MWF, 11-12:10, Baskin Engineering 169

Stochastic differential equations (SDE) arise in many branches of science and engineering. This is an introduction to SDE, requiring only upper division probability and differential equations, since we will approach the analysis of questions about SDE through the associated differential equations.Approximate topical outline (with some papers that are worth knowing about as a way to get into the literature)

I have provided some important and key citations to various topics in the course. Papers that do not have a link here are available at the California Digital Library.

The Basics

Brownian Motion and White Noise

Brown,R. 1828. A brief account of microscopical observations made in the months of June, July, and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. The Philosophical Magazine 4:161-173

Gray, A. 1874. Notice [comparing Darwin and Brown]. American Naturalist 8:473-479

Oster, G.2002. Darwin's motors Nature 417:25 [He conludes: "Darwin may have had the best idea that anyone ever had. Think about it."]

Raser, J.M. and E.K. O'Shea. 2005. Noise in gene expression: Origins, consequences and control. Science 309:2010

Reed, M. C. 2004. Why is mathematical biology so hard? Notices of the AMS 51:338-342.

The Gambler's Ruin

Ornstein Uhlenbeck Process

Higham, D.J. 2001. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review 43:525

Il"in, A.M. and R.Z. Khas'minskii. 1964. On equations of Brownian motion. Theory of Probability and Its Applications 9:421-444

Prajneshu and R. Venugopalan. 1999. von Bertalanffy growth model in a random environment. Canadian Journal of Fisheries and Aquatic Sciences 56:1026

Ito and Stratonovich Calculi

Stratonovich, R.L. 1966. A new representation for stochastic integrals and equations. SIAM Journal on Control 4:362-371

Wong, E. and M. Zakai. 1965. On the relation between ordinary and stochastic differential equations. International Journal of Engineering Science 3:213-229

Kolmogorov backward and forward (aka Fokker Planck) equations

Feynman-Kac formula and path integrals

Brush, S. 1961. Functional integrals and statistical physics. Reviews of Modern Physics 33:79

Graham, R. 1977. Path integral formulation of general diffusion processes. Zeitshcrift fur Physik B 26:281

Keller, J.B. and D.W. McLaughlin. 1975. The Feynman integral. American Mathematical Monthly 82:451

Ludwig, D. 1975. Persistence of dynamical systems under random perturbations. SIAM Review 17:605

Stratonovic, R.L. 1971. On the probability functional of diffusion processes. Selected Translations in Statistics and Probability 10:273

Poisson Increment SDE and the Generalized Ornstein Uhlenbeck Process

Kac, M. 1974. A stochastic model related to the telegrapher's equation. Rocky Mountain Journal of Mathematics 4:497

Applications (mainly based on approximate and asymptotic solutions of SDE)

The Einstein-Smoluchowski theory of Brownian motion

Daniels, H.E. 1960. Approximate solutions of Green's type for univariate stochastic processes. Journal of the Royal Statistical Society B 22:376

Papanicolaou, G.C. 1978. Asymptotic analysis of stochastic equations. pg 111-179 in Studies in Mathematics, Volume 18: Studies in Probability Theory (M. Rosenblatt, editor); Mathematical Association of America

The diffusion (Kramers) theory of reaction rates and escape from a domain of attraction

Lande, R. 1985. Expected time for random genetic drift of a population between stable phenotypic states. Proceedings of the National Academy of Science 82:7641

Kramers, H.A. 1940. Brownian motion in a field of force and the diffusion theory of chemical reactions. Physica 7:284

Matkowsky, B. and Z. Schuss. 1977. The exit problem for randomly perturbed dynamical systems. SIAM Journal on Applied Mathematics 33:365

Newman, C.M., Cohen, J.E. and C. Kipnis. 1985. Neo-darwinian evolution implies punctuated equilibria. Nature 315:400


Fluctuations in threshold systems

Frank, F.C. 1951. On spontaneous asymmetric synthesis. Biochemica et Biophysica Acta 11:459-463

Mangel, M. Barrier transitions driven by fluctuations, with application to ecology and evolution. Theoretical Population Biology 45:16

Mangel, M. and D. Ludwig. Probability of extinction in a stochastic competition. 1977. SIAM Journal on Applied Mathematics 33:256

Yildirim, N., Santillan, M., Horike, D. and M. Mackey. 2004. Dynamics and bistability in a reduced model of the lac operon. Chaos 14:279


Gillespie's tau-method for chemical kinetics

Gibson, M.A. and J. Bruck. 2000. Efficient exact stochastic simulation of chemical systems with many species and many channels. Journal of Chemical Physics A 104:1876

