In this thesis, high order stochastic Runge-Kutta methods are developed for the numerical solution of (Stratonvich) stochastic differential equations and numerical results are presented. The problems associated with non-communativity of stochastic differential equation systems are addressed and stochastic Runge-Kutta methods particularly suited for such systems are derived. The thesis concludes with a discussion on various implementation issues, along with numerical results from variable stepsize implementation of a stochastic embedded pair of Runge-Kutta methods.

Thesis (Ph. D.)--Auburn University, 2008. / Abstract. Vita. Includes bibliographical references (p. 113).

Thesis (doctoral)--Universität Bonn, 1988. / Includes bibliographical references (p. 99-102).

Thesis (Ph.D.) - University of Queensland, 2003. / Includes bibliography.

>Magister Scientiae - MSc / The sustainable use of natural resources is of utmost importance for every community. In particular, it is important for every given generation to plan in such a way that proper provision is made for future generations. The scientific understanding of resources use and appreciation for its life-supporting capacity is therefore essential. Mathematical modeling has proved useful to inform the planning and management of strategies for sustainable use of natural resources. Some specific topics in resource management has been studied intensively through many decades. In particular, mining, fisheries, forestry and water resources are among these. Instead of presenting a study of the latter topics, this dissertation presents a variety of cases of mathematical modeling in resource management. The aim is to improve the general understanding of the relevant problems. We expand on existing literature, papers of other authors, and add to such studies by focusing on specific items in the work, illuminating it with further explanations and graphs, or by modifying the models through the introduction of stochastic perturbations. In particular this dissertation makes contributions by giving more explanation, on the so-called environmental Fisher information or EFI for brevity (Section 2.4 and Chapter 6), and by introducing stochasticity into a pest control model (Chapter 4) and into a savanna vegetation model (Chapter 5). In Chapter 3 we present a model from the literature pertaining to the problem of shifting cultivation, i.e, the use of forest land when used for subsistence level agricultural purposes, until the land is so degraded that the occupants abandon it and move on to a new stand. The model used to study the shifting period is similar to the forest rotation problem. A model, already in the literature, for biological control of a pest is studied in Chapter 4. Onto the deterministic model we impose a stochastic perturii bation, so that we obtain a stochastic differential equation model. We prove stochastic stability of the disease-free state, when the basic reproduction number of the pest is below unity. We have performed simulations of solutions of the stochastic system. In Chapter 5 we review an existing ordinary differential equation model for the competition between trees and grass in savanna environment. The competition between them is for soil water, fed by annual rainfall. On the other hand, trees and grass are perturbed by fire, and some other environmental forcings such as herbivores. For this ODE model, we introduce stochastic perturbations. The stochastic perturbations are in the form of three mutually independent Brownian motions. Simulations to illustrate the effect of the stochasticity are shown. We present a three-tiered predator-prey model and consider its stability in terms of Fisher information. This appears as Chapter 6. The Fisher information is defined on the basis of the so-called sustainable measures hypotheses. The model is already in the literature and in the dissertation we present several computations to show the influence of carrying capacity of prey and of mortality rate on EFI. Another problem that we consider, in Chapter 7, is that of lake eutrophication caused by excessive phosphorus inflow. The computation illustrates the management of the runoff nutrients into or out of the lake. Necessary and the sufficient conditions for an optimal utility management are obtained using standard optimal control theory. The results of this dissertation demonstrate the modeling techniques in the sustainable use of natural resources. Sustainability is the quest for equal opportunities over all generations. The manner in which this sustainability is quantified in models is being debated and improved all the time. The discourse on sustainability is especially important in view of a growing world population, and with forcings such as climate change. The most important original contribution in this dissertation is the stochastic analysis on the pest control model and the savanna model.

The concept of backward stochastic differential equation (BSDE) was initially brought up by Bismut when studying the stochastic optimal control problem. And it has been applied to describe various problems particularly to those in finance. After the fundamental work by Pardoux and Peng who proved the well-posedness of the nonlinear BSDE, the BSDE has been investigated intensively for both theoretical and practical purposes. In this thesis, we are concerned with a class of numerical methods for solving BSDEs, especially the one proposed by Zhao et al.. For this method, the convergence theory of the semi-discrete scheme (the scheme that discretizes the equation only in time) was already established, we shall further provide the analysis for the fully discrete scheme (the scheme that discretizes in both time and space). Moreover, using the BSDE as the adjoint equation, we shall construct the numerical method for solving the stochastic optimal control problem. We will discuss the situation when the control is deterministic as well as when the control is feedback.

