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Brownian motion, stochastic calculus in finance

Recommended for the 2nd year of Specialist’s program.

Start: February 8, 2022

Methods of Instruction
Lectures: Tuesday, 4.45-6.10pm
Seminars: Tuesday, 6.30-7.40pm

Language: Russian/English
Format: online

The purpose of the proseminar (which includes a course of lectures and seminars) is to introduce students to the basic concepts of the theory of Wiener process, also called Brownian motion, the associated theory of stochastic integration and their use for solving financial problems. 

Key concepts: Itô's lemma, stochastic exponential, Girsanov theorem, Wiener process

Topics:

 

 

Topic

 1 Wiener process. Construction using a random Fourier series.
 2 Remarkable properties of the trajectories of the Wiener process.
 3 Law(s) of the iterated logarithm.
 4 Continuity modulus. 
 5 Filtration generated by the Wiener process. 
 6 Stochastic integrals. Isometrics and localization. 
 7 Ito formula and its applications.
 8 Levi's theorem (martingale characterization of the Wiener process). 
 9 Stochastic exponent. Girsanov's theorem.
 10 Stochastic equations. Strong decisions.
 11 Girsanov's theorem and weak solutions of stochastic equations. 
 12 The Novikov condition for uniform integrability of the stochastic exponential. 
 13 The predictable representation theorem. 
 14 Black-Scholes model and option price. 
 15 Theoretical and practical aspects of BS formulas. 
 16 Optimal portfolio management. Bellman equation (HJB). 
 17 Inverse Stochastic Equations (BSDE). 
 18 Nonlinear BSDEs. 
 19 Markov BSDEs and their relation to PDEs.
 20 Ornstein-Uhlenbeck process and pair trading. 
 21 Introduction to the theory of arbitration.