Course of Stochastic Differential Equations with Numerics

Master in Mathematical Engineering
School of Engineering
University of Padova

Teacher: Tiziano Vargiolu

• Preliminaries
  Probability measure as measure, and expectation as integral. Conditional expectation and limit theorems: monotone convergence, Fatou's lemma and dominated convergence.

• Brownian motion
  Definition. Covariation function of a Brownian motion. Construction of Brownian motion from a symmetric random walk. Non-differentiability of Brownian motion paths. Measuring variability: total (first) variation and quadratic variation.

• Martingales
  Definition of stochastic processes and of filtration, stochastic process adapted to a filtration and filtration generated by a stochastic process. Definition of (super-)(sub-)martingales. Examples: conditioning, sums or products of i.i.d. random variables. Brownian martingales: Brownian motion, exponential and square of a Brownian motion.

• Stochastic integral
  Motivations and an example. Stochastic integrals of simple processes and properties: linearity, expectation and the Ito isometry. Stochastic integrals of adapted processes in M^2. Stochastic integrals of deterministic functions. Dependence on the integration interval: continuity, quadratic variation and martingale property.

• Stochastic calculus
  Ito processes and stochastic differential. The Itō-Doeblin formula. Area under a Brownian motion path. Geometric Brownian motion. The Wiener integral.

• Stochastic differential equations
  Definitions: drift and diffusion, solution of a SDE, existence and uniquess theorem. The Markov property. The infinitesimal generator of a Markov process. Links between SDEs and PDEs: the Kolmogorov backward equation and the Feynman-Kac representation.

• Change of probability and the Girsanov theorem
  Absolutely continuous and equivalent probability measures. Measures defined by a density. The Radon-Nykodym theorem. Conditional expectation under a new probability measure. The Girsanov theorem: drift transformation in the Brownian motion. A characterization of Brownian motion. The martingale representation theorem. An inverse of the Girsanov theorem.

• Multidimensional Brownian motion
  Random vectors. The vector of means and the covariance matrix. Characteristic function of a random vector. Gaussian random vectors. Definitions of standard and correlated Brownian motions. Correlation matrix and Cholesky decomposition. The cross variation. Cross variation of two Browian motions, in the cases of independence and of correlation.

• Multidimensional stochastic calculus
  Multidimensional stochastic integrals: linearity, mean and covariance matrix, martingale property, cross variation.

• Stopping times and martingales
  Definition of stopping time, entrance and exit time. Stopped stochastic processes and their properties. Definition of local martingale. Construction of the stochastic integral for a process with locally square integrable sample paths.
  Stochastic differential equations in multidimensional spaces. The Markov property. The infinitesimal generator of a Markov process.
  Links between SDEs and PDEs: the Kolmogorov backward equation and the Feynman-Kac representation.
  The Girsanov theorem and the martingale representation theorem in the case of a multidimensional Brownian motion.

• Numerical implementations
  Simulation of continuous and discrete random variables: the pseudo-inverse method.
  Random walk:
  
  Monte Carlo methods. Variance reduction techniques: antithetic variables, control variables, importance sampling.
  Variance reduction in Monte Carlo schemes,
  
  Markov chain approximations of stochastic processes: a general result. Applications: the Cox-Ross-Rubinstein model, the Euler scheme for stochastic differential equations.
  Monte Carlo Euler scheme,
  Parallelized Monte Carlo Euler scheme,
  
  Tree methods. Building computationally simple trees: method of Nelson-Rawaswamy. Implementation of an option written on an exponential Ornstein-Uhlenbeck process. Parallel computing in the case of Monte Carlo schemes and of tree methods.
  Computationally simple trees,

• Steven E. Shreve, Stochastic Calculus for Finance II - Continuous-Time Models, Springer 2004

• Bernt Oksendal, Stochastic Differential Equations - An introduction with applications, Springer 1998

• Ovidiu Calin, An informal introduction to stochastic calculus with applications, World Scientific 2015
• Ubbo F. Wiersema, Brownian motion calculus, Wiley 2008

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