An Extended McKean–Vlasov Dynamic Programming Approach to Robust Equilibrium Controls Under Ambiguous Covariance Matrix

Abstract

This paper studies a general class of time-inconsistent stochastic control problems under ambiguous covariance matrix. The time inconsistency is caused in various ways by a general objective functional and thus the associated control problem does not admit Bellman’s principle of optimality. Moreover, we model the state by a McKean–Vlasov dynamics under a set of non-dominated probability measures induced by the ambiguous covariance matrix of the noises. We apply a game-theoretic concept of subgame perfect Nash equilibrium to develop a robust equilibrium control approach, which can yield robust time-consistent decisions. We characterize the robust equilibrium control and equilibrium value function by an extended optimality principle and then we further deduce a system of Bellman–Isaacs equations to determine the equilibrium solution on the Wasserstein space of probability measures. The proposed analytical framework is illustrated with its applications to robust continuous-time mean-variance portfolio selection problems with risk aversion coefficient being constant or state-dependent, under the ambiguity stemming from ambiguous volatilities of multiple assets or ambiguous correlation between two risky assets. The explicit equilibrium portfolio solutions are represented in terms of the probability law.

Appendix A: Differentiability on Wasserstein Space

The derivative relies on the lifting of functions of $u:{\mathcal {P}}_2(\mathbb {R}^d)\mapsto \mathbb {R}$ into $\mathbb {R}$-valued functions U defined on $L^2(\Omega ,{\mathcal {G}},\mathbb {P};\mathbb {R^d}^d)$ by $U(X)=u(\mathbb {P}_X)$. If the lift U is Fréchet differentiable (respectively, Fréchet differentiable with continuous derivatives) on $L^2({\mathcal {G}};\mathbb {R}^d)$, then we call that u is differentiable (respectively, $C^1$) on ${\mathcal {P}}_2(\mathbb {R}^d)$. By the Riesz’ representation theorem, we can find an element DU(X) of $L^2({\mathcal {G}};\mathbb {R}^d)$ such that the Fréchet derivative [DU](X) satisfies the relation $[DU](X)(Y)=\mathbb {E}[DU(X)Y]$. Moreover, it can be represented as $DU(X)=\partial _\nu u(\mathbb {P}_X)(X)$, where the function $\partial _\nu u(\mathbb {P}_X):\mathbb {R}^d\mapsto \mathbb {R}^d$ is called the derivative of u at $\nu =\mathbb {P}_X$. If we denote by $L^2_\nu (\mathbb {R}^d)$ the set of measurable functions $\varphi :\mathbb {R}^d\mapsto \mathbb {R}^d$, which are square-integrable with respect to $\nu$, then there is $\partial _\nu u(\nu )\in L^2_\nu (\mathbb {R}^d)$ for $\nu \in {\mathcal {P}}_2(\mathbb {R}^d)=\{\mathbb {P}_X:X\in L^2({\mathcal {G}};\mathbb {R}^d)\}$. We say that u is partially $C^2$ if

• it is $C^1$;

• for any $\nu \in {\mathcal {P}}_2(\mathbb {R}^d)$, the mapping $(\nu ,x)\in {\mathcal {P}}_2(\mathbb {R}^d)\times \mathbb {R}^d\mapsto \partial _\nu u(\nu )(x)$ is continuous at any point $(\nu ,x)$ such that $x\in Supp(\nu )$;

• for any $\nu \in {\mathcal {P}}_2(\mathbb {R}^d)$, the mapping $x\in \mathbb {R}^d\mapsto \partial _\nu u(\nu )(x)$ is differentiable in the standard sense, and its derivative is jointly continuous at any point $(\nu ,x)$ such that $x\in Supp(\nu )$.

then we denote the gradient as $\partial _x\partial _\nu u(\nu )(x)\in \mathbb {R}^{d\times d}$.

