2}{2(1-\gamma )}-\frac{{\widetilde{\phi }}^V_j(1-\rho _j^2)\sigma _j^2}{2(1-\gamma )},\\ c_j=&(1-\gamma )(\beta ^S_j\lambda _1+\beta ^V_j\lambda _2)-\frac{1}{2}\gamma (1-\gamma )((\beta ^S_j)^2+(\beta ^V_j)^2)\\&- \frac{(1-\gamma )(\beta ^S_j)^2{\widetilde{\phi }}^S_j}{2}-\frac{(1-\gamma )(\beta ^V_j)^2{\widetilde{\phi }}^V_j}{2}. \end{aligned} \end{aligned}$$

$\square$

Appendix E Proof of Proposition 3.5

Proof

Recall that the utility loss $L^\varPi =1-\exp \big \{\frac{1}{1-\gamma }\big [(H_1^\varPi -H_1)v_1+(H_2^\varPi -H_2)v_2+(h^\varPi -h)\big ]\big \}$. If the investor chooses strategy $\varPi _1$, then $H_2^{\varPi _1}=H_2$ since the strategy does not affect on the uncertainty about the second component of the ambiguity. Assume that $\beta ^S_1=p^S_1+p^S_2{\widehat{H}}_1$ and $\beta ^V_1=p^V_1+p^V_2{\widehat{H}}_1$, where $p^S_j$ and $p^V_j$$(j=1,2)$ are constants. Let ${\widehat{H}}_1$ be the function satisfying the equation $({\widehat{H}}_1)'={\widehat{a}}{\widehat{H}}_1+{\widehat{b}}\left( {\widehat{H}}_1\right) ^2+{\widehat{c}}$, where ${\widehat{a}},{\widehat{b}}$ and ${\widehat{c}}$ are all constants, and we obtain the following system of equations in terms of $H^{\varPi }_1$ and ${\widehat{H}}_1$:

$$\begin{aligned} \left\{ \begin{aligned}&(H^{\varPi }_1)'=P_1+P_2{\widehat{H}}_1+P_3(H_1^{\varPi })^2+P_4H_1^{\varPi }+P_5{\widehat{H}}_1H_1^{\varPi }+P_6(H_1^{\varPi })^2, \quad&H_1^{\varPi }(0)=0,\\&({\widehat{H}}_1)'={\widehat{a}}{\widehat{H}}_1+{\widehat{b}}{\widehat{H}}_1+{\widehat{c}}, \quad&{\widehat{H}}_1(0)=0, \end{aligned}\right. \end{aligned}$$

(68)

where

$$\begin{aligned} \left\{ \begin{aligned} P_1&=(1-\gamma )(\lambda _1p_1^S+\lambda _2p^V_2-\frac{1}{2}(\gamma +{\widehat{\phi }}_1^S))(p_1^S)^2-\frac{1}{2}(\gamma +{\widehat{\phi }}_1^V))(p_1^V)^2,\\ P_2&=(1-\gamma )(\lambda _1p_1^S+\lambda _2p^V_2-p_1^Sp_2^S(\gamma +{\widehat{\phi }}_1^S)-p_1^Vp_2^V(\gamma +{\widehat{\phi }}_1^V)),\\ P_3&=-\frac{1}{2}(1-\gamma )((\gamma +{\widehat{\phi }}_1^S)(p_2^S)^2+(\gamma +{\widehat{\phi }}_1^V)(p_2^V)^2),\\ P_4&=-\kappa _1+\sigma _1(\rho _1(1-\gamma -{\widetilde{\phi }}_1^S)p_1^S+\sqrt{1-\rho ^2_1}(1-\gamma -{\widetilde{\phi }}_1^V)p_1^V),\\ P_5&=\sigma _1(\rho _1(1-\gamma -{\widetilde{\phi }}_1^S)p_2^S+\sqrt{1-\rho ^2_1}(1-\gamma -{\widetilde{\phi }}_1^V)p_2^V),\\ P_6&=\frac{1}{2(1-\gamma )}\sigma _1^2(1-\gamma -\rho _1^2{\widetilde{\phi }}_1^S-(1-\rho _1^2){\widetilde{\phi }}_1^V). \end{aligned} \right. \end{aligned}$$

(69)

It follows from (59) and (68) that

$$\begin{aligned} (H^{\varPi _1}_1-H_1)'+E(t)(H^{\varPi _1}_1-H_1)'=F(t), \end{aligned}$$

