Simulating Brownian motion ( BM ) and geometric Brownian
نویسنده
چکیده
2) and 3) together can be summarized by: If t0 = 0 < t1 < t2 < · · · < tk, then the increment rvs B(ti) − B(ti−1), i ∈ {1, . . . k}, are independent with B(ti) − B(ti−1) ∼ N(0, ti − ti−1) (normal with mean 0 and variance ti − ti−1). In particular, B(ti) − B(ti−1) is independent of B(ti−1) = B(ti−1)−B(0). If we only wish to simulate B(t) at one fixed value t, then we need only generate a unit normal Z ∼ N(0, 1) and set B(t) = √ tZ. But typically, we will want to simulate (say) k values at times t0 = 0 < t1 < t2 < · · · < tk to get the entire vector (with correlated coordinates):
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