Extensions and BSDE's

The value function of Problem (1.1.1) can be reformulated in terms of the $Y$ component of the solution $(Y,Z,A)$ of the Backward Stochastic Differential Equation

$$\begin{eqnarray*} Y_{t} &=&g(S_{T})-\int_{t}^{T}Z_{t}^{*}dW_{t}+A_{T}-A_{t} \\ Y_{t} &\ge &g(S_{t}) \end{eqnarray*}$$

where $A$ is a non-decreasing process satisfying

$$\int_{0}^{T}(Y_{t}-g(S_{t}))dA_{t}=0,$$

see e.g. [KKPPQ].

This leads us to consider the more general problem of approximating the solution of Reflected Forward - Backward Stochastic Differential Equations of the form

$$\begin{eqnarray*} X_{t} &=&X_{0}+\int_{t}^{T}b(t,X_{t},Y_{t},Z_{t})dt+ \int_{t}^... ...nt_{t}^{T}Z_{t}^{*}dW_{t}+A_{T}-A_{t} \\ Y_{t} &\ge &g(t,X_{t}) \end{eqnarray*}$$

where $A$ is a non-decreasing process satisfying

$$\int_{0}^{T}(Y_{t}-g(t,X_{t}))dA_{t}=0.$$

This framework partially includes control problems associated to HJB equations. It is related to non-linear PDE's of the form

$$\begin{eqnarray*} && \min \{-v_{t}(t,x)-\frac{1}{2}\mbox{Trace}[v_{xx}(t,x)aa^{*... ...;\hspace{8cm}; \;v(t,x)-g(t,x)\}\; =\;0 \\ &&v(T,.) \;=\;g(T,.) \end{eqnarray*}$$

where $\theta (t,x)$ solves

$$v_{x}(t,x) a(t,x,v(t,x),\theta(t,x))=\theta(t,x)\;,$$

through the relations

$$v(t,X_{t})=Y_{t} \;\;\mbox{ and }\;\; \theta(t,X_{t})^{*}=Z_{t}\;.$$

See e.g. [MY].

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