(70)

where $E=-P_4-P_6(H^{\varPi _1}_1+H_1)-P_5{\widehat{H}}_1$ and $F=P_1-c+(P_4-a)H_1+(K_6-b)H_1^2+P_2{\widehat{H}}_1+P_3{\widehat{H}}_1^2+P_5{\widehat{H}}_1H_1.$

Solving (70), we have

$$\begin{aligned} H^{\varPi _1}_1(t)-H_1(t)=e^{-\int _{t}^{T}E(s)ds}\int _{t}^{T}e^{\int _{s}^{T}E(\tau )d\tau }F(s)ds. \end{aligned}$$

Under the assumption that the suboptimal strategies are admissible, the integral $\displaystyle e^{-\int _{t}^{T}E(s)ds}$ is bounded and positive. Furthermore, the choice of strategy $\varPi _1$ means that ${\widetilde{\phi }}_1^S = 0$ and ${\widetilde{\phi }}_1^V = 0$. Thus,

$$\begin{aligned} \begin{aligned} F=&\frac{\gamma -1}{2\gamma ^2(\phi ^S_1+\gamma )} \left( \lambda _1\phi ^S_1+(\gamma +\phi ^S_1)\rho _1\sigma _1({\widehat{H}}_1-\frac{\gamma (\gamma +\phi _1^S-1)}{(\gamma -1)(\gamma +\phi _1^S)}H_1)\right) ^2\\&+\frac{\gamma -1}{2\gamma ^2(\phi ^V_1+\gamma )} \left( \lambda _1\phi ^V_1+(\gamma +\phi ^V_1)\sqrt{1-\rho _1^2}\sigma _1({\widehat{H}}_1-\frac{\gamma (\gamma +\phi _1^V-1)}{(\gamma -1)(\gamma +\phi _1^V)}H_1)\right) ^2>0. \end{aligned} \end{aligned}$$

This leads to $H_1^\varPi -H_1>0$. Following (59), we obtain $\displaystyle h^{\varPi _1}-h= \kappa _1\theta _1\int _0^t \left( H^{\varPi _1}_1(s)-H_1(s)\right) ds>0.$ Therefore, we have

$$\begin{aligned} L^{\varPi _1}=1-\exp \left\{ \frac{1}{1-\gamma }\left[ (H_1^\varPi -H_1)v_1+(h^\varPi -h)\right] \right\} >0. \end{aligned}$$

(71)

Due to symmetry, $L^{\varPi _2}>0$ can be proved in a similar way.

If $L^{\varPi _3}$ is chosen, from Propositions 3.1 and 3.3, we have

$$\begin{aligned} H^{\varPi _3}_j(t)-H_j(t)=e^{-\int _{t}^{T}E_j(s)ds}\int _{t}^{T}e^{\int _{s}^{T}E_j(\tau )d\tau }F_j(s)ds, \ j=1, 2, \end{aligned}$$

where $F_j=\frac{\gamma -1}{2(\gamma +\phi ^V_j)}\left( \lambda _j+\frac{\sigma _jH^{\varPi _3}_j\sqrt{1-\rho _j^2}(\gamma +\phi ^V_j-1)}{\gamma -1}\right) .$ It is straightforward to see that $F_j>0$, and thus, $L^{\varPi _3}>0$. $\square$

Appendix F HJB in Sect. 4

In complete markets, the value function (47) satisfies the robust HJB PDE:

