Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control
Let \(X\) be a stochastic process with values in a Hilbert space \(H\), which solves some stochastic evolution equation driven by a cylindrical Brownian motion \(W\) in another Hilbert space \(\Theta\), and let \({\mathcal L}_t\) be the generator of the process \(X\). The authors of the present paper investigate the generalized backward Kolmogorov equation \[ \partial_t v(t,x)+{\mathcal L}_t[v(t,.)](x)= \psi(t, x,v(t, x), G(t,x)^*\nabla_x v(t,x)),\;(t,x)\in [0,T]\times H,\;u(T,x)= \varphi(x), \] where \(\psi: [0,T]\times H\times R\times\Theta\to R\) is a given function, and \(\nabla_x v(t,x)\) is the Gâteaux derivative of \(v\) w.r.t. \(x\). In view of applications to the control theory, the authors are interested in mild solutions of this equation. They prove existence and uniqueness by assuming existence and some growth condition on the first Gâteau derivatives of \(\varphi\) and \(\psi\). As the main ingredient in their approach they extend backward stochastic differential equations (BSDE) \[ dY_t= \psi(t, X_t, Y_t, Z_t)dt+ \langle Z_t,dW_t\rangle_\Theta,\;t\in [0,T],\quad Y_T= \varphi(X_T), \] to their framework [cf. \textit{E. Pardox} and \textit{S. G. Peng}, Syst. Control Lett. 14, No. 1, 55-61 (1990; Zbl 0692.93064), the first paper on finite-dimensional, nonlinear BSDE of the above form, and \textit{Y. Hu} and \textit{S. Peng}, Stochastic Anal. Appl. 9, No. 4, 445-459 (1991; Zbl 0736.60051), study of BSDEs in infinite dimension, but in another framework than that which is needed here], and they derive a Feynman-Kac type formula which relates the solution of the backward stochastic differential equation with the mild solution of the Kolmogorov equation. Finally, applications to stochastic control are presented; they illustrate the effectiveness of the authors' results on nonlinear Kolmogorov equations.
backward stochastic differential equations in infinite dimension
Kolmogorov equation in infinite dimension
scientific article; zbMATH DE number 1906090
scientific article