Sufficient condition for near-optimal control of general controlled linear forward–backward stochastic differential equations

Abstract

This article studies sufficient conditions for near-optimal stochastic control for systems governed by general linear controlled forward–backward stochastic differential equations (FBSDEs in short). The control is allowed to enter into both drift and diffusion coefficients. We prove that under certain additional conditions on the Hamiltonian, the near-maximum condition on the Hamiltonian function in the integral form is sufficient for near-optimality. As an applications, an example is given to illustrate our theoretical results.

Appendix

The following result gives the definition and some basic properties of the Clarke’s generalized gradient.

Definition 7

Let F be a convex set in ${\mathbb {R}}^{n}$ and let $f:F\rightarrow {\mathbb {R}}$ be a locally Lipschitz function. The generalized gradient of f at $\widehat{x}\in F$, denoted by $\partial _{x}f\left( \widehat{x}\right)$, is a set defined by

$$\begin{aligned} \partial _{x}f\left( \widehat{x}\right) =\left\{ \xi \in {\mathbb {R}}^{n}:\left\langle \xi ,\upsilon \right\rangle \le f^{\circ }\left( \widehat{x},\upsilon \right) ,\text { for any }\upsilon \in {\mathbb {R}}^{n}\right\} , \end{aligned}$$

where $f^{\circ }\left( \widehat{x},\upsilon \right) =\lim \sup _{y\rightarrow \widehat{x},t\rightarrow 0}\frac{1}{t}\left( f\left( y+t\upsilon \right) -f\left( y\right) \right)$.

Proposition 8

If $f:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ is locally Lipschitz at $x\in {\mathbb {R}}^{n}$, then the following statements holds

• 1. $\partial _{x}f\left( x\right)$ is nonempty, compact and convex set in ${\mathbb {R}}^{n}$.

• 2. $\partial _{x}\left( -f\right) \left( x\right) =-\partial _{x}\left( f\right) \left( x\right)$.

• 3. $\partial _{x}f\left( x\right) \ni 0$ if f attains a local minimum or maximum at x.

• 4. If f is continuously differentiable at x, then $\partial _{x}f\left( x\right) =\left\{ f^{\prime }\left( x\right) \right\}$.

• 5. If f, $g:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ are locally Lipschitz functions at $x\in {\mathbb {R}}^{d}$, then $\partial _{x}\left( f+g\right) \left( x\right) \subset \partial _{x}f\left( x\right) +\partial _{x}g\left( x\right)$.

For the detailed proof of the above Proposition see Clarke [16] or the book by Yong and Zhou ([13] Lemma 2.3).

Lemma 9

(Ekeland’s Variational Principle [15]) Let (F, $d_{F})$ be a complete metric space and $f:F\rightarrow \overline{{\mathbb {R}}}$ be a lower semi-continuous function which is bounded from below. For a given $\varepsilon >0$, suppose that $u^{\varepsilon }\in F$ satisfying $f\left( u^{\varepsilon }\right) \le \inf _{u\in F}f(u)+\varepsilon$. Then for any $\delta >0$, there exists $u^{\delta }\in F$ such that

• 1. $f\left( u^{\delta }\right) \le f\left( u^{\varepsilon }\right)$.

• 2. $d_{F}\left( u^{\delta },u^{\varepsilon }\right) \le \delta$.

• 3. $f\left( u^{\delta }\right) \le f\left( u\right) +\dfrac{ \varepsilon }{\delta }d_{F}\left( u,u^{\delta }\right) ,\,$ for all $u\in F$.