Stochastic variational formula for fundamental solutions of parabolic PDE (Q1067195)

Let us consider a solution of the parabolic PDE \[ \partial g/\partial t=\Delta g+b(x)\nabla g,\quad t>0,\quad g(x,0)=g^ 0(x)>0. \] Under suitable hypotheses it is known [\textit{W. H. Fleming}, Appl. Math. Optimization 4, 329-346 (1978; Zbl 0398.93068)] that \(I(T,x)=-\log g(T,x)\) is the optimal cost of the stochastic control problem: \[ \min imize\quad I(T,x;u)=Ex\{\int^{T}_{0}| b(\xi_ t)-u_ t|^ 2dt-\log g^ 0(\xi_ t)\} \] where \(d\xi_ t=u_ tdt+dW_ t\), \(\xi_ 0=x\). Under the condition that \(b\in C_ b^{\infty}\) the same variational representation is given for the fundamental solution p(t,x,y) that is when \(g^ 0(x)=\delta (x-y)\). In this case, however, the limiting form of the above control problem has \(I\equiv +\infty\) because the endpoint is fixed at y. To overcome this difficulty, for each \(\alpha >0\), the cost is computed up to time T-\(\alpha\) with a suitable penalty function \(F_{\alpha}(\xi_{T-\alpha})\). Then by letting \(\alpha\) \(\to 0\) a new cost function is obtained whose minimum value is shown to be \(I(T,x,y)=-\log p(t,x,y)\). The optimal control is computed in the standard dynamic programming way from I. The proof is a nice application of a result of Molchanov about the asymptotic behavior of p(t,x,y) for small t. The same strategy is then applied to solve a stochastic control problem with a more general loss function L (but with less than quadratic growth) with fixed endpoints.