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Stochastic Hamiltonians associated with stochastic differential equations and non-smooth final value. (English) Zbl 1150.60398
Summary: For a given Lipschitz continuous function $$\varphi(x):\mathbb R^d\to\mathbb R$$ admitting a weak gradient $$\partial_x\varphi(x):\mathbb R^d\to\mathbb R^d$$ we associate the random variable $$\varphi(x(T))$$, where $$x(t),t\in[0,T]$$ is the solution of a stochastic differential system with Lipschitz continuous coefficients and first order continuously differentiable difussion coefficients. It is proved that the random variable $$\varphi(x(T))$$ can be represented as a final value $$S(T,x(T))=\varphi(x(T))$$ using a continuous function $$S(t,x):[0,T]\times\mathbb R^d\to\mathbb R$$ which admits a weak gradient $$\partial_x S(t,x):[0,T]\times\mathbb R^d\to\mathbb R^d$$ and $$S(t,x(t)),\;t\in[0,T]$$, fulfills a first order stochastic differential equation (the stochastic differential $$d_t[S(t,x(t))]$$ equals a stochastic hamiltonian). It can be meaningful for getting feedback optimal control associated with stochastic control problems and for describing an admissible feedback strategy involved in a financial market.
##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 93C42 Fuzzy control/observation systems