In this paper we derive a new handy integral equation for the free boundary of infinite time horizon, continuous time, stochastic, irreversible investment problems with uncertainty modeled as a one-dimensional, regular diffusion X0,x. The new integral equation allows to explicitly find the free boundary b(·) in some so far unsolved cases, as when X0,x is a three-dimensional Bessel process or a ...

On the basis of integral representations we propose fast numerical methods to solve the Cauchy problem for the stochastic wave equation without boundaries and with the Dirichlet boundary conditions. The algorithms are exact in a probabilistic sense. 1 Statement of the problem and integral representations Consider the stochastic wave equation ∂X ∂t (t, x)− a2 X ∂x (t, x) = g(t, x) + f(t, x) dW (...

We investigate simulations of exact solutions of the stochastic Dr. Herman Numerical Realizations of Solutions f the Stochastic KdV Equation Exact Solution of Stochastic KdV Wadati 1983 The One Soliton Solution Under Noise Statistical Averages The Exact Solution for < u(x, t) > via the Diffusion Equation Solving the Diffusion Equation The Damped Stochastic KdV Asymptotics Numerical Simulation o...

In this thesis, we study several stochastic partial differential equations (SPDE’s) in the spatial domain R, driven by multiplicative space-time white noise. We are interested in how rough and unbounded initial data affect the random field solution and the asymptotic properties of this solution. We first study the nonlinear stochastic heat equation. A central special case is the parabolic Ander...

Stochastic differential equation (SDE) and Fokker-Planck equation (FPE) are two general approaches to describe the stochastic drift-diffusion processes. Solving SDEs relies on the Monte Carlo samplings of individual system trajectory, whereas FPEs describe the time evolution of overall distributions via path integral alike methods. The large state space and required small step size are the majo...

Contaminant transport by liquid ow in a porous medium is modeled by the addition of a stochastic term to Darcy's ow equation. The resulting stochastic diierential equation is studied using results from the theory of diiusions as embodied in the Dynkin formula. The resulting integral equation for the probability distribution of uid elements is solved for the case of a spatially homogeneous mediu...

In this article,we present a wavelet method for solving stochastic Volterra integral equations based on Haar wavelets. First, we approximate all functions involved in the problem by Haar Wavelets then, by substituting the obtained approximations in the problem, using the It^{o} integral formula and collocation points then, the main problem changes into a system of linear or nonlinear equation w...

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved. A class o...

We extend Walsh’s theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulatio...

In this paper, we introduce the Petrov-Galerkin method for solution of stochastic Volterra integral equations. Here, we use continues Lagrange-type k-0 elements, since these elements have simple structure and via them, the solution of stochastic Volterra integral equation is reduced to algebraic equations. Also the error analysis of this method is done. In Comparison with other methods, this me...