B0319 - STOCHASTIC CALCULUS I

Academic Year 2022/2023

  • Moduli: Stefano Pagliarani (Modulo 1) Andrea Pascucci (Modulo 2)
  • Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)

Learning outcomes

At the end of the course, the students know the fundamentals of the theory of stochastic processes in discrete and continuous time that naturally intervene in applications in Physics, Economics, Biology and Engineering.

Course contents

1 - Stochastic processes and martingales:
Definition in the paths space, and law of a process. Equivalence among stochastic processes: equivalence in law, modifications and indistinguishability. Construction: Kolmogorov extension theorem. Continuous processes and Kolmogorov continuity theorem. Brownian motion and heat equation: definition, existence, Markov property, transition density and fundamental solution. Martingales and properties. Doob’s decomposition. Stopping times and stopped processes. Optional sampling theorem. Doob’s maximal inequality. Brownian martingales. Spaces of continuous martingales. The Usual Hypotheses. Stopping times and martingales.

2 - Ito’s stochastic integration:
Bounded variation functions. Riemann-Stieltjes integral. Deterministic Ito’s formula. Quadratic variation of Brownian motion. Simple processes in L^2. Wiener’s integral. Properties of the stochastic integral: martingality and Ito isometry. Extension of the integral to L^2. Integral and stopping times. Quadratic variation and bounded variation trajectories of the stochastic integral. Integral in L^2_loc. Local martingales. Ito processes: unique representation and differential notation. Ito formula for the Brownian motion. Geometric Brownian motion. General one-dimensional Ito formula. Multidimensional Brownian motion. Multidimensional Ito formula.

3 - Stochastic Differential Equations (SDE) and their link with PDEs: Standard assumptions: existence and uniqueness of strong solutions. Exit times. Infinitesimal generator. Probabilistic representation of the solution of Dirichlet and Cauchy-Dirichlet problem. Examples of probabilistic representations: Dirichlet problem for the Laplace operator, Cauchy- Dirichlet problem for the heat equation, characteristic method. Fundamental solution and transition density. Linear stochastic equations: Langevin equation and Kolmogorov operator, Kalman condition and controllability.

Prerequisites: differential calculus, Lebesgue integration theory, basic probability theory.

Readings/Bibliography

- Pascucci (2010), PDE and Martingale Methods in Option Pricing. Chapters 3, 4, 5, 9 and 12

- Varadhan, Stroock (1997) Multidimensional Diffusion Processes. Chapters 1, 2, 4 and 5

Teaching methods

Frontal lectures on the board

Assessment methods

Oral examination to asses the knowledge and the skills acquired by the students, who will be asked to respond to theoretical questions (definitions, theorems, proofs, etc.), and possibly to solve brief exercises.

Teaching tools

Lecture notes in pdf covering some parts of the program.

Office hours

See the website of Stefano Pagliarani

See the website of Andrea Pascucci