## "Theoretical and practical aspects of the numerical solution of stochastic differential equations" (Diploma thesis, 2012)

Author:            Martin Zinner

Supervisors:   Dr. rer. nat. Jan Rudl (Technische Universität Dresden)

Prof. Dr. rer.nat. René Schilling (Technische Universität Dresden)

Short description:

The interest rate model of Cox, Ingersoll and Ross ("CIR model") can be expressed by the following stochastic differential equation driven by a Brownian motion.

It can be shown, that the solution of this stochastic differential equation never attains negative values. Unfortunately, an explicit solution cannot be provided, thus a numerical approximation scheme has to be used. We use the Euler scheme to approximate the solution, what gives rise to two problems. Since the iterations in the Euler scheme attain negative values with positive probability, we may obtain negative values, which is not wanted in the simulation of the interest rate. In addition, the iteration after the occurrence of a negative value cannot be calculated, since the square root is defined for positive values. We illustrate how to resolve these problems in Section 3.5.

Chapter 2 and Chapter 3 develope the theory for one dimensional stochastic differential equations driven by a Brownian motion. The first part (Chapter 2) shows the existence and uniqueness of (strong) solutions under the usual Lipschitz conditions. The Euler scheme will be used to approximate the (strong) solution uniformly and the order of the uniform convergence will be proven.

The second part (Chapter 3) consists of proving the existence and uniqueness of (strong) solutions under a Yamada condition and demonstrating how the Euler scheme approximates the solution under these assumptions. Since the square root function is not Lipschitz continuous in the second variable but satisfies the Yamada condition, this weaker condition has to be used for the CIR model.

Finally we apply the general theoretical results from Chapter 3 to the CIR model and discuss how various recent papers handle these problems in the special case of the CIR model.