23 research outputs found
A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion
In this thesis, we mainly study some properties for certain stochastic diāµer-ential equations.The types of stochastic diāµerential equations we are interested in are (i) stochastic diāµerential equations driven by Brownian motion, (ii) stochastic functional diāµerential equations driven by fractional Brownian motion, (iii) McKean-Vlasov stochastic diāµerential equations driven by Brownian motion,(iv) McKean-Vlasov stochastic diāµerential equations driven by fractional Brownian motion.The properties we investigate include the weak approximation rate of Euler-Maruyama scheme, the central limit theorem and moderate deviation principle for McKean-Vlasov stochastic diāµerential equations. Additionally, we investigate the existence and uniqueness of solution to McKean-Vlasov stochastic diāµerential equations driven by fractional Brownian motion, and then the Bismut formula of Lionās derivatives for this model is also obtained.The crucial method we utilised to establish the weak approximation rate of Euler-Maruyama scheme for stochastic equations with irregular drift is the Girsanov transformation. More precisely, giving a reference stochastic equa-tions, we construct the equivalent expressions between the aim stochastic equations and associated numerical stochastic equations in another proba-bility spaces in view of the Girsanov theorem.For the Mckean-Vlasov stochastic diāµerential equation model, we ļ¬rst construct the moderate deviation principle for the law of the approxima-tion stochastic diāµerential equation in view of the weak convergence method. Subsequently, we show that the approximation stochastic equations and the McKean-Vlasov stochastic diāµerential equations are in the same exponen-tially equivalent family, and then we establish the moderate deviation prin-ciple for this model.Based on the result of Well-posedness for Mckean-Vlasov stochastic diāµer-ential equation driven by fractional Brownian motion, by using the Malliavin analysis, we ļ¬rst establish a general result of the Bismut type formula for Lions derivative, and then we apply this result to the non-degenerate case of this model
Weak convergence of SFDEs driven by fractional Brownian motion with irregular coefficients
In this paper, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with H\"older continuous drift driven by fractional Brownian motion with Hurst index . The methods used in this paper are Girsanov's transformation and the property of the corresponding reference stochastic differential equations
Central Limit Theorem and Moderate Deviation Principle for McKean-Vlasov SDEs
Abstract: Under a Lipschitz condition on distribution dependent coefficients, the central limit theorem and the moderate deviation principle are obtained for solutions of McKean-Vlasov type stochastic differential equations, which generalize the corresponding results for classical stochastic differential equations to the distribution dependent setting
Large deviations for neutral stochastic functional differential equations
In this paper, under a one-sided Lipschitz condition on the drift coefficient
we adopt (via contraction principle) a exponential approximation argument to
investigate large deviations for neutral stochastic functional differential
equations.Comment: 17page
Estimate of Heat Kernel for Euler-Maruyama Scheme of SDEs Driven by {\alpha}-Stable Noise and Applications
In this paper, the discrete parameter expansion is adopted to investigate the
estimation of heat kernel for Euler-Maruyama scheme of SDEs driven by
{\alpha}-stable noise, which implies krylov's estimate and khasminskii's
estimate. As an application, the convergence rate of Euler-Maruyama scheme of a
class of multidimensional SDEs with singular drift( in aid of Zvonkin's
transformation) is obtained.Comment: 22page
Weak convergence of Euler scheme for SDEs with low regular drift
In this paper, we investigate the weak convergence rate of Euler-Maruyamaās approximation for stochastic differential equations with low regular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in some fractional Sobolev space