No description available.

It has been shown that backward doubly stochastic differential equations (BDSDEs) provide a probabilistic representation for a certain class of nonlinear parabolic stochastic partial differential equations (SPDEs). It has also been shown that the solution of a BDSDE with Lipschitz coefficients can be approximated by first discretizing time and then calculating a sequence of conditional expectations. Given fixed points in time and space, this approximation has been shown to converge in mean square. In this thesis, we investigate the approximation of solutions of BDSDEs with coefficients that are measurable in time and space using a time discretization scheme with a view towards applications to SPDEs. To achieve this, we require the underlying forward diffusion to have smooth coefficients and we consider convergence in a norm which includes a weighted spatial integral. This combination of smoother forward coefficients and weaker norm allows the use of an equivalence of norms result which is key to our approach. We additionally take a brief look at the approximation of solutions of a class of infinite horizon BDSDEs with a view towards approximating stationary solutions of SPDEs. Whilst we remain agnostic with regards to the implementation of our discretization scheme, our scheme should be amenable to a Monte Carlo simulation based approach. If this is the case, we propose that in addition to being attractive from a performance perspective in higher dimensions, such an approach has a potential advantage when considering measurable coefficients. Specifically, since we only discretize time and effectively rely on simulations of the underlying forward diffusion to explore space, we are potentially less vulnerable to systematically overestimating or underestimating the effects of coefficients with spatial discontinuities than alternative approaches such as finite difference or finite element schemes that do discretize space. Another advantage of the BDSDE approach is that it is possible to derive an upper bound on the error of our method for a fairly broad class of conditions in a single analysis. Furthermore, our conditions seem more general in some respects than is typically considered in the SPDE literature.

Stochastic delay differential equations naturally arise in models of complex natural phenomena, yet continue to resist efforts to find analytical solutions to them: general solutions are limited to linear systems with additive noise and a single delayed term. In this work we solve the case of distributed delays in linear systems with additive noise. Key to our solution is the development of a consistent interpretation for integrals over stochastic variables, obtained by means of a virtual discretization procedure. This procedure makes no assumption on the form of noise, and would likely be useful for a wider variety of cases than those we have considered. We show how it can be used to map the distributed delay equation to a known multivariate system, and obtain expressions for the system's time-dependent mean and autocovariance. These are in the form of series over the system's natural modes and completely define the solution. — An interpretation of the system as an amplitude process is explored. We show that for a wide range of realistic parameters, dynamics are dominated by only a few modes, implying that most of the observed behaviour of stochastic delayed equations is constrained to a low-dimensional subspace. — The expression for the autocovariance is given particular attention. A recurring problem for stochastic delay equations is the description of their temporal structure. We show that the series expression for the autocovariance does converge over a meaningful range of time lags, and therefore provides a means of describing this temporal structure.

Let H be a separable Hilbert space. Suppose (Ω, F, Ft, P) is a complete stochastic basis with a right continuous filtration and {Wt,t ∈ R} is an H-valued cylindrical Brownian motion with respect to {Ω, F, Ft, P). U(t, s) denotes an almost strong evolution operator generated by a family of unbounded closed linear operators on H. Consider the semilinear stochastic integral equation [formula omitted] where • f is of monotone type, i.e., ft(.) = f(t, w,.) : H → H is semimonotone, demicon-tinuous, uniformly bounded, and for each x ∈ H, ft(x) is a stochastic process which satisfies certain measurability conditions. • gs(.) is a uniformly-Lipschitz predictable functional with values in the space of Hilbert-Schmidt operators on H. • Vt is a cadlag adapted process with values in H. • X₀ is a random variable. We obtain existence, uniqueness, boundedness of the solution of this equation. We show the solution of this equation changes continuously when one or all of X₀, f, g, and V are varied. We apply this result to find stationary solutions of certain equations, and to study the associated large deviation principles. Let {Zt,t ∈ R} be an H-valued semimartingale. We prove an Ito-type inequality and a Burkholder-type inequality for stochastic convolution [formula omitted]. These are the main tools for our study of the above stochastic integral equation. / Science, Faculty of / Mathematics, Department of / Graduate