Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1995. / Includes bibliographical references (p. 112-119).

In this thesis, we are interested in the stochastic differential equation with jumps under regime switching. Firstly, we investigate a continuous-time version of the mean-variance portfolio selection model with jumps under regime switching. The portfolio selection proposed and analyzed for a market consisting of one bank account an d multiple stocks. The random regime switching is assumed to be independent of the underlying Brownian motion and jump processes. Secondly, we consider the problem of pricing contigent claims on a stock whose price process is modeled by a Levy process. Since the market is incomplete and there is not a unique equivalent martingale measure. We study approaches to pricing options. Finally, we investigate a continuous-time version Markowitz's mean-variance portfolio selection problem which is studied in a market with one bank account, one stock and proportional transaction costs. This is a singular stochastic control problem. Via a series of transformations, the problem is turned into a double obstacle problem.

This thesis is in two parts. Part I concerns simulation of random fields using the circulant embedding method, and Part II studies the numerical solution of stochastic differential equations (SDEs).

Stochastic Differential Equations (SDE) offer a rich framework to model the probabilistic evolution of the state of a system. Numerical approximation methods are typically needed in evaluating relevant Quantities of Interest arising from such models. In this dissertation, we present novel effective methods for evaluating Quantities of Interest relevant to computational finance when the state of the system is described by an SDE.

Within the context of rank-based interacting particle systems, we study the fluctuations in a symmetric ensemble around its stable distribution. This system is inspired by the classic Atlas model but represents its opposite pole since both the highest- and lowest-ranked particles will have non-zero drifts. In the first part of the dissertation, we derive a fine asymptotic analysis that includes a Law of Large Numbers. The lack of monotonicity of the ensemble requires that we develop alternative tools to those traditionally used in the analysis of the Atlas model. In the second part of the dissertation, we characterize the system’s fluctuations and show that, as the number of particles goes to infinity, they converge weakly to the mild solution of the Additive Stochastic Heat Equation on the real line with a symmetric initial condition. To establish this result, we use the technique proposed by Dembo and Tsai, 2017, where the Empirical Measure Process is used as a proxy for the ensemble’s fluctuations. We expect that a combination of our work, and the available knowledge about the Atlas model, could help draw a full picture of how a finite rank-based interacting particle system with a general drift structure fluctuates around its stationary distribution as the number of particles goes to infinity, a long-standing open question in the field.

In this dissertation, we investigate a very interesting class of quasi-linear stochastic partial differential equations. The main purpose of this article is to prove an existence result for such type of stochastic differential equations with non-standard growth conditions. The main difficulty in the present problem is that the existence cannot be easily retrieved from the well known results under Lipschitz type of growth conditions [42]. / Dissertation (MSc)--University of Pretoria, 2010. / Mathematics and Applied Mathematics / unrestricted

Novel analytical nonperturbative techniques are developed in the area of nonlinear and linear stochastic differential equations and applications are considered to a variety of physical problems. First, a method is introduced for deriving first- and second-order moment equations for a general class of stochastic nonlinear equations by performing a renormalization at the level of the second moment. These general results, when specialized to the weak-coupling limit, lead to a complete set of closed equations for the first two moments within the framework of an approximation corresponding to the direct-interaction approximation. Additional restrictions result in a self-consistent set of equations for the first two moments in the stochastic quasi-linear approximation. The technique is illustrated by considering two specific nonlinear physical random problems: model hydrodynamic and Vlasov-plasma turbulence. The equations for the phenomenon of hydrodynamic turbulence are examined in more detail at the level of the quasi-linear approximation, which is valid for small turbulence Reynolds numbers. Closed form solutions are found for the equations governing the random fluctuations of the velocity field under the assumption of special time-dependent, uniform or sheared, mean flow profiles. Constant, transient and oscillatory flows are considered. The smoothing approximation for solving linear stochastic differential equations is applied to several specific physical problems. The problem of a randomly perturbed quantum mechanical harmonic oscillator is investigated first using the wave kinetic technique. The equations for the ensemble average of the Wigner distribution function are defined within the framework of the smoothing approximation. Special attention is paid to the so-called long-time Markovian approximation, where the discrete nature of the quantum mechanical oscillator is explicitly visible. For special statistics of the random perturbative potential, the dependence of physical observables on time is examined in detail. As a last example of the application of the stochastic techniques, the diffusion of a scalar quantity in the presence of a turbulent fluid is investigated. An equation corresponding to the smoothing approximation is obtained, and its asymptotic long-time version is examined for the cases of zero-mean flow and linearly sheared mean flow. / Ph. D.

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)

In this thesis, we address two problems arising from the application of stochastic differential equations (SDEs). The first one pertains to the detection of asset bubbles, where the price process solves an SDE. We combine the strict local martingale model together with a statistical tool to instantaneously check the existence and severity of asset bubbles through the asset’s historical price process. Our approach assumes that the price process of interest is a CEV process. We relate the exponent parameter in the CEV process to an asset bubble by studying the future expectation and the running maximum of the CEV process. The detection of asset bubbles then boils down to the estimation of the exponent. With a dynamic linear regression model, inference on the exponent can be carried out using historical price data. Estimation of the volatility and calibration of the parameters in the dynamic linear regression model are also studied. When using SDEs in practice, for example, in the detection of asset bubbles, one often would like to simulate its paths using the Euler scheme to study the behavior of the solution. The second part of this thesis focuses on the convergence property of the Euler scheme under the assumption that the coefficients of the SDE are locally Lipschitz and that the solution has no finite explosion. We prove that if a numerical scheme converges uniformly on any compact time set (UCP) in probability with a certain rate under the globally Lipschitz condition, then when the globally Lipschitz condition is replaced with a locally Lipschitz one plus a no finite explosion condition, UCP convergence with the same rate holds. One contribution of this thesis is the proof of √n-weak convergence of the asymptotic normalized error process. The limit error process is also provided. We further study the boundedness for the second moment of the weak limit process and its running maximum under both the globally Lipschitz and the locally Lipschitz conditions. The convergence of the Euler scheme in the sense of approximating expectations of functionals is also studied under the locally Lipschitz condition