Hybrid stochastic functional differential equations with infinite delay : approximations and numerics

This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is established for a large class of highly nonlinear hybrid SFDEs with infinite delay. Our new theory makes it possible to numerically approximate the solution of the hybrid SFDE with infinite delay, via the numerical solution of the corresponding hybrid SFDE with finite delay

Computational methods for various stochastic differential equation models in finance

This study develops efficient numerical methods for solving jumpdiffusion stochastic delay differential equations and stochastic differential equations with fractional order. In addition, two novel algorithms are developed for the estimation of parameters in the stochastic models. One of the algorithms is based on the implementation of the Bayesian inference and the Markov Chain Monte Carlo method, while the other one is developed by using an implicit numerical scheme integrated with the particle swarm optimization

Aproksimacije rešenja stohastičkih diferencijalnih jednačina primenom Taylor-ovih redova

The subject of the doctoral dissertation is the application of the Taylor formula for the coefficients of various types of stochastic differential equations, for the purpose of the approximation of theirs solutions under non standard conditions, such as the global Lipschitz condition and the linear growth condition. Under certain assumptions, the almost sure convergence and the convergence in the p-th mean, p>0, of the sequence of approximate solutions towards the solution of the initial equation, is shown. The rate of the Lp convergence increases as the orders of the Taylor approximations of the coefficients of the initial equation increase. Shown results are illustrated through the examples which are designed such that the global Lipschitz condition and/or the linear growth condition for the drift and diffusion coefficients are not satisfied. That way, the need for the shown results is satisfied. Techniques used in the proofs are determined by the type of the considered equation, as well as by the conditions which are assumed for the coefficients of the equations