$$\begin{aligned} \begin{aligned}&\sup _{\beta ^S_i,\beta ^V_i}\inf _{e^S_i,e^V_i} \bigg \{ J_t+x\big (r+\sum _{j=1}^{2}(\beta ^{S}_j\lambda _jv_j-\beta ^{S}_ju_je^S_j \\&\quad +\beta ^{V}_j\mu _jv_j-\beta ^{V}_ju_je^V_j)\big )J_x +\sum _{j=1}^{2}\big (\kappa _j(\theta _j-v_j)-\rho _j\sigma _ju_je^S_j\\&\quad -\sqrt{1-\rho _j^2}\sigma _ju_je^V_j\big )J_{v_j}-\sum _{j=1}^{2}\left( \mu _j^U-\rho _j\psi ^U_je^S_j-\sqrt{1-\rho _j^2}\psi ^U_je^V_j\right) J_{u_j} \\&\quad +\big (\mu ^Y-\rho _2\psi _2^Ue^S_2u_1-\rho _1\psi _1^Ue^S_1u_2\\&\quad -\sqrt{1-\rho _2^2}\psi _2^Ue^V_2u_1-\sqrt{1-\rho _1^2}\psi _1^Ue^V_1u_2\big )J_y+\frac{1}{2}x^2\sum _{j=1}^{2}[(\beta ^{S}_j)^2\\&\quad +(\beta ^{V}_j)^2]v_jJ_{xx} +\rho \rho _1\psi ^U_1\beta ^S_2u_2xJ_{xu_1} \\&\quad +\sum _{j=1}^{2}\sigma _jv_jx\big (\beta ^S_j\rho _j+\beta ^V_j\sqrt{1-\rho _j^2}\big )J_{x v_j}+\rho \rho _2\psi ^U_2\beta ^S_1u_1xJ_{xu_2}\\&\quad +x\bigg ( \beta ^S_1u_1(\rho \rho _2u_1\psi ^U_2+\rho _1u_2\psi ^U_1)\\&\quad +\beta ^S_2u_2(\rho _2\psi _2^Uu_1+\rho \rho _1\psi ^U_1u_2)+\sqrt{1-\rho _1^2}\beta ^V_1\psi ^U_1v_1\\&\quad +\sqrt{1-\rho _2^2}\beta ^V_2\psi ^U_2v_2 \bigg )J_{xy} +\frac{1}{2}\sum _{j=1}^{2}{\varvec{\varSigma }}_{jj}J_{v_jv_j}+ {\varvec{\varSigma }}_{12}J_{v_1v_2}\\&\quad +\sum _{j=1}^{2}{\varvec{\varSigma }}_{j,j+2}J_{v_ju_j}+\varvec{\varSigma _{14}}J_{v_1u_2}+{\varvec{\varSigma }}_{23}J_{v_2u_1}\\&\quad +{\varvec{\varSigma }}_{15}J_{v_1y} +{\varvec{\varSigma }}_{25}J_{v_2y}+\frac{1}{2}\sum _{j=3}^{4}{\varvec{\varSigma }}_{jj}J_{u_ju_j} +{\varvec{\varSigma }}_{34}J_{u_1u_2}+{\varvec{\varSigma }}_{35}J_{u_1y}\\&\quad +{\varvec{\varSigma }}_{45}J_{u_2y}+{\varvec{\varSigma }}_{55}J_{yy}+\sum _{j=1}^{2}\frac{(e^S_j)^2}{2\varPsi ^S_j} +\frac{(e^V_j)^2}{2\varPsi ^V_j}\bigg \}=0, \end{aligned} \end{aligned}$$

(72)

where the symmetrical matrix ${\varvec{\varSigma }}(t)$ is given as follows:

$$\begin{aligned} \begin{bmatrix} \frac{1}{2}\sigma _1^2v_1 &{} \rho \rho _1\rho _2\sigma _1\sigma _2&{} \sigma _1u_1\psi ^U_1&{} \rho \rho _1\sqrt{1-\rho _2^2}\sigma _1\psi ^U_2u_1&{} \sigma _1\rho _1u_1(\rho _1u_2\psi ^U_1+\rho \rho _2u_1\psi ^U_2)+(1-\rho _1^2)\sigma _1\psi _1^U\\ &{} \frac{1}{2}\sigma _2^2v_2 &{} \rho \rho _1\rho _2\sigma _2\psi ^U_1u_2 &{} \sigma _2u_2\psi _2^U &{} \sigma _2\rho _2u_2(\rho _2u_1\psi ^U_2+\rho \rho _1u_2\psi ^U_1)+(1-\rho _2^2)\sigma _2\psi _2^U\\ &{} &{} \frac{1}{2}(\psi ^U_1)^2 &{} \rho \rho _1\rho _2\psi ^U_1\psi ^U_2 &{} (\psi ^U_1)^2u_2+\rho \rho _1\rho _2\psi _1^U\psi _2^Uu_1\\ &{} &{} &{}\frac{1}{2}(\psi ^U_2)^2&{} (\psi ^U_2)^2u_1+\rho \rho _1\rho _2\psi _1^U\psi _2^Uu_2\\ &{} &{} &{} &{} \frac{1}{2}\left( u_1^2(\psi ^U_2)^2+u_2^2(\psi ^U_1)^2+\rho \rho _1\rho _2\psi ^U_1\psi ^U_2\right) \end{bmatrix}. \end{aligned}$$

