
Finally, they will analyze the simulation data according to the theories presented at the beginning of course.Īt the end of the course, we will analyze the dynamical data of more complicated systems, such as financial markets or meteorological data, using the basic theory of stochastic processes. Then, they will use these theories to develop their own python codes to perform numerical simulations of small particles diffusing in a fluid. The students will first learn the basic theories of stochastic processes. A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics. It is freely available for Windows, Mac, and Linux through the Anaconda Python Distribution. We will use the Jupyter (iPython) notebook as our programming environment. We treat both discrete and continuous time settings, emphasizing the importance of right-continuity of the sample path and ltration in the latter. This course is an introduction to stochastic processes through numerical simulations, with a focus on the proper data analysis needed to interpret the results. Chapter 4 deals with ltrations, the mathematical notion of information pro-gression in time, and with the associated collection of stochastic processes called martingales. This is in stark contrast to the deterministic motion of planets and stars, which can be perfectly predicted using celestial mechanics. (e) Derivation of the Black-Scholes Partial Dierential Equation.

Therefore, such motions must be modeled as stochastic processes, for which exact predictions are no longer possible. (c) Stochastic dierential equations and Ito’s lemma. The motion of falling leaves or small particles diffusing in a fluid is highly stochastic in nature.
