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Unformatted text preview: IEOR 4106, Spring 2011, Professor Whitt Brownian Motion, Martingales and Stopping Times Thursday, April 21 1 Martingales A stochastic process { Y ( t ) : t ≥ } is a martingale (MG) with respect to another stochastic process { Z ( t ) : t ≥ } if E [ Y ( t )  Z ( u ) , ≤ 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 ≥ } 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 ≥ } is a martingale (MG), without saying “with respect to.” Example 1.1 Let B ( t ) : t ≥ } be standard ( μ = 0, zerodrift, and σ 2 = 1, unit variance) Brownian motion (BM). We will use the fact that standard BM { B ( t ) : t ≥ } is a martingale with respect to itself. Then we just say that standard BM is a martingale. The MG property of BM, like almost everything else, is proved by applying the property of stationary and independent increments . We show that E [ B ( t )  B ( u ) , ≤ u ≤ s ] = B ( s ) for 0 ≤ s < t : E [ B ( t )  B ( u ) , ≤ u ≤ s ] = E [ B ( s ) + B ( t ) B ( s )  B ( u ) , ≤ u ≤ s ] (add and subtract) = E [ B ( s )  B ( u ) , ≤ u ≤ s ] + E [ B ( t ) B ( s )  B ( u ) , ≤ u ≤ s ] (conditional expectation of sum is sum of conditional expectations) = B ( s ) + E [ B ( t ) B ( s )  B ( u ) , ≤ u ≤ s ] because ( E [ B ( s )  B ( u ) , ≤ u ≤ s ] = B ( s ) we know B ( s ), because it is in the condition, so nothing random) = B ( s ) + E [ B ( t ) B ( s )] (independent increments) = B ( s ) + 0 = B ( s ) (BM has mean 0) . But we shall be interested in martingales with respect to BM that are themselves appro priate functions of BM. Example 1.2 An example considered below is the stochastic process { B ( t ) 2 t : t ≥ } . We let Y ( t ) = B ( t ) 2 t and Z ( t ) = B ( t ) in the definition above. Thus we say that { B ( t ) 2 t : t ≥ } is a MG with respect to BM { B ( t ) : t ≥ } . However, it can be shown that { B ( t ) 2 t : t ≥ } is also a MG with respect to { B ( t ) 2 t : t ≥ } . And similarly for other functions of BM that we will consider. 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. 2 Stopping Times A nonnegative random variable T is a stopping time relative to a continuoustime stochastic process { Z ( t ) : t ≥ } if, for any time t , the event { T ≤ t } depends on Z ( s ) only for 0 ≤ s ≤ t ....
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This note was uploaded on 01/16/2012 for the course IEOR 4106 taught by Professor Whitward during the Spring '11 term at Columbia College.
 Spring '11
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