Course title: Markov Semigroup of Operators and Application in PDE
Course coordinator: Anindya Goswami
Pre-requisites: Measure theoretic probability theory, Stochastic Calculus (Find a brief lecture note on prerequisite measure theory, mathematical model of probability theory, and stochastic calculus in this hyperlink. This note includes links to video recordings of lectures, too.)
Objectives (relevance, goals, type of students for whom useful, outcome, etc):
Postgraduate or doctoral students who will participate in a Mathematical finance course and a course on Ito calculus might benefit from this course. It details Markov processes, their generators, martingale problems, and associated PDE, PIDE, etc.
Course content {# of lectures, each lecture is of 1.5 hours}:
Module-1 {1-5}:-
Recollection of conditional expectation given a σ-algebra (Reading Assignment), Definition of Markov Process on a Polish space, Time homogeneity and inhomogeneity, Discrete State Markov chain(MC) on Discrete Time (DT) and Continuous Time (CT), embedded chain of CT-MC, exponential distribution, conditional distribution of arrival times. {1}
Chapman Kolmogorov's equation, Semigroup property of Transition function, Rate matrix, Derivation of an Integral equation for transition functions. Differentiability of transition function for countable state MC. {1}
Kolmogorov's backward equation. Expression of rate matrix, examples. The explosion of a Markov chain. Reuter’s criterion for non-explosions and its proof. {1}
Rate matrix of Birth & Death process. Application of Reuter's criterion: Sufficient condition of non-explosion. Necessary condition of non-explosion for B&D process. Branching process: derivation of the forward and backward equation of transition functions. Two-state B&D processes: expression of transition function by solving the backward equation. Derivation of transition function for a general pure birth process. {2}
Module-2 {6-11}:-
Semimartingale representation of discrete-state Markov chain and Ito’s formula. {1}
Semigroup of bounded linear operators, strongly continuous and uniformly continuous semigroup, Infinitesimal generator. The infinitesimal generator of a uniformly continuous semigroup is bounded and vice versa. Infinitesimal generator of the semigroup of translation in a Euclidean space is unbounded, a first-order linear PDE. Infinitesimal generator of a finite state continuous time Markov chain is bounded {1}
Brownian motion and its quadratic variation, application of Ito’s rule on Ito process, harmonic function and mean value property {1}
Maximum principle of harmonic function and its application, Dirichlet problem, Stochastic representation of the bounded solution to a Dirichlet problem, regular point of boundary {2}
Zaremba's cone condition, Continuity of candidate solution at regular boundary points with detailed proofs {1}
Module -3 {12-16}
Stochastic representation of a solution to a heat equation, Sufficient condition on unbounded initial data for the existence of a classical solution to the heat equation, Tychonoff's uniqueness theorem for the heat equation, Widder’s result, and its extension on the positive solution to the heat equationThe solution to the mixed initial boundary value problem, Feynman Kac formula. {2}
Stochastic representation of the solution to the Cauchy problem for the backward heat equation with continuous terminal data, potential function, and Lagrangian {1}
Weak solution of a stochastic differential equation, PDE with variable coefficient elliptic operators, Elliptic and uniformly elliptic operators, Operator associated with weak solution to SDE and FSDE, Statement of Dirichlet problems with variable coefficient elliptic operators. Cauchy Problem with variable coefficients: Feynman-Kac formula and its proof. {2}
Module-4 {17-20}:-
Growth property of C0 semigroup. Unique semigroup generated by a bounded linear operator. Properties of C0 semigroup and its application in solving homogeneous initial value problem (hIVP), Mild solution to hIVP. {1}
Definition of resolvent operator, Statement of Hille-Yoshida theorem, Yoshida approximation, Mild solution to linear evolution problems, Inhomogeneous initial value problem, the formula of variations of constants, a sufficient condition for existence of classical solution {1}
The relation between the Feynman-Kac formula and the formula of variations of constants {1}
homogeneous non-autonomous evolution problems and solution operator, evolution system, inhomogeneous nonautonomous evolution problem and mild/generalized solution {1}
Module-5 {21-22}:-
An extension of the Hille-Yoshida theorem and its application to solve inhomogeneous non-autonomous evolution problem, Semilinear evolution problem with Lipschitz additive term and its mild solution, the integral equation of the mild solution. Existence and uniqueness of a mild solution to (sEP) {1}
Regularity of mild solution, a sufficient condition for the existence of a classical solution, a detailed proof. {1}
Suggested readings (with full list of authors, publisher, year, edition, etc.)
Introduction to Stochastic Processes: P. G. Hoel, S. C. Port and C.J. Stone (1986) Waveland Press Inc.
Siegrist, K. (2021). Probability, Mathematical Statistics, and Stochastic Processes. University of Alabama in Huntsville.
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)
Ioannis Karatzas and Steven Shreve, Brownian Motion and Stochastic Calculus, GTM, Springer-Verlag New York, 1998.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York, 1983
The lecture notes are accessible from the following link.
https://drive.google.com/drive/folders/1xsbq6dr4EP-Mok7WwpocVhw5E3ZVRkhP?usp=sharing