# C8.1 Stochastic Differential Equations - Material for the year 2020-2021

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Integration theory: Riemann-Stieljes and Lebesgue integral and their basic properties

Probability and measure theory: $\sigma$-algebras, Fatou lemma, Borel-Cantelli, Radon-Nikodym, $L^p$-spaces, basic properties of random variables and conditional expectation,

Martingales in discrete and continuous time: construction and basic properties of Brownian motion, uniform integrability of stochastic processes, stopping times, filtrations, Doob's theorems (maximal and $L^p$-inequalities, optimal stopping, upcrossing, martingale decomposition), martingale (backward) convergence theorem, $L^2$-bounded martingales, quadratic variation;

Stochastic Integration: Ito’s construction of stochastic integral, Ito’s formula.

16 lectures

### Assessment type:

- Written Examination

Stochastic differential equations (SDEs) model evolution of systems affected by randomness. They offer a beautiful and powerful mathematical language in analogy to what ordinary differential equations (ODEs) do for deterministic systems. From the modelling point of view, the randomness could be an intrinsic feature of the system or just a way to capture small complex perturbations which are not modelled explicitly. As such, SDEs have found many applications in diverse disciplines such as biology, physics, chemistry and the management of risk.

Classic well-posedness theory for ODEs does not apply to SDEs. However, when we replace the classical Newton-Leibnitz calculus with the (Ito) stochastic calculus, we are able to build a new and complete theory of existence and uniqueness of solutions to SDEs. Ito formula proves to be a powerful tool to solve SDEs. This leads to many new and often surprising insights about quantities that evolve under randomness. This course is an introduction to SDEs. It covers the basic theory but also offers glimpses into many of the advanced and nuanced topics.

By the end of this course, students will be able to analyse if a given SDEs admits a solution, characterise the nature of solution and explain if it is unique or not. The students will also be able to solve basic SDEs and state basic properties of the diffusive systems described by these equations.

Recap on martingale theory in continuous time, quadratic variation, stochastic integration and Ito's calculus.

Levy's characterisation of Brownian motion, stochastic exponential, Girsanov theorem and change of measure, Burkholder-Davis-Gundy, Martingale represenation, Dambis-Dubins-Schwarz.

Strong and weak solutions of stochastic differential equations, existence and uniqueness.

Examples of stochastic differential equations. Bessel processes.

Local times, Tanaka formula, Tanaka-Ito-Meyer formula.

- J. Obloj,
*Continuous martingales and stochastic calculus*online notes.

Students are encouraged to study all the material up to and including the Ito formula prior to the course. - D. Revuz and M. Yor,
*Continuous martingales and Brownian motion*(3rd edition, Springer).

- I. Karatzas and S. E. Shreve,
*Brownian Motion and Stochastic Calculus*, Graduate Texts in Mathematics 113 (Springer-Verlag, 1988). - L. C. G. Rogers & D. Williams,
*Diffusions, Markov Processes and Martingales Vol 1 (Foundations) and Vol 2 (Ito Calculus)*(Cambridge University Press, 1987 and 1994). - R. Durrett,
*Stochastic Calculus*(CRC Press). - B. Oksendal,
*Stochastic Differential Equations: An introduction with applications*(Universitext, Springer, 6th edition). - N. Ikeda & S. Watanabe,
*Stochastic Differential Equations and Diffusion Processes*(North--Holland Publishing Company, 1989). - H. P. McKean,
*Stochastic Integrals*(Academic Press, New York and London, 1969).