An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise
The aim of the paper is to study the existence and uniqueness of the solution to a nonlinear monotone stochastic differential equation \[ dX(t)+ A(t)X(t)dt = X(t)dW(t) \] with linear multiplicative noise in a real separable Hilbert space. The approach is to introduce the rescaling transform \(X(t)= e^{W(t)}y(t)\) and to reduce the above equation to the random differential equation \[ \frac{dy}{dt}(t) + e^{-W(t)}A(t)(e^{W(t)}y(t)) +\mu y(t)=0,\qquad t\in [0,T], \] which is treated as an operational equation in a convenient Hilbert space and can be rewritten as a monotone-type random equation in an appropriate space of infinite-dimensional stochastic processes on \([0,T]\). When the nonlinear operator is the subgradient of a convex function, the problem reduces to a convex optimization problem. One of the advantages of the approach adopted in this paper is that it gives sharp pointwise estimates and new pathwise regularity for the solutions.
stochastic partial differential equations
scientific article