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.
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.
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