Existence And Uniqueness Of Solutions Of Stochastic Differential Equations Driven By A Fractional Brownian Motion

Before the1990s, Brownian motion and the theory of SDE based on it occupied animportant position in the stochastic analysis, and they were widely applied to the fieldssuch as finance, random networks, and so on. However, with the further research, manyscholars find some phenomena in financial markets such as the fluctuation of stock pricefollows a " biased random walk" process and the rate of return has "decile-shaped fat tail"distribution, which can be only exactly reasonable explained by the self-similarity andlong-term memory of fractional Brownian motion (FBM). So, the mathematical theory ofthe fractional Brownian motion is an important tool to study the fractal market.The main work: The first is to improve and the existing fractional stochasticdifferential equations (FSDE) and fractional It formulas; Second, we give the generalsolutions of the corresponding linear FSDE with Hurst exponent H∈(1/3,1/2)andH∈(1/n,1/(n-1))respectively; Finally, the proof of existence and uniqueness forsolutions of FSDE based on the fractional Maruyama notations with Hurst indexH∈(1/3,1/2)is given. These results can go further to improve the theory of FSDE.This thesis is divided into five chapters. Chapter one describes the backgrounds andsignificance of this question, the domestic and international research profiles in this area,the main content and architecture of this paper. In the second chapter, we first give thepreparatory knowledge, introduce FBM, fractional integral and FSDE, then improve andpromote the existing FSDE and the fractional It formulas to the new ones. Chapter threededuces the general solution of the corresponding linear FSDE with Hurst exponentH∈(1/3,1/2)and H∈(1/n,1/(n-1))respectively. In the fourth chapter, we use thePicard successive approximation method to prove the existence and uniqueness ofsolutions of FSDE with Hurst index H∈(1/3,1/2). Then, make an error estimate on theapproximate solution. In the last chapter, we summarize the full-text content andinnovations and put forward the next research work.