Gillespie, D.T. 2000. The chemical Langevin equation. Journal of Chemical Physics 113:297

Gillespie, D.T. 2001. Approximate accelerated sotchastic simulation of chemically reacting systems. Journal of Chemical Physics 115:1716

Gillespie, D.T. and L.R. Petzold. 2003. Improved leap-size selection for accelerated stochastic simulation. Journal of Chemical Physics 119:8229

Knessel, C., Mangel, M., Matkowsky, B.J., Schuss, Z. and C. Tier. 1984. Solution of Kramers-Moyals equations for problems in chemical physics. Journal of Chemical Physics 81:1285-1293

Moyal, J.E. 1949. Stochastic processes and statistical physics. Journal of the Royal Statistical Society B 11:150-210


There is no text, but I will suggest some books and papers that might interest you. Grades will be determined by homework (assigned during class periods, with homework from one week due the following Fridayd), a take home final and participation (I expect students to attend class).

But then, in any science, the 'noise' might prove to be not merely something to get rid of, but the essential phenomenon of interest. It seems curious (at least, to a physicist) that this was first seen clearly not in physics, but in biology. In the late 19th century many biologists saw it as the major task confronting them to confirm Darwin's theory by exhibiting the detailed mechanism by which evolution takes place...Biologists have a mechanistic picture of the world because, being trained to believe in causes, they continue to use the full power of their brains to search for them--and so they find them.

Jaynes, E.T.. 2003. Probability Theory. The logic of science. Cambridge University Press (pg 230...328)

There is no official text for the class (your notes place that role) , but some books that I like are

Dixit, A.K. and R.S. Pindyck. 1994. Investment Under Uncertainty. Princeton University Press

Glasserman, P. 2003. Monte Carlo Methods in Financial Engineering. Springer Verlag

Grasman, J. and O.A. van Herwaarden. 1999. Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications. Springer Verlag.

Karlin, S. and H.M. Taylor. 1981. A Second Course in Stochastic Processes. Academic Press.

Kloeden, P.E. and E. Platen. 1999. Numerical Solution of Stochastic Differential Equations. Springer Verlag.

Risken, H. 1996. The Fokker-Planck Equation. Springer Verlag.

Schuss, Z. 1980. Theory and Application of Stochastic Differential Equations. Wiley

Wax, N. Selected Papers on Noise and Stochastic Processes. Dover [inexpensive and worth owning]

Wong, E. Stochastic Processes in Information and Dynamical Systems. McGraw Hill


and here are some other ones


Arnold, L. 1974. Stochastic Differential Equations: Theory and Applications. Wiley

Costantino, R.F. and R.A. Desharnais. 1991. Population Dynamics and the Tribolium Model: Genetics and Demography. Springer

Fleming, W.H. and R.W. Rishel. 1975. Deterministic and Stochastic Optimal Control. Springer

Gihman, I. and A.V. Skorokhod. 1972. Stochastic Differential Equations. Springer Verlag

Gillespie, D.T. 1992. Markov Processes. An Introduction for Phyiscal Scientists. Academic Press

Jaynes, E.T.. 2003. Probability Theory. The logic of science. Cambridge University Press

Kac, M. Probability and Related Topics in the Physical Sciences. American Mathematical Society

Keizer, J. 1987. Statistical Thermodynamics of Nonequilibrium Processes. Springer

Ludwig, D. 1974. Stochastic Population Theories. Springer.

Riccardi, L. 1977. Diffusion Processes and Related Topics in Biology. Springer

Riccardi, L. 1990 (editor) Lectures in Applied Mathematics and Informatics, Manchester University Press.

Schulman, L. 1981. Techniques and Applications of Path Integrals. Wiley

Skorokhod, A.V., Hoppenstead, F.C., and H. Salehi. 2002. Random Perturbation Methods. Springer Verlag

Varadhan, S.R.S. 1980. Diffusion Problems and Partial Differential Equations. Springer

This book has its roots in two different areas of mathematics: pure mathematics, where structures are discovered in the context of other mathematical structures and investigated, and applications of mathematics, where mathematical structures are suggested by real-world problems arising in science and engineering, investigated, and then used to address the motivating problem. While there are philosophical differences between applied and pure mathematical scientists, it is often difficult to sort them out.

Skorokhod, A.V., Hoppenstead, F.C., and H. Salehi. 2002. Random Perturbation Methods. Springer Verlag (pg v)