Floyd B. Hanson

Applied Stochastic Processes and Control for Jump-Diffusions

This self-contained, practical, entry-level text integrates the basic principles of applied mathematics, applied probability, and computational science for a clear presentation of stochastic processes and control for jump-diffusions in continuous time. The author covers the important problem of controlling these systems and, through the use of a jump calculus construction, discusses the strong role of discontinuous and nonsmooth properties versus random properties in stochastic systems. The book emphasizes modeling and problem solving and presents sample applications in financial engineering and biomedical modeling. Computational and analytic exercises and examples are included throughout. While classical applied mathematics is used in most of the chapters to set up systematic derivations and essential proofs, the final chapter bridges the gap between the applied and the abstract worlds to give readers an understanding of the more abstract literature on jump-diffusions. An additional 160 pages of online appendices are available on a Web page that supplements the book. Audience This book is written for graduate students in science and engineering who seek to construct models for scientific applications subject to uncertain environments. Mathematical modelers and researchers in applied mathematics, computational science, and engineering will also find it useful, as will practitioners of financial engineering who need fast and efficient solutions to stochastic problems. Contents List of Figures; List of Tables; Preface; Chapter 1. Stochastic Jump and Diffusion Processes: Introduction; Chapter 2. Stochastic Integration for Diffusions; Chapter 3. Stochastic Integration for Jumps; Chapter 4. Stochastic Calculus for Jump-Diffusions: Elementary SDEs; Chapter 5. Stochastic Calculus for General Markov SDEs: Space-Time Poisson, State-Dependent Noise, and Multidimensions; Chapter 6. Stochastic Optimal Control: Stochastic Dynamic Programming; Chapter 7. Kolmogorov Forward and Backward Equations and Their Applications; Chapter 8. Computational Stochastic Control Methods; Chapter 9. Stochastic Simulations; Chapter 10. Applications in Financial Engineering; Chapter 11. Applications in Mathematical Biology and Medicine; Chapter 12. Applied Guide to Abstract Theory of Stochastic Processes; Bibliography; Index; A. Online Appendix: Deterministic Optimal Control; B. Online Appendix: Preliminaries in Probability and Analysis; C. Online Appendix: MATLAB Programs

Persistence times of populations with large random fluctuations.

Abstract The growth of populations which undergo large random fluctuations can be modelled with stochastic differential equations involving Poisson processes. The problem of determining the persistence time is that of finding the time of first passage to some small critical population size. We consider in detail a simple model of logistic growth with additive Poisson disasters of fixed magnitude. The expectation and variability of the persistence time are obtained as solutions of singular differential-difference equations. The dependence of the persistence time of a colonizing species on the parameters of the model is discussed. The model may also be viewed as random harvesting with fixed quotas and a comparison is made between the mean extinction time and those for deterministic models.

Logistic Growth with Random Density Independent Disasters

Abstract A stochastic differential equation for a discontinuous Markov process is employed to model the magnitude of a population which grows logistically between disasters which are proportional to the current population size (density independent disasters). The expected persistence or extinction time satisfies a singular differential-difference equation. When the number of disasters, in the absence of recovery, between carrying capacity and extinction is two, analytical expressions are found for the mean persistence time. A comparison is made with the previously studied case of decrements of constant magnitude. When the two problems are suitably normalized, the mean survival times are quite different for the two models, especially in a critical range of initial population sizes near extinction. The expected survival time of a colonizing species is discussed quantitatively in terms of the parameters of the model. Insight into the nature of the probability density of the survival time is obtained by means of computer simulations. The densities resemble gamma densities and long tails appear when the disasters are density independent, implying a small change of long term survival. When the number of consecutive disasters which take the population from carrying capacity to extinction is large, a singular decomposition is employed to solve the differential-difference equation for the mean persistence time. The results are discussed in terms of population strategies in hazardous environments.

Persistence in density dependent stochastic populations

Abstract Persistence, as measured by time to extinction, is studied in a density dependent population that is subject to small environmental and demographic randomness. Diffusion processes are formally derived from branching processes in constant and random environments. The moment generating function of the extinction time, which satisfies a second order ordinary differential equation, is found asymptotically in the limit of small diffusion and is related to the diffusion limit of the Galton-Watson process and the Ornstein-Uhlenbeck process. Extinction occurs with probability one, though the mean and variance of the extinction time are found to be exponentially large and suggest the extinction time is exponentially distributed. The notion of persistence is compared with other qualitative measures of stability. Four examples are studied and compared.

