Weak convergence for a spatial approximation of the nonlinear stochastic heat equation

We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation has a solution only in one space dimension. In order to obtain results for higher dimensions, colored noise is considered here, besides white noise in one dimension. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.Comment: 19 page

Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values

In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the considered SEE. We use this framework to establish existence, uniqueness, and regularity results for mild solutions of parabolic semilinear SEEs with singularities at the initial time. We also provide several counterexample SEEs that illustrate the optimality of our results

On the differentiability of solutions of stochastic evolution equations with respect to their initial values

In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are $n$-times continuously Fr\'{e}chet differentiable, then the solutions of the considered SEEs are also $n$-times continuously Fr\'{e}chet differentiable with respect to their initial values. Moreover, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes

Convergence of a robust deep FBSDE method for stochastic control

In this paper, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemmingfrom stochastic control. It is a modification of the deep BSDE method in which the initial value to thebackward equation is not a free parameter, and with a new loss function being the weighted sum of the costof the control problem, and a variance term which coincides with the mean squared error in the terminalcondition. We show by a numerical example that a direct extension of the classical deep BSDE methodto FBSDEs, fails for a simple linear-quadratic control problem, and motivate why the new method works.Under regularity and boundedness assumptions on the exact controls of time continuous and time discretecontrol problems, we provide an error analysis for our method. We show empirically that the methodconverges for three different problems, one being the one that failed for a direct extension of the deep BSDEmethod

Weak convergence for a spatial approximation of the nonlinear stochastic heat equation

We find the weak rate of convergence of approximate solutions of the nonlinear stochastic heat equation, when discretized in space by a standard finite element method. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation only has a solution in one space dimension. In order to get results for higher dimensions, colored noise is considered here, besides the white noise case where considerably weaker assumptions on the noise term is needed. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence

Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE

We introduce a new family of refined Sobolev-Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the It\={o} formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.Comment: 32 page

An energy-based deep splitting method for the nonlinear filtering problem

The purpose of this paper is to explore the use of deep learning for the solution of the nonlinear filtering problem. This is achieved by solving the Zakai equation by a deep splitting method, previously developed for approximate solution of (stochastic) partial differential equations. This is combined with an energy-based model for the approximation of functions by a deep neural network. This results in a computationally fast filter that takes observations as input and that does not require re-training when new observations are received. The method is tested on three examples, one linear Gaussian and two nonlinear. The method shows promising performance when benchmarked against the Kalman filter and the bootstrap particle filter.Comment: 20 pages, 5 figure