Introduction to Financial Mathematics

1. Basic problems and approaches to financial mathematics

• • • Excursus into the history of financial mathematics

    • Asset and derivative pricing

    • Portfolio optimization. Risk management

2. Random Processes: Basic Concepts

• Filtrations. Adapted and predictable processes

• Brownian motion

• Quadratic variation of processes

3. Martingales and semimartingales

4. Stochastic integration

• Ito’s formula

• Doob–Meyer decomposition theorem

5. Stochastic differential equations

• Ito processes

• Strong and weak solutions

• Ito’s theorem on existence and uniqueness of strong solutions

6. Stochastic exponentials, growth-optimal portfolios

7. Markov processes. Kolmogorov’s backward and forward equations

8. Arbitrage Pricing Theory

• Girsanov’s theorem

• Risk-neutral measure

• Martingale representation theorem

• Fundamental theorems of asset pricing

• Black-Scholes model. Fair price of European contingent claims

• Delta hedging

Stochastic Optimal Control

1. Diffusion controlled processes and classical PDEs

• Dynamic programming principle. Hamilton-Jacobi-Bellman equation

• Merton’s portfolio problem

• Investment-consumption problem with random time horizon 

• Production-consumption model on infinite horizon

• Stochastic optimization problem with linear-quadratic regulator

2. Viscosity Solutions in Stochastic Optimal Control

• Non-smoothness of the value function and other motivations for the viscosity approach. Definitions and examples of viscosity solutions. Connection with classical solutions

• Construction of solutions using the vanishing viscosity method

• From dynamic programming principle to viscosity solutions of HJB. Optimality criterion in terms of viscosity solutions. Comparison principle for viscosity solutions

• Model of irreversible investments

• Super-replication cost in models with uncertain volatility

3. Optimal Stopping Problems

• Formulation of the optimal stopping problem. Dynamic programming and viscosity principle

• Methods for solving the optimal stopping problem in one-dimensional case

• Fair price of American put options

• Formulation of the optimal switching problem. Explicit solution in one-dimensional two-regime case. Examples from economics

4. Backward Stochastic Differential Equations

• Existence and uniqueness results. Comparison principle. Connection with PDEs

• Pontryagin’s maximum principle. Optimization of a family of controlled BSDEs

• Reflected BSDEs and optimal stopping problems

• Applications in option hedging

Optional and further topics

• Duality methods in stochastic optimization

• Differential games

• Mean field games

Also planned

• Review of the latest articles on mathematical finance topics

• Analysis of articles on market microstructure using the studied mathematical apparatus

[1] Pham H., Continuous-time Stochastic Control and Optimization with Financial Applications. Springer Berlin, Heidelberg, 2009

[2] Fleming W.H., Soner H.M., Controlled Markov processes and viscosity solutions. Springer Science & Business Media, 2006

[3] Oksendal B., Stochastic differential equations: an introduction with applications. Springer Science & Business Media, 2013

[4] Shreve S., Stochastic Calculus for Finance II: Continuous-Time Models. Spring

er Finance, 2010

[5] Karatzas I., Shreve S., Methods of Mathematical Finance, Springer New York, 2016

[6] Protter P., Stochastic integration and differential equations. Springer, 2005