Appendix G Detection-Error Probabilities

Define the conditional characteristic functions

$$\begin{aligned} f_1(\omega ,t,T)=\mathrm {E}^{{\mathbb {P}}}[\exp (i\omega \xi _1(T))|{\mathcal {F}}_t^{S,V_1,V_2}]=\mathrm {E}^{{\mathbb {P}}}[({\mathcal {Z}}_1(T))^{i\omega }|{\mathcal {F}}_t^{S,V_1,V_2}] \end{aligned}$$

and

$$\begin{aligned} f_2(\omega ,t,T)= & {} \mathrm {E}^{{\mathbb {P}}^e}[\exp (i\omega \xi _1(T))|{\mathcal {F}}_t^{S,V_1,V_2}]=\mathrm {E}^{{\mathbb {P}}^e}[({\mathcal {Z}}_1(T))^{i\omega }|{\mathcal {F}}_t^{S,V_1,V_2}]\\= & {} \mathrm {E}^{{\mathbb {P}}}[({\mathcal {Z}}_1(T))^{i\omega +1}|{\mathcal {F}}_t^{S,V_1,V_2}], \end{aligned}$$

where $i^2=-1$ and $\omega$ is the transform variable. Denote $\mathbf {e_t}=[e^S_1,e^S_2,e_1^V,e^V_2]$ and

$$\begin{aligned} {\varvec{\sigma }}=\left[ \sigma _1\sqrt{V_1}\rho _1,\sigma _2\sqrt{V_2}\rho _2,\sigma _1\sqrt{V_1}\sqrt{1-\rho _1^2},\sigma _2\sqrt{V_2}\sqrt{1-\rho _2^2}\right] . \end{aligned}$$

Since the conditional characteristic functions are martingales, the Feynman–Kac theorem implies that $f_1$ and $f_2$ satisfy the same PDE:

$$\begin{aligned}&\frac{\partial f}{\partial t}+\sum _{j=1}^{2}\kappa _j(\theta _j-V_j)\frac{\partial f}{\partial V_j}+\frac{1}{2}{\mathcal {Z}}_1(t)^2||\mathbf {e_t}||^2 \frac{\partial ^2 f}{\partial {\mathcal {Z}}_1^2}+\frac{1}{2}\sum _{j=1}^{2}\sigma ^2_jV_j \frac{\partial ^2 f}{\partial V_j^2}\nonumber \\&-\sum _{j=1}^{2}{\mathcal {Z}}_1(t){\varvec{\sigma }}{\mathbf {e_t}}{}^{T}\frac{\partial ^2 f}{\partial {{\mathcal {Z}}_1}\partial V_j}=0 \end{aligned}$$

(73)

with different terminal conditions $f_1(\omega ,T,T)={\mathcal {Z}}_1^{i\omega }(T)$ and $f_2(\omega ,T,T)={\mathcal {Z}}_1^{i\omega +1}(T)$, respectively. Conjecturing the solution in the form $f={\mathcal {Z}}_1^{i\omega }(t)\exp (C_1(t)V_1+C_2(t)V_2+D(t))$ and substituting it into (73), we obtain

$$\begin{aligned}&C^{'}_{j}(t)-\kappa _jC_j(t)+\frac{1}{2}i\omega (i\omega -1)(q^S_j+q^V_j)^2\\&\quad +\frac{1}{2}\sigma _j^2C_j^2(t)-i\omega C_j(t)\sigma _i(q^S_j\rho _j+q^V_j\sqrt{1-\rho _j^2})=0,\ C_j(T)=0,\ j=1, 2, \\&D^{'}(t)+\kappa _1\theta _1C_1(t)+\kappa _2\theta _2C_2(t)=0,\ D(T)=0. \end{aligned}$$

Similarly, we can obtain the solution for $f_2$ by replicating $i\omega$ with $i\omega +1$ in the above equations. Thus, we have the detection-error probability

$$\begin{aligned} \varepsilon _T(\phi ^S_1,\phi ^S_2,\phi ^V_1,\phi ^V_2)=\frac{1}{2}-\frac{1}{2\pi }\int _{0}^{\infty }\left( \mathrm {Re}\left[ \frac{f_1(\omega ,0,T)}{i\omega }\right] -\mathrm {Re}\left[ \frac{f_2(\omega ,0,T)}{i\omega }\right] \right) d\omega . \end{aligned}$$