Transcription of 1 Simulating Brownian motion (BM) and …
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1 Simulating Brownian motion (BM) and …
Copyrightc 2013 by Karl Sigman1 Simulating Brownian motion (BM) and geometric Brownianmotion (GBM)For an introduction to how one can construct BM, see the Appendix at the end ofthese stochastic processB={B(t) :t 0}possessing (wp1) continuous sample paths is calledstandard Brownian motion (BM) (0) = both stationary and independent (t) B(s) has a normal distribution with mean 0 and variancet s,0 s < )and3)together can be summarized by: Ift0= 0< t1< t2< < tk, then the incrementrvsB(ti) B(ti 1), i {1,..k}, are independent withB(ti) B(ti 1) N(0,ti ti 1)(normal with mean 0 and varianceti ti 1). In particular,B(ti) B(ti 1) is independent ofB(ti 1) =B(ti 1) B(0).If we only wish to simulateB(t) at one fixed valuet, then we need only generate a unitnormalZ N(0,1) and setB(t) = tZ. But typically, we will want to simulate (say)kvaluesat timest0= 0< t1< t2< < tkto get the entire vector (with correlated coordinates):(B(t1).
1.1 BM with drift X(t) = ˙B(t) + twill denote the BM with drift 2R and variance term ˙>0. It has continuous sample paths and is de ned by 1. X(0) = 0.
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