# Stochastic analysis II, spring 2017

# Stochastic analysis II, spring 2017

**Teacher:** Dario Gasbarra

**Scope:** 5 cr

**Type:** Advanced studies

**Teaching:**

**Topics:**Kolmogorov extension theorem and stochastic processes, Kolmogorov continuity criterium, Paul Levy Construction of Brownian motion. Processes with jumps: Poisson process and counting processes, random measures. General theory in continuous time, filtration, predictable σ-algebra, stopping times and predictable times. Stochastic integrals with respect to processes with locally finite variation. Continuous time martingales, existence of right continuous modification, predictable and dual predictable projections, compensator and Doob-Meyer decomposition, locally square integrable martingales, quadratic and predictable variation. Stochastic integration with respect to continuous martingales, Kunita-Watanabe inequality, Ito isometry and Ito integral, Ito formula and generalizations. Burkholder Davis Gundy inequalities. Ito-Tanaka formula and local times. Change of measure and Girsanov theorem. Stochastic differential equations, weak and strong solutions. Partial differential equations and Feynman-Kac formula. Applications: stochastic filtering, stochastic control, option pricing in mathematical finance.

**Prerequisites:** Probability theory, stochastic analysis I

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## News

## Teaching schedule

Weeks 11-18, lectures on Wednesday 12-14 in B120 and Thursday 12-14 in room D123, and exercises on Tuesday 10-12 in room D123

Easter holiday 13.-19.4.

## Exams Home exam assignment, latex

## Course material lecture notes (last update on 10.5.2017)

### https://wiki.helsinki.fi/download/attachments/198320251/levybm.m?version=2&modificationDate=1484767144667&api=v2

## Main references:

Richard Bass, Stochastic processes, Cambridge University Press 2011.

Fabrice Baudoin, Diffusion Processes and Stochastic Calculus. European Mathematical Society Ems Textbooks in Mathematics 2014.

Alexander Gushchin, Stochastic calculus for quantitative finance. ISTE Press, Optimization in insurance and finance 2015.

## Registration

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## Exercises

### Assignments

- (28.3.2017)
- (4.4.2017)
- (11.4.2017)
- (2.5.2017)

### Exercise classes

Group | Day | Time | Room | Instructor |
---|---|---|---|---|

1. | Tuesday | 10-12 | D123 | Dario Gasbarra |

## Course feedback

Course feedback can be given at any point during the course. Click here.