Limit theorem for controlled backward SDEs and homogenization of Hamilton-Jacobi-Bellman equations
type: Item
instance of: scholarly article
title: Limit theorem for controlled backward SDEs and homogenization of Hamilton-Jacobi-Bellman equations
zbMATH Open document ID: 1109.60055
author: Rainer Buckdahn
author: Naoyuki Ichihara
published in: Applied Mathematics and Optimization
publication date: 2006-11-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 5069396
zbMATH Keywords: backward stochastic differential equation
zbMATH Keywords: viscosity solution
reviewed by: Maria Stolarczyk
MaRDI profile type: Publication
full work available at URL: s00245-004-0805-3
OpenAlex ID: W1994167177
DOI: 10.1007/S00245-004-0805-3
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description: scientific article; zbMATH DE number 5069396