BACKGROUND The acceptance of virtual preclinical testing of control algorithms is growing and thus also the need for robust and reliable models. Models based on ordinary differential equations (ODEs) can rarely be validated with standard statistical tools. Stochastic differential equations (SDEs) offer the possibility of building models that can be validated statistically and that are capable o...

In this paper we discuss the stochastic differential equation (SDE) susceptibleinfected-susceptible (SIS) epidemic model with demographic stochasticity. First we prove that the SDE has a unique nonnegative solution which is bounded above. Then we give conditions needed for the solution to become extinct. Next we use the Feller test to calculate the respective probabilities of the solution first...

In this paper, we study the following stochastic differential equation (SDE) in R: dXt = dZt + b(t, Xt)dt, X0 = x, where Z is a Lévy process. We show that for a large class of Lévy processes Z and Hölder continuous drift b, the SDE above has a unique strong solution for every starting point x ∈ R . Moreover, these strong solutions form a C-stochastic flow. As a consequence, we show that, when Z...

Abstract We provide a support theorem for the law of solution to stochastic differential equation (SDE) with jump noise. This applies quite general Lévy-driven SDEs and is illustrated by examples rather degenerate noises, where leads an informative description support.

This chapter is an introduction and survey of numerical solution methods for stochastic differential equations. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial systems. We include a review of fundamental concepts, a description of elementary numerical methods and the concepts of convergence and order for stochas...

In typical stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô’s formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advant...

The temporal autocorrelation (AC) function associated with monitoring order parameters characterizing conformational fluctuations of an enzyme is analyzed using a collection of surrogate models. The surrogates considered are phenomenological stochastic differential equation (SDE) models. It is demonstrated how an ensemble of such surrogate models, each surrogate being calibrated from a single t...

We generalise the current theory of optimal strong convergence rates for implicit Euler-based methods by allowing for Poisson-driven jumps in a stochastic differential equation (SDE). More precisely, we show that under one-sided Lipschitz and polynomial growth conditions on the drift coefficient and global Lipschitz conditions on the diffusion and jump coefficients, three variants of backward E...

We consider a class of non-local Schrödinger operators and, using the ground state of such an operator, we define a random process generated by a unitary equivalent Lévy-type operator with unbounded coefficients. We construct this càdlàg process and show that it satisfies a related stochastic differential equation with jumps. Making use of this SDE we derive and prove the multifractal spectrum ...

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...