# Fiches de cours 2017-2018

## Stochastic calculus II

Filipovic Damir

English

#### Summary

This course gives an introduction to fundamental notions and techniques of stochastic calculus in continuous time necessary for applications in finance such as option pricing and hedging.

#### Content

Topics include :

• Ito calculus
• Stochastic differential equations
• Martingale representation
• Girsanov theorem
• Optimal stochastic control
• Jump processes
• Numerical simulation

#### Keywords

Ito calculus, diffusion, martingale representation, change of measure, Brownian motion, compound Poisson process

#### Learning Prerequisites

##### Required courses

Stochastic calculus I

#### Learning Outcomes

By the end of the course, the student must be able to:
• Explain the stochastic integral with respect to a Brownian motion
• Explain the notion of an Ito processes with finite activity jumps and its quadratic variation
• Apply Ito's formula to multivariate Ito processes with finite activity jumps
• Compute the stochastic exponential of an Ito process with finite activity jumps
• Explain the notion of a stochastic differential equation, the existence, uniqueness, and Markov property of its solution
• Apply the Feynman-Kac theorem on the stochastic representation of solutions to partial differential equations
• Solve a stochastic differential equation formally, for the linear case, and numerically, for the general case
• Derive the HJB equation for some basic stochastic optimal control problems
• Compute the optimal strategy for some basic optimal portfolio choice and consumption problems, via the HJB equation and the martingale method
• Explain the three pillars of stochastic calculus: Ito's formula, Girsanov's theorem, and the martingale representation theorem

#### Transversal skills

• Use a work methodology appropriate to the task.

#### Teaching methods

Lectures, exercises, homework

#### Assessment methods

• 20% homework
• 30% midterm exam
• 50% final exam

Midterm and final exams are open book (only lecture notes)

#### Supervision

 Office hours No Assistants Yes Forum No

#### Resources

No

##### Bibliography

Björk, T. (2004), "Arbitrage Theory in Continuous Time", Oxford University Press

Glasserman, P. (2004), "Monte Carlo Methods in Financial Engineering", SpringerVerlag

Lamberton, D. and Lapeyre, B. (2000), "Introduction to Stochastic Calculus Applied to Finance", Chapman&Hall/CRC

Oksendal, B. (2007), "Stochastic Differential Equations. An Introduction with Applications", Springer Verlag

Shreve, S. (2004), "Stochastic Calculus for Finance II. Continuous-Time Models", Springer Verlag

#### Prerequisite for

Credit risk
Derivatives (taken concurrently)
Fixed income analysis
Real options and financial structuring

### Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11  EXTRANEF_126
11-12
12-13
13-14EXTRANEF_126  EXTRANEF_126
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand