This paper discusses theoretical properties, shows the performance and presents some extensions of techniques used for simulation of stochastic differential equation applied on the financial data modeling. There are realized comparisons of different approaches for discretization schemes and their performances from the simulation convergence point of view. This study shows that, depending on the...

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.

Using a class of stochastic Euler and Runge-Kutta methods, we numerically solve a reaction-diffusion equation with additive random excitation. By discretizing the space and the associated stochastic differential system, we present a comparison of the diffusibility behaviors between the schemes above. The model presented here consists of reaction-diffusion equations describing the evolution of t...

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by Lévy white noise ”obtained by subordination of a Gaussian white noise”. Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general cádlág modifications. General results are applied to equations with fractional Laplacian. ...

We examine a certain class of nonstationary inverse problems. We view them as a state estimation problem. The time evolution of the state of the system is modeled by a stochastic differential equation. The observation equation is linear with additive measurement noise. We introduce the time discrete infinite dimensional state estimation system concerning the problem. For computational reasons t...

We provide the exact analytic solution of the stochastic Schrödinger equation describing a harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models for infinite-dimensional systems. We compute the Green's...

In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problem with non standard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a finite dimensional dynamics, which describes the boundary conditions of the internal system. In other terms, we are concerned with non standard boundary conditi...

The nonlinear optical Galton board (NLOGB), a quantum walk like (but nonlinear) discrete time quantum automaton, is shown to admit a complex evolution leading to long time thermalized states. The continuous limit of the Galton board is derived and shown to be a nonlinear Dirac equation (NLDE). The (Galerkin-truncated) NLDE evolution is shown to thermalize toward states qualitatively similar to ...

We present a stochastic approach for solving the quantumkinetic equation introduced in Part I. A Monte Carlo method based on backward time evolution of the numerical trajectories is developed. The computational complexity and the stochastic error are investigated numerically. Variance reduction techniques are applied, which demonstrate a clear advantage with respect to the approaches based on s...