Maximum Principle for Mean-Field SDEs Under Model Uncertainty

Abstract

This paper is concerned with necessary and sufficient conditions for optimal control problem of mean-field type under model uncertainty. Since the classical variational method is not enough to get the results in this case, we should draw support from weak convergence method. As an application in finance, we study the time-inconsistent mean-variance portfolio selection problem in this model uncertainty setup.

Appendix: The Uniform Boundedness of the Riccati Equation

Lemma A.1

Suppose that $\Phi (t,\theta ^*)$ is a solution to the Riccati equation (42). Then, it must be uniformly bounded for $t\in [0,T]$ and $\theta ^*\in [0,1]$.

Proof

It is known from [14, 42] that the Riccati equation (42) relates to the following optimal LQ control problem: Minimize the cost functional

$$\begin{aligned} J(t,x_0;u(\cdot ))=2\pi \mathbb {E}\left[ x(T)^{\top }\Theta ^*x(T)\right] \end{aligned}$$

subject to the dynamics

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}dx(s)=\Big (e(s)x(s)+A^{\top }(s)u(s)\Big )ds+\sum \limits _{j=1}^{d}(\beta ^j(s))^{\top }u(s)dB^j(s),\\ &{}x(t)=x_0, \end{array}\right. } \end{aligned}$$

where the initial $(t,x_0)\in [0,T] \times \mathbb {R}^2$. If $\Phi (t,\theta ^*)$ solves (42), then, recalling Theorem 2.2 in [42], the optimal value function is given by

$$\begin{aligned} \inf _{u(\cdot ) \in \mathcal {U}[0, T]} J\left( t,x_{0}; u(\cdot )\right) =x_0^{\top }\Phi (t,\theta ^*)x_0. \end{aligned}$$

(45)

Let $x^0(\cdot )$ be the solution to the dynamics corresponding to the admissible control $u(\cdot )\equiv 0$. Then, (45) implies

$$\begin{aligned} x_0^{\top }\Phi (t,\theta ^*)x_0 \le 2\pi \mathbb {E}\left[ x^0(T)^{\top }\Theta ^*x^0(T)\right] . \end{aligned}$$

It follows from a prior estimate of linear SDE that there exists a constant $C>0$ such that

$$\begin{aligned} 0 \le |x_0^{\top }\Phi (t,\theta ^*)x_0| \le C |x_0^{\top } x_0|. \end{aligned}$$

The arbitrariness of the initial state $x_0 \in \mathbb {R}^2$ then gives the uniform boundedness of $\Phi (t,\theta ^*)$ on $[0,T] \times [0,1]$.

Lemma A.2

Let $\Phi (t,\theta ^*)$ be a solution to the Riccati equation (42). Then, $\bigg (\sum \limits _{j=1}^{d}\beta ^j \Phi (\theta ^*) (\beta ^j)^{\top }\bigg )^{-1}$ is uniformly bounded with respect to $\theta ^* \in [0,1]$.

Proof

If $\theta ^* \in [0,\frac{1}{2}]$, we have $\left( \begin{matrix} \theta ^* &{} 0 \\ 0 &{} 1-\theta ^* \end{matrix} \right) \ge \left( \begin{matrix} 0 &{} 0 \\ 0 &{} \frac{1}{2} \end{matrix} \right)$. Using the comparison theorem of Riccati equation (see Corollary 3.4 in [42]), we could get that

$$\begin{aligned} \Phi (t,\theta ^*)\ge \Phi _{\frac{1}{2}}(t)=\left( \begin{matrix} 0&{} 0 \\ 0 &{} \pi \exp \Big ( \int ^T_t 2e(s)-A^{\top }_2(s)(\beta _2(s)\beta ^{\top }_2(s))^{-1}A_2(s)ds\Big ) \end{matrix} \right) , \end{aligned}$$

where $\Phi _{\frac{1}{2}}$ denotes the solution to (42) with $\Phi _{\frac{1}{2}}(T)=\left( \begin{matrix} 0 &{} 0 \\ 0 &{} \pi \end{matrix} \right)$. Note that $\Phi (t,\theta ^*) \in C([0,T];{S}^2_+)$ and

$$\begin{aligned} 0< \left( \sum \limits _{j=1}^{d}\beta ^j \Phi (\theta ^*) (\beta ^j)^{\top }\right) ^{-1} \le \left( \sum \limits _{j=1}^{d}\beta ^j \Phi _{\frac{1}{2}} (\beta ^j)^{\top }\right) ^{-1}. \end{aligned}$$

Then, the uniform boundedness of $(\sum \limits _{j=1}^{d}\beta ^j \Phi _{\frac{1}{2}} (\beta ^j)^{\top })^{-1}$ comes immediately from the nondegeneracy condition (34), which further implies the required result.

If $\theta ^* \in (\frac{1}{2},1]$, we have $\left( \begin{array}{ll} \theta ^* &{} 0 \\ 0 &{} 1-\theta ^* \end{array} \right) \ge \left( \begin{array}{ll} \frac{1}{2} &{} 0 \\ 0 &{} 0 \end{array} \right)$. Then, the rest of analysis can be proceeded similarly as above.