The objective of this course is to give an introduction to the probabilistic techniques required to understand some of the most widely used models of continuous mathematical finance.

The first part of this course, given by D. Kreher, discusses stochastic integrals with respect to Brownian motion and selected topics in stochastic analysis such as the change of variable formula (Girsanov's theorem), the martingale representation theorem, and stochastic differential equations. We further discuss the general theory of risk-neutral pricing in continuous time.

The second part, given by U. Horst, provides an introduction into stochastic optimal control with applications to utility maximization and portfolio optimization. We discuss both the traditional Hamilton-Jacobi-Bellman approach that derives value functions and optimal strategies through PDE methods as well as more recent approaches using backward stochastic differential equations (BSDEs). We apply to optimal control techniques to portfolio optimization and portfolio liquidation problems.

Here is a tentative outline (the focus may vary with student interest):

- Review of Brownian motion and the pathwise Ito-Calculus (known from Stochastic Finance I)
- Ito calculus for Brownian motion

a) The Ito Integral

b) The martingale representation theorem

c) The geenralized Black-Scholes model - Topics in diffusion theory

a) Stochastic diferential equations

b) Girsanov Theorem - Risk neutral derivative pricing

- Stochastic optimal control and PDEs

a) The dynamic programing principle

b) The HJB equation

c) Viscosity theory

d) Application to portfolio optimization ("Merton problem") - The BSDE approach to optimal control

a) Backward stochastic differential equations

b) Comparision principle for BSDEs

c) Applications to LQ problems

We meet **Monday, 9-11 (RUD 26, 0.311)** and **Wednesday, 9-11 (RUD 26, 0.307)**. Here are tentative Lecture Notes (we start with Section 3).

The **tutorials** take place, Mon, 11-13 (RUD 25, 3.008). There will be no exercises. Instead, students are expected to give group presentations on topics related to and complementing the course material. Possible topics include: diffusion processes with jumps, exotic options (barrier, lookback, ...), linear-quadratic control problems, large investor models, mean-field games, ...

Students taking this course are expected to have a strong background in probability theory (at least at the level of "Stochastik II").

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