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University of Kansas

1. Liu, Yanghui. Numerical solutions of rough differential equations and stochastic differential equations.

Degree: PhD, Mathematics, 2016, University of Kansas

URL: http://hdl.handle.net/1808/21866

In this dissertation, we investigate time-discrete numerical approximation schemes for rough differential equations and stochastic differential equations (SDE) driven by fractional Brownian motions (fBm). The dissertation is organized as follows. In Chapter 1, we introduce the basic settings and define time-discrete numerical approximation schemes. In Chapter 2, we consider the Euler scheme for SDEs driven by fBms. For a SDE driven by a fBm with Hurst parameter H> ½ it is known that the existing (naive) Euler scheme has the rate of convergence n^{1-2H}. Since the limit H → ½ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^{o} SDEs for H=½, the convergence rate of the naive Euler scheme deteriorates for H → ½. The new (modified Euler) approximation scheme we are introducing in this chapter is closer to the classical Euler scheme for Stratonovich SDEs for H=½ and it has the rate of convergence γ_{n}^{-1}, where γ_{n}=n^{ 2H-½} when H ½ it is known that the existing (naive) Euler scheme has the rate of convergence n^{1-2H}. Since the limit H → ½ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^{o} SDEs for H=½, the convergence rate of the naive Euler scheme deteriorates for H → ½. The new (modified Euler) approximation scheme we are introducing in this chapter is closer to the classical Euler scheme for Stratonovich SDEs for H=½ and it has the rate of convergence γ_{n}^{-1}, where γ_{n}=n^{ 2H-½} when H \frac34. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {X_{t}, 0 ≤ t ≤ T} is the solution of a SDE driven by a fBm and if {X_{t}^{n}, 0 ≤ t ≤ T} is its approximation obtained by the new modified Euler scheme, then we prove that γ_{n} (X^{n-X}) converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when H∈ ( ½, \frac34]. In the case H \frac 34, we show the L^{p} convergence of n(X^{n}_{t-X}_{t}) and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme. In Chapter 3, we consider the Crank-Nicolson method for a SDE driven by a m-dimensional fBm. We consider the Crank-Nicolson method in three cases: (i) m1; (ii) m=1 and and the drift term is equal to non-zero; and (iii) m=1 and the drift term is equal to zero. We will show that the convergence rate of the Crank-Nicolson method is n^{ 1/2-2H}, n^{-1/2-H} and n^{-2H}, respectively, in these three cases, and these convergence rates are exact in the sense that the error process for the Crank-Nicolson method…
*Advisors/Committee Members: Nualart, David (advisor), Hu, Yaozhong (advisor), Nualart, David (cmtemember), Hu, Yaozhong (cmtemember), Feng, Jin (cmtemember), Tu, Xuemin (cmtemember), Zhang, Jianbo (cmtemember), Soo, Terry (cmtemember).*

Subjects/Keywords: Mathematics; fourth moment theorem; fractional Brownian motions; Numerical solutions; rough differential equations; stochastic differential equations

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APA (6^{th} Edition):

Liu, Y. (2016). Numerical solutions of rough differential equations and stochastic differential equations. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/21866

Chicago Manual of Style (16^{th} Edition):

Liu, Yanghui. “Numerical solutions of rough differential equations and stochastic differential equations.” 2016. Doctoral Dissertation, University of Kansas. Accessed December 16, 2017. http://hdl.handle.net/1808/21866.

MLA Handbook (7^{th} Edition):

Liu, Yanghui. “Numerical solutions of rough differential equations and stochastic differential equations.” 2016. Web. 16 Dec 2017.

Vancouver:

Liu Y. Numerical solutions of rough differential equations and stochastic differential equations. [Internet] [Doctoral dissertation]. University of Kansas; 2016. [cited 2017 Dec 16]. Available from: http://hdl.handle.net/1808/21866.

Council of Science Editors:

Liu Y. Numerical solutions of rough differential equations and stochastic differential equations. [Doctoral Dissertation]. University of Kansas; 2016. Available from: http://hdl.handle.net/1808/21866