In this paper a stochastic differential equation (SDE) with infinite memory is considered. The drift coefficient of the equation is a nonlinear functional of the past history of the solution. Sufficient conditions for existence and uniqueness of stationary solution are given. This work is motivated by recent papers [1] and [2] where stochastically forced nonlinear equations of hydrodynamics wer...

The benefit of monetary assets cannot be over emphasized because it stands as an engine room to every investment which accumulates wealth such daily, weekly, monthly and yearly etc. In this study, a closed form solution Stochastic Differential Equation (SDE) was successfully exploited for the analysis asset values other stock market quantities. solutions variables were critically observed by si...

In this paper, we present the numerical solution of ordinary differential equations (or SDEs), from each order especially second-order with time-varying and Gaussian random coefficients. We indicate a complete analysis for second-order equations in special case of scalar linear second-order equations (damped harmonic oscillators with additive or multiplicative noises). Making stochastic differe...

In this chapter we overview a Bayesian approach to a wide range of signal processing problems in which the goal is to find the signal, which is a solution of an ordinary or stochastic differential equation, given noisy observations of its solution. In the case of ordinary differential equations (ODEs) this gives rise to a finite dimensional probability measure for the initial condition, which t...

We consider a nonlinear stochastic heat equation in spatial dimension $d=2$, forced by white-in-time multiplicative Gaussian noise with correlation length $\varepsilon>0$ but divided factor of $\sqrt{\log\varepsilon^{-1}}$. impose condition on the Lipschitz constant nonlinearity so that problem is "weak noise" regime. show that, as $\varepsilon\downarrow0$, one-point distribution solution conve...

A stochastic algorithm is proposed for the global optimization of nonconvex functions subject to linear constraints. Our method follows the trajectory of an appropriately defined Stochastic Differential Equation (SDE). The feasible set is assumed to be comprised of linear equality constraints, and possibly box constraints. Feasibility of the trajectory is achieved by projecting its dynamics ont...

In [5] the authors obtained Mean-Field backward stochastic differential equations (BSDE) associated with a Mean-field stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, corresponding to a large number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such Mean-Field BSD...

The expressions of solutions for general n × m matrix-valued inhomogeneous linear stochastic differential equations are derived. This generalizes a result of Jaschke (2003) for scalar inhomogeneous linear stochastic differential equations. As an application, some IR vector-valued inhomogeneous nonlinear stochastic differential equations are reduced to random differential equations, facilitating...

Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f , the Feynman-Kac formula gives a condition for f(t,X) to be a local martingale. We generalize the Feynman-Kac formula in two main ways. First, it is extended to nondifferentiable functions. Second, the process X is not required to satisfy...

dXt = b(t,Xt) dt + dBt, s, t ∈ R, Xs = x ∈ R. The above SDE is driven by a bounded measurable drift coefficient b : R × Rd → Rd and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow φs,t(·) of the SDE lives in the space L2(Ω;W 1,p(Rd, w)) for all s, t and all p > 1, where W 1,p(Rd, w) denotes a weighted Sobolev space with weight w possessing a p-th moment wi...