1. Girsanov’s Theorem: From Game Theory to Finance Anatoliy Swishchuk Math & Comp Finance Lab Dept of Math & Stat, U of C “Lunch at the Lab” Talk December 6, 2005

2. Outline Simplest Case: Girsanov’s Theorem in Game Theory GT for Brownian Motion Applications GT in Finance Discrete-Time (B,S)-Security Markets Continuous-Time (B,S)-Security Markets Other Models in Finance: Merton (Poisson), Jump-Diffusion, Diffusion with SV General Girsanov’s Theorem Conclusion

3. Original Girsanov’s Paper Girsanov, I. V. (1960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory Probability and Its Applications, 5, 285-301. Extension of Cameron-Martin Theorem (1944) for multi-dimensional shifted Brownian motion

4. Cameron-Martin Theorem

5. Girsanov’s Theorem

6. Game Theory. I.

7. Game Theory. II.

8. Girsanov’s Theorem in Game Theory Take p=1/2-probability of success or to win- to make game fair, or (the same) to make total gain X_n martingale in nth game p=1/2 is a martingale measure (simpliest)

9. Discrete-Time (B,S)-Security Market. I.

10. Discrete-Time (B,S)-Security Market. II.

11. Discrete-Time (B,S)-Security Market. III.

12. GT for Discrete-Time (B,S)-SM Change measure from p to p^*=(r-a) / (b-a). Here: p^* is a martingale measure (discounted capital is a martingale)

13. GT for Discrete-Time (B,S)-SM: Density Process

14. Continuous-Time (B,S)-Security Market. I.

15. Continuous-Time (B,S)-Security Market. II.

16. GT for Continuous-Time (B,S)- SM. I.

17. GT for Continuous-Time (B,S)- SM. II.

18. GT for Other Models. I: Merton (Poisson) Model

19. GT for Other Models. II: Diffusion Model with Jumps

20. GT for Other Models. II: Diffusion Model with Jumps (contd)

21. GT for Other Models. III. Continuous- Time (B,S)-SM with Stochastic Volatility

22. GT for Other Models. III. Continuous- Time (B,S)-SM with Stochastic Volatility (contd)

23. General Girsanov’s Theorem (Transformation of Drift)

24. The End Thank You for Your Attention and Time! Merry Christmas!