Elements of stochastic analysis: Gaussian processes, Brownian motion, probabilistic representation for the solution to partial differential equations, stochastic integration and stochastic differential equations.
Curriculum
teacher profile teaching materials
Brownian motion (part II). Multidimensional Brownian motion. Harmonic function and Dirichlet problem. Solution of Dirichlet problem via Brownian motion (on smooth domains). Poisson problem and its solution on smooth domains. Law of the iterated logarithm. Skorohod embedding. Donsker's invariance principle with applications.
Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Ito's formula with applications. Multidimensional Ito's formula, general formula for stochastic integral.
Stochastic differential equations (SDE). Linear SDE with solutions. Theorem for existence and uniqueness of solutions of SDE. Partial differential equations. Feynman-Kac formula. Applications to financial mathematics (introduction to the Black-Scholes model).
- An introduction to Stochastic Differential Equations (Evans)
- Brownian Motion and Stochastic Calculus (Karatzas and Shreve, 1998) https://www.springer.com/gp/book/9780387976556
- An Introduction to Stochastic Calculus with Applications to Finance (Ovidiu Calin)
https://people.emich.edu/ocalin/Teaching_files/D18N.pdf
Mutuazione: 20410457 CP430 - CALCOLO STOCASTICO in Matematica LM-40 CANDELLERO ELISABETTA
Programme
Brownian motion (part I). Definition, properties and explicit construction of the Brownian motion. Markov property. Strong Markov property and reflection principle.Brownian motion (part II). Multidimensional Brownian motion. Harmonic function and Dirichlet problem. Solution of Dirichlet problem via Brownian motion (on smooth domains). Poisson problem and its solution on smooth domains. Law of the iterated logarithm. Skorohod embedding. Donsker's invariance principle with applications.
Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Ito's formula with applications. Multidimensional Ito's formula, general formula for stochastic integral.
Stochastic differential equations (SDE). Linear SDE with solutions. Theorem for existence and uniqueness of solutions of SDE. Partial differential equations. Feynman-Kac formula. Applications to financial mathematics (introduction to the Black-Scholes model).
Core Documentation
- Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf- An introduction to Stochastic Differential Equations (Evans)
- Brownian Motion and Stochastic Calculus (Karatzas and Shreve, 1998) https://www.springer.com/gp/book/9780387976556
- An Introduction to Stochastic Calculus with Applications to Finance (Ovidiu Calin)
https://people.emich.edu/ocalin/Teaching_files/D18N.pdf
Type of delivery of the course
60 hours of lectures in class (included exercise sessions)Type of evaluation
There will be no intermediate tests/exams, the final exam will consist of 2 separate parts. Written part: There will be some questions about definitions and main results, and a few exercises to solve (proofs are not required for this part). This part will take about 2 hours and will consist of 4 exercises, each one worth 8 points. Oral part: definitions, main results and proofs. The first question for everyone will be "explain a topic of your choice". This part will take approximately 45 minutes. teacher profile teaching materials
Brownian motion (part II). Multidimensional Brownian motion. Harmonic function and Dirichlet problem. Solution of Dirichlet problem via Brownian motion (on smooth domains). Poisson problem and its solution on smooth domains. Law of the iterated logarithm. Skorohod embedding. Donsker's invariance principle with applications.
Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Ito's formula with applications. Multidimensional Ito's formula, general formula for stochastic integral.
Stochastic differential equations (SDE). Linear SDE with solutions. Theorem for existence and uniqueness of solutions of SDE. Partial differential equations. Feynman-Kac formula. Applications to financial mathematics (introduction to the Black-Scholes model).
- An introduction to Stochastic Differential Equations (Evans)
- Brownian Motion and Stochastic Calculus (Karatzas and Shreve, 1998) https://www.springer.com/gp/book/9780387976556
- An Introduction to Stochastic Calculus with Applications to Finance (Ovidiu Calin)
https://people.emich.edu/ocalin/Teaching_files/D18N.pdf
Mutuazione: 20410457 CP430 - CALCOLO STOCASTICO in Matematica LM-40 CANDELLERO ELISABETTA
Programme
Brownian motion (part I). Definition, properties and explicit construction of the Brownian motion. Markov property. Strong Markov property and reflection principle.Brownian motion (part II). Multidimensional Brownian motion. Harmonic function and Dirichlet problem. Solution of Dirichlet problem via Brownian motion (on smooth domains). Poisson problem and its solution on smooth domains. Law of the iterated logarithm. Skorohod embedding. Donsker's invariance principle with applications.
Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Ito's formula with applications. Multidimensional Ito's formula, general formula for stochastic integral.
Stochastic differential equations (SDE). Linear SDE with solutions. Theorem for existence and uniqueness of solutions of SDE. Partial differential equations. Feynman-Kac formula. Applications to financial mathematics (introduction to the Black-Scholes model).
Core Documentation
- Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf- An introduction to Stochastic Differential Equations (Evans)
- Brownian Motion and Stochastic Calculus (Karatzas and Shreve, 1998) https://www.springer.com/gp/book/9780387976556
- An Introduction to Stochastic Calculus with Applications to Finance (Ovidiu Calin)
https://people.emich.edu/ocalin/Teaching_files/D18N.pdf
Type of delivery of the course
60 hours of lectures in class (included exercise sessions)Type of evaluation
There will be no intermediate tests/exams, the final exam will consist of 2 separate parts. Written part: There will be some questions about definitions and main results, and a few exercises to solve (proofs are not required for this part). This part will take about 2 hours and will consist of 4 exercises, each one worth 8 points. Oral part: definitions, main results and proofs. The first question for everyone will be "explain a topic of your choice". This part will take approximately 45 minutes.