Pre-requisite

• 236A Working knowledge of probability and differential equations.
• 236B basic material (Brownian motion, Ito integral, SDE) covered in 236A (see outline below).

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Outlines

• Brownian motion -- characterization, representation, martingale property, quadratic variation, first passage time.
• Ito integral -- construction, Ito representation theorem, martingale representation theorem, Stratonovich integral.
• Stochastic differential equations -- Ito formula, solution methods, existence, uniqueness, weak and strong solutions.
• Derivative pricing -- delta-hedging, self-financing portfolio, Black-Scholes equation, put-call parity.
• Linear filtering and estimation -- Kolmogorov-Wiener filter, Kalman-Bucy filter (discrete and continuous in time).

• Diffusion theory -- Markov property, generator, Dynkin formula, Kolmogorov backward equation, the resolvent, Feynman-Kac formula, the Girsanov theorem.
• Applications to boundary value problems for partial differential equations.
• Risk-Neutral pricing -- risk-neutral probability, Girsanov and martingale representation theorems in finance
• Optimal stopping and American derivatives.
• Jump-diffusion processes

Homework assignments

Grading
Weekly assignments.