lect0428

# lect0428 - IEOR 4106 Introduction to Operations Research...

This preview shows pages 1–2. Sign up to view the full content.

IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2009, Professor Whitt Brownian Motion, Gambler’s Ruin and Martingales Tuesday, April 28 Sections 10.1-10.3. Brownian motion We here discuss the gambler’s ruin problem for Brownian motion, using martingales. 1. Basic Deﬁnitions Let B ( t ) : t 0 } be standard ( μ = 0, zero-drift, and σ 2 = 1, unit variance) Brownian motion (BM). A stochastic process { Y ( t ) : t 0 } is a martingale (MG) with respect to another stochas- tic process { Z ( t ) : t 0 } if E [ Y ( t ) | Z ( u ) , 0 u s ] = Y ( s ) for 0 < s < t . As an extra technical regularity condition, we require that E [ | Y ( t ) | ] < for all t as well. The stochastic process { Z ( t ) : t 0 } above is giving relevant information. The segment { Z ( s ) : 0 s t } gives the history up to time t . Often the information process Z is just the given stochastic process Y . Then we just say that { Y ( t ) : t 0 } is a martingale (MG), without saying “with respect to.” We will use the fact that standard BM { B ( t ) : t 0 } is a martingale with respect to itself. Then we just say that standard BM is a martingale. But we shall be interested in martingales with respect to BM that are themselves appropriate functions of BM. It can be shown that they too are simply martingales (with respect to their own “internal” history), but we will simply show that they are martingales with respect to BM. . A nonnegative random variable T is a stopping time relative to a stochastic process { Z ( t ) : t 0 } if, for any time t , the event { T t } depends on Z ( s ) only for 0 s t . Stopping before time t depends only upon the history up to time t . The event that a stopping time T is less than or equal to t cannot depend on the future of the reference stochastic process { Z ( s ) : s 0 } after time t . The

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/26/2011 for the course IEOR 4106 taught by Professor Whitt during the Spring '08 term at Columbia.

### Page1 / 5

lect0428 - IEOR 4106 Introduction to Operations Research...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online