- Lecturer: Michael Monoyios
Course term: Michaelmas
Course lecture information: 16 hours of lectures in MT
Course overview:
The course gives the mathematical theory underlying continuous-time and continuous-process models that are used in finance. It begins with the definition and properties of Brownian motion (BM), leading to construction and properties of the Ito stochastic integral with respect to BM (then discussion of the integral with respect to a continuous (local) martingale and with respect to an Ito process). The centrepiece of the course is Ito's (change of variable) formula for functions of BM (and for functions of Ito processes). Important classes of processes, such as exponential martingales, are introduced, and Levy's characterisation of BM is given. Stochastic differential equations (SDEs), whose solutions are Markov diffusion processes, are discussed, with different notions of solution, and conditions for existence and uniqueness of solutions. Theorems with application in finance are presented: the martingale representation theorem, Girsanov's theorem (connecting absolutely continuous changes of probability measure to absolutely continuous change of drift), and the connection to partial
differential equations (PDEs), as exemplified by the Feynman-Kac
theorem.
differential equations (PDEs), as exemplified by the Feynman-Kac
theorem.
Course synopsis:
Motivation: financial models based on differential equations with
randomness; Brownian motion (BM): definition; limit of a random walk; quadratic and total variation properties; non-differentiability of paths; Markov property (strong Markov property discussed); reflection principle.
Construction of the stochastic integral with respect to BM: simple integrands to general adapted integrands; properties of the integral: the Ito isometry, quadratic variation, martingale and local martingale properties; extension to integral with respect to continuous local martingales and with respect to Ito processes; the Ito formula.
Stochastic differential equations (SDEs); strong and weak solutions; Markov property; diffusions.
Martingale representation theorem; Girsanov theorem; Feynman-Kac
theorem and connection to PDEs.
Reading List:
1. Steven Shreve, Stochastic calculus for finance II: continuous-time models, Springer 2004
2. Marek Capinski, Ekkehard Kopp and Janusz Traple, Stochastic calculus for finance, Cambridge University Press 2012
3. Fima C. Klebaner, Introduction to stochastic calculus with applications, Third edition, Imperial College Press 2012
4. Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, Springer 1999
5. Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, Second Edition, Springer 1991
6. Jean-Francois Le Gall, Brownian motion, martingales, and stochastic calculus, Springer 2016
randomness; Brownian motion (BM): definition; limit of a random walk; quadratic and total variation properties; non-differentiability of paths; Markov property (strong Markov property discussed); reflection principle.
Construction of the stochastic integral with respect to BM: simple integrands to general adapted integrands; properties of the integral: the Ito isometry, quadratic variation, martingale and local martingale properties; extension to integral with respect to continuous local martingales and with respect to Ito processes; the Ito formula.
Stochastic differential equations (SDEs); strong and weak solutions; Markov property; diffusions.
Martingale representation theorem; Girsanov theorem; Feynman-Kac
theorem and connection to PDEs.
Reading List:
1. Steven Shreve, Stochastic calculus for finance II: continuous-time models, Springer 2004
2. Marek Capinski, Ekkehard Kopp and Janusz Traple, Stochastic calculus for finance, Cambridge University Press 2012
3. Fima C. Klebaner, Introduction to stochastic calculus with applications, Third edition, Imperial College Press 2012
4. Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, Springer 1999
5. Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, Second Edition, Springer 1991
6. Jean-Francois Le Gall, Brownian motion, martingales, and stochastic calculus, Springer 2016