Optimal Harvesting with Both Population and Price Dynamics

We consider the effects of large inflationary price fluctuations on the computed optimal harvest strategy for a randomized Schaefer model. Both prices and population sizes are assumed random with both background (Wiener) and jump (Poisson) components. Population fluctuations are assumed to be density independent, i.e., relative changes are independent of population size. Stochastic dynamic programming is employed to find the optimal harvesting effort and economic return for a realistic set of bioeconomic data for Pacific halibut. It is found that inflationary effects have a pronounced influence on the optimal return, even in a hazardous or disastrous environment. However, optimal harvesting effort levels are much less sensitive to inflationary effects.

Kinetic Models for a Gas with Internal Structure

Kinetic model equations for a polyatomic gas have been obtained by employing a diagonal approximation in the truncation of the linearized operator of the Wang Chang‐Uhlenbeck equation. These models possess an H theorem, and a Chapman‐Enskog analysis of these equations yields the correct transport coefficients, such as bulk viscosity, viscosity, and thermal conductivity for translational and internal degrees of freedom. The restrictions of the Wang Chang‐Uhlenbeck equation are discussed, and it is believed that these model equations should be most useful in the study of those kinetic problems involving scalar transport phenomena, in particular, sound propagation, heat transfer, and shock structure.

Techniques in Computational Stochastic Dynamic Programming

Differential dynamic programming (DDP) is a variant of dynamic programming in which a quadratic approximation of the cost about a nominal state and control plays an essential role. The method uses successive approximations and expansions in differentials or increments to obtain a solution of optimal control problems. The DDP method is due to Mayne [11, 8]. DDP is primarily used in deterministic problems in discrete time, although there are many variations. Mayne [11] in his original paper did give a straight-forward extension to continuous time problems, while Jacobson and Mayne [8] present several stochastic variations. The mathematical basis for DDP is given by Mayne in [12], along the relations between dynamic programming and the Hamiltonian formulation of the maximum principle. A concise, computationally oriented survey of DDP developments is given by Yakowitz [16] in an earlier volume of this series and the outline for deterministic control problems in discrete time here is roughly based on that chapter. Earlier, Yakowitz [15] surveys the use of dynamic programming in water resources applications, nicely placing DDP in the larger perspective of other dynamic programming variants. Also, Jones, Willis and Yeh [9], and Yakowitz and Rutherford [17] present brief helpful summaries with particular emphasis on the computational aspects of DDP.

Option pricing for a stochastic-volatility jump-diffusion model with log-uniform jump-amplitudes

An alternative option pricing model is proposed, in which the stock prices follow a diffusion model with square root stochastic volatility and a jump model with log-uniformly distributed jump amplitudes in the stock price process. The stochastic-volatility follows a square-root and mean-reverting diffusion process. Fourier transforms are applied to solve the problem for risk-neutral European option pricing under this compound stochastic-volatility jump-diffusion (SVJD) process. Characteristic formulas and their inverses simplified by integration along better equivalent contours are given. The numerical implementation of pricing formulas is accomplished by both fast Fourier transforms (FFTs) and more highly accurate discrete Fourier transforms (DFTs) for verifying results and for different output

Optimal consumption and portfolio control for jump-diffusion stock process with log-normal jumps

A computational solution is found for a optimal consumption and portfolio policy problem in which the underlying stock satisfies a geometric jump-diffusion in which both the diffusion and jump amplitude are log-normally distributed. The optimal objective is to maximize the expected, discounted utility of terminal wealth and the cumulative discounted utility of instantaneous consumption. The jump-diffusion allows for a more realistic distribution, skewed toward negative jumps and having leptokurtic behavior in which the tails are thicker so that the distribution is more slender around the peak than normal. Computational issues pertinent to jump-diffusion calculations are discussed.

Stochastic Analysis of Jump-Diffusions for Financial Log-Return Processes

A jump-diffusion log-return process with log-normal jump amplitudes is presented. The probability density and other properties of the theoretical model are rigorously derived. This theoretical density is fit to empirical log-returns of Standard & Poor’s 500 stock index data. The model repairs some failures found from the log-normal distribution of geometric Brownian motion to model features of realistic financial instruments: (1) No large jumps or extreme outliers, (2) Not negatively skewed such that the negative tail is thicker than the positive tail, and (3) Non-leptokurtic due to the lack of thicker tails and higher mode.