Now, we give examples to explain the notion of the derivative when $d=1$. If we suppose $u(\nu )=\int _\mathbb {R}h(x)\nu (dx)$, then its lifted function is $U(X)=\mathbb {E}[h(X)]$ and the Fréchet derivative of which is $[DU](X)(Y)=\mathbb {E}[D_xh(X)Y]$, which shows that $\partial _\nu u(\nu )=D_xh$ and $\partial _x\partial _\nu u(\nu )=D^2_x h$. In our paper, we study the Markowitz mean-variance portfolio selection problems as examples. Its objective functional consists of a mean-term $\mathbb {E}(X_T)$ and a variance-term $\hbox {Var}(X_T)$, which measure the expectation of return and the risk at the terminal time T, respectively. Their corresponding derivatives are given as follows:

• When $h(x)=x$, that is, $u(\nu )={\overline{\nu }}:=\int _\mathbb {R}x\nu (dx)$, then $U(X)=\mathbb {E}(X)$, $\partial _\nu u(\nu )(x)=1$ and $\partial _x\partial _\nu u(\nu )(x)=0$;

• When $h(x)=(x-{\overline{\nu }})^2$, it means that $u(\nu )=\hbox {Var}(\nu ):=\int _\mathbb {R}(x-{\overline{\nu }})^2\nu (dx)$. Then $U(X)=\hbox {Var}(X)$, and thus $\partial _\nu u(\nu )(x)=2(x-{\overline{\nu }})$ and $\partial _x\partial _\nu u(\nu )(x)=2$,

which show that both ${\overline{\nu }}$ and $\hbox {Var}(\nu )$ are $C^2$.

1.1 A.1 Itô’s Formula in Wasserstein Space

Next, we introduce the chain rule for functions defined on ${\mathcal {P}}_2(\mathbb {R}^d)$ proven independently by Buckdahn et al. [15] and Chassagneux et al. [17]. Let us consider a stochastic process with the form of (2.4) and a partially $C^2$ function u on ${\mathcal {P}}_2(\mathbb {R}^d)$, then for any fixed $(t,\xi )\in [0,T]\times L^2({\mathcal {F}}_t;\mathbb {R}^d)$ and fixed $(\alpha ,\sigma )\in {\mathcal {A}}\times {\mathcal {V}}_\Theta$, we have

$$\begin{aligned} \begin{aligned}&~u\left( s,\rho ^{t,\mu ,\alpha ,\sigma }_s\right) =u\big (t,\mu \big )+\int ^s_t \partial _\tau u\left( \tau ,\rho ^{t,\mu ,\alpha ,\sigma }_\tau \right) d\tau \\&+\int ^s_t\mathbb {E}^\sigma \left[ \partial _\mu u\left( \tau ,\rho ^{t,\mu ,\alpha ,\sigma }_\tau \right) \big (X^{t,\xi ,\alpha }_\tau \big )\cdot b_\tau +\frac{1}{2}\textrm{tr}\left( \partial _x\partial _\mu u\left( \tau ,\rho ^{t,\mu ,\alpha ,\sigma }_\tau \right) \big (X^{t,\xi ,\alpha }_\tau \big )\cdot a_\tau \right) \right] d\tau \end{aligned} \end{aligned}$$

for $s\in [t,T]$, where $a_\tau =h_\tau \Sigma _\tau h_\tau ^{\top }$. Moreover, the first integral-term corresponds to the time variation from t to s, while the second one corresponds to the variation of measures from $\mu =\rho ^{t,\mu ,\alpha ,\sigma }_t$ to $\rho ^{t,\mu ,\alpha ,\sigma }_s$.

When $d=1$, by Itô’s lemma along the flow of conditional measures, it is clear that

• $\overline{\rho ^{t,\mu ,\alpha ,\sigma }_s}={\overline{\mu }}+\int ^s_t\mathbb {E}^\sigma \left[ b_\tau \right] d\tau$;

• $\hbox {Var}(\rho ^{t,\mu ,\alpha ,\sigma }_s)=\hbox {Var}\left( \mu \right) +\int ^s_t\mathbb {E}^\sigma [h^2_\tau \Sigma _\tau ]d\tau$.