- Lecturer: Michael Monoyios
General Prerequisites:
Course Term: Michaelmas
Course Lecture Information: 16 lectures
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 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 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.
Learning Outcomes:
Course Synopsis:
Motivation: financial models based on differential equations with
randomness; Brownian motion (BM): definition; limit of a random walk;
(rigorous construction as background); quadratic and total variation
properties; non-differentiability of paths; Markov property (strong
Markov property as background); 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.
randomness; Brownian motion (BM): definition; limit of a random walk;
(rigorous construction as background); quadratic and total variation
properties; non-differentiability of paths; Markov property (strong
Markov property as background); 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.