Limit theorem for controlled backward SDEs and homogenization of Hamilton-Jacobi-Bellman equations
Let \((W_s)_{s\geq 0}\) be a \(d\)-dimensional Brownian motion, \((\alpha_s)_{0\leq s\leq T}\) a control process and \(\varepsilon> 0\) small parameter. The backward stochastic differential equation (BSDE): \[ dX^{\varepsilon,\alpha}_s= g(\varepsilon^{-1} X^{\varepsilon,\alpha}_s, \alpha_s) ds+ \varepsilon^{-1} b(\varepsilon^{-1} X^{\varepsilon,\alpha}_s,\alpha_s)ds+ \sigma(\varepsilon^{-1} X^{\varepsilon,\alpha}_s, \alpha_s) dW_s, \] \[ dY^{\varepsilon,\alpha}_s= -f(\varepsilon^{-1} X^{\varepsilon,\alpha}_s, X^{\varepsilon,\alpha}_s, Y^{\varepsilon,\alpha}_s, Z^{\varepsilon,\alpha}_s, \alpha_s) ds+ Z^{\varepsilon,\alpha}_s dW_s+ dM^{\varepsilon,\alpha}_s, \] \[ X^{\varepsilon,\alpha}_s= x,\quad Y^{\varepsilon,\alpha}_s= h(X^{\varepsilon,\alpha}_s),\quad M^{\varepsilon,\alpha}_s= 0,\quad \langle M^{\varepsilon,\alpha}_s, W\rangle= 0, \] is considered. The authors define a function \[ u^\varepsilon(t, x)= \text{ess-inf}_\alpha\,Y^{\varepsilon,\alpha}_s.\tag{1} \] It is shown that \(u^\varepsilon(t, x)\) coincides with the continuous viscosity solution to the following parabolic partial differential equation, Hamilton-Jacobi-Bellman equation (HJB): \[ \begin{cases} -\partial_t u^\varepsilon+ H(\varepsilon^{-1} x,x,u^\varepsilon, Du^\varepsilon, \varepsilon^{-1} Du^\varepsilon, D^2 u^\varepsilon)= 0\quad &\text{in }[0,T)\times \mathbb{R}^d,\\ u^\varepsilon(T,\cdot)= h(\cdot)\quad &\text{on }\mathbb{R}^d,\end{cases}\tag{2} \] with the Hamiltonian \[ \begin{multlined} H(\eta,x,y,p,q,X)= \sup_{\alpha\in U}\{-\text{tr}\{\tfrac 12(\sigma\sigma^*)(\eta,\alpha) X\}- \langle g(\eta,\alpha), p\rangle- \langle b(\eta, \alpha), q\rangle-\\ f(\eta, x,y,\sigma^*(\eta,\alpha)p,\alpha)\}.\end{multlined} \] The authors prove that \(u^\varepsilon(t, x)\) converges to \(u^0(t, x)\) for every \((t,x)\) as \(\varepsilon\) tends to zero and the convergence problem of (1) is equivalent to the homogenization problem for HJB equation (2).
backward stochastic differential equation
scientific article; zbMATH DE number 5069396