Math 526: Brownian Motion Notes
نویسنده
چکیده
Theorem 1. If (Xt) is a Brownian motion, then there exist constants μ, σ 2 such that Xt − Xs ∼ N ((t− s)/μ, (t− s)σ). Proof. Without loss of generality take s = 0 and pick some n ∈ Z and write Xt −X0 = (Xt/n −X0) + (X2t/n −Xt/n) + . . .+ (Xt −X(n−1)t/n). Each term on the right hand side is independent and identically distributed. Letting n→∞, by the central limit theorem, the right hand side must converge to some normally distributed random variable (the continuity of (Xt) makes sure that one can apply the CLT, as the increments must become smaller and smaller as n increases).
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