Inder K Rana: An Introduction to Measure and Integration. (First Edition) Narosa Publishers
1977.
II. Stochastic Optimal Control by M.K. Ghosh
We study controlled diffusion processes governed by controlled
stochastic differential equations of Ito's type. We introduce the notion
of admissible controls, Markov
controls etc. We discuss various cost evaluation criteria over
finite and infinite horizon.
We derive Hamilton-Jacobi-Bellman (HJB) equations using dynamic programming
principle. We discuss the existence and uniqueness of solutions of these equations.
Finally we give a characterization of optimal controls as minimizing selectors
of appropriate (pseudo) Hamiltonians associated with the HJB equations.
III. Brownian Motion by A.Goswami, B.Rajeev and S.Athreya
Existence, basic properties
- like scaling , independent increments, finite
dimensional distributions, Martingales associated
with BM, statementof Levy's theorem,
path properties (nowhere differentiability, unbounded
variation, quadratic variation).
Markov Property and applications : Ordinary
Markov property, Strong Markov property, Reflection principle,
and Zero set.
Brownian motion on Path space, weak convergence and the
invariance principle.
IV. Markov Chains by K.B. Athreya
Review of Markov chains in discrete time and countable
state space and the limit theory of such chains.
Markov chains on general state spaces - basic definitions and examples. Chapman-Kolmogorov
equations. Harris irreducibility, recurrence and minorization. Limit theorems for regenerative
sequences. Limit theory of Harris recurrent Markov chains. Markov chains on metric spaces -
Feller continuity, Stationary measures and convergence questions.
Pre-requisities:
Basic theory of Markov chains in discrete time and countable state space. Basic real analysis,
convergence of sequences in Ceasaro mean, power series, interchange of limit and summation,
interchange of limit and integration.
IV. Abelian Sand Pile Model by S.Athreya
We will discuss the Abelian sandpile model in the box $\Lam_n = [-n, n]^d
\cap \Zd$ for $d \ge 1$. We shall discuss its basic set-up and main
properties. We will present simulations for the same. Time permitting we
will discuss current research and some unsolved problems in the area.
V. Percolation theory Course outline by Rahul Roy
We will study percolation on trees and on the square lattice.
On the tree, we will define the critical parameter and the critical
exponents and obtain them explicitly. Here we will need results from
branching processes.
On the square lattice, we will sketch the proof of the fact that the
critical parameter for bond percolation on the square lattice is 1/2.
For this we will need to develop some correlation inequalities as well
as study the growth of the percolation cluster.
VI. Stochastic Calculus by B.V. Rao
In these lectures we start with
the Wiener Integral and then proceed to the definition
of Ito Integral. We shall prove a version of Ito Formula,
a chain rule for Brownian differentials.
We shall briefly discuss the existence of local time
using Tanaka Formula.