# An introduction to stochastic calculus, statistics of stochastic processes and application to finance

## Presentation

Finance:  List of Lectures :

• A review on discrete-time market models
• Towards continuous-time models
• Stochastic processes in continuous-time : elements of theory
• Stochastic stochastic integral and Itô calculus
• Stochastic Differential Equations
• Black-Scholes model and pricing in the complete markets realm

Statistics :

The following chapters will be studied :

• Basic material on stochastic calculus
• SDE
• Maximum likelihood estimation for diffusions

Organisation:

Finance :

Lectures, Tutorials and LabWork will be given in English (independently of the audience). Students will be given a copy of the Lecture Notes at the beginning of the term.

Statistics:

lecture notes will be provided to students at the beginning of the course.

## Objectives

Finance:

The student will be able to:

• Use the fundamental techniques related to the Itô calculus/integration theory.
• Apply and use the basic definitions of Mathematical Finance and Arbitrage theory such as the notions of portfolio, arbitrage, equivalent martingale measure and completeness.
• Compute the arbitrage price of European type contingent claims on various market models (in particular on the Black-Scholes model).

Statistics:

The aim of this part is to understand the relevance of classical Stochastic Differential Equations (SDEs) driving the various financial models, and the estimation of the underlying parameters of interest.

The student will be able to :

• Solve explicitly the classical SDE arising in financial mathematics.
• Proceed to the statistical estimation of the underlying parameters, in particular the maximum likelihood estimation.
• Show the various properties of these estimators via the study of limit theorems for Markovian diffusions.
• Discretize a SDE by Euler scheme.

## Needed prerequisite

Discret time martingales
Finance market and models in discrete time

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...