Explicit strong solutions of SPDE's with applications to nonlinear filtering
An explicit strong solution is given for a type of Cauchy problems such as the Zakai equation (the solution of it being the unnormalized conditional density of a nonlinear filtering problem): \[ \begin{aligned} du(t,x,\omega) &=(L+c)u(t,x,\omega)dt+h(x,t) u(t,x,\omega) dY_t (\omega),\\ u(0,x,\omega) &=u_0(x), \quad (t,x,\omega) \in[0,T] \times R\times \Omega,\end{aligned}\tag{1} \] \(L\) is a second order differential operator, \(c,h\) and \(u_0\) are regular functions. The authors give an improvement to previuos results, giving assumptions to obtain strong solutions for (1) and their result applies for a class of unbounded, degenerate diffusion and drift coefficients. The proof is based essentially on Kolmogorov's continuity theorem for random fields and a differential calculus with respect to space-time parameters. The main theorem gives technical assumption so that (1) admits a strong solution which can be represented explicitly by such a backward Feynman-Kac formula. Finally, a concrete and numerical application is done in mathematical finance: a volatility estimator given the past is computed.
stochastic partial differential equations
scientific article; zbMATH DE number 1160087