1 Probability and Statistics- 2 Probability and Stochastic Processes- 3 Ito Stochastic Calculus- 4 Stochastic Differential Equations- 5 Stochastic Taylor Expansions- 6 Modelling with Stochastic Differential Equations- 7 Applications of Stochastic Differential Equations- 8 Time Discrete Approximation of Deterministic Differential Equations- 9 Introduction to Stochastic Time Discrete Approximation- 10 Strong Taylor Approximations- 11 Explicit Strong Approximations- 12 Implicit Strong Approximations- 13 Selected Applications of Strong Approximations- 14 Weak Taylor Approximations- 15 Explicit and Implicit Weak Approximations- 16 Variance Reduction Methods- 17 Selected Applications of Weak Approximations- Solutions of Exercises- Bibliographical Notes

Numerical Solution of Stochastic Differential Equations

D. B. Owen: Handbook of Statistical Tables. London: Pergamon Press; Reading, Massachusetts: Addison‐Wesley, 1962. Pp. xii+580. 70s.

Handbook of Statistical Tables.

Natural systems are undeniably subject to random fluctuations, arising from either environmental variability or thermal effects. The consideration of those fluctuations supposes to deal with noisy quantities whose variance might at times be a sizable fraction of their mean levels. It is known that, under these conditions, noisy fluctuations can interact with the system's nonlinearities to render counterintuitive behavior, in which an increase in the noise level produces a more regular behavior. In systems with spatial degrees of freedom, this regularity takes the form of spatiotemporal order. An overview is presented of the mechanisms through which noise induces, enhances, and sustains ordered behavior in passive and active nonlinear media, and different spatiotemporal phenomena are described resulting from these effects. The general theoretical framework used in the analysis of these effects is reviewed, encompassing the theory of stochastic partial differential equations and coupled sets of ordinary stochastic differential equations. Experimental observations of self-organized behavior arising out of noise are also described, and details on the numerical algorithms needed in the computer simulation of these problems are given.

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Spatiotemporal order out of noise

A method is proposed for the numerical solution of Ito stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler iterative scheme. It requires the Runge–Kutta iterative scheme to be applied to a different stochastic differential equation obtained by subtraction of a correction term from the given one.

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The numerical solution of stochastic differential equations

The Status of Population Balances

A simulation procedure is presented in this work which can analyze the behavior of any dispersed phase system consisting of particles whose random behavior is specified in terms of probability functions. The procedure distinguishes itself from its predecessors in being free from arbitrary discretization of time or any other parameter along which the system evolves, and its ability to predict the behavior of randomly behaving small populations. Where population balance equations, which describe the behavior of particulate systems, cannot be readily solved, the simulation technique presented herein represents an efficient alternative to the modeling of such systems. Tables and a flow chart are given which would enable the use of the method for any system with specified particle behavior.

Simulation of particulate systems using the concept of the interval of quiescence

An algorithm is derived for solving a large class of Ito random integral equations. The derivation of the algorithm involves approximate discretization of the given Ito equation. The Ito integrals arising out of discretization are expressed as functions of normal random variables. The algorithm gives a sample pathwise solution and is readily implementable on a digital computer.

Numerical Solution of Ito Integral Equations

A puristic analysis of population balance—II.

Mathematical models of cell populations which recognize them to be segregated into individual cells differing from one another in state and behavior lead to complicated integro-differential equations. Analysis of fluctuations in small populations leads to the even more complex situation of coupled integro-differential equations. An early technique by Kendall for simulating the behavior of populations which obey the birth-and-death model has been extended in this work to include the factor of variation in individual cellular state and behavior. The resulting algorithm is probabilistically exact in the sense that the random variables to be generated have exactly derived distribution functions, and involves no arbitrary discretization of the time interval, characteristic of simulation techniques in general. The simulated systems include populations distributed according to their age in a birth-and-death process and a batch culture of cells distributed according to their mass which grow and reproduce by binary fission.

J.D. Borwanker

Papers

Simulation of particulate systems using the concept of the interval of quiescence

Numerical Solution of Ito Integral Equations

A puristic analysis of population balance—II.

Monte Carlo simulation of microbial population growth

Analysis of population balance—III: Agglomerating populations