A Wiener process has and independent Gaussian increments

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Non-equilibrium statistical mechanics and stochastic thermodynamics

Error, Chance and the Machine, Ch Random processes in continuous time

Problem

A Wiener process W(t)W(t) has ⟨W(t)2⟩=t\langle W(t)^2\rangle = t and independent Gaussian increments. Partition [0,T][0,T] into NN equal subintervals of length Δt=T/N\Delta t = T/N, and for a power p>0p>0 define SN(p)=∑k=1N∣W(tk)−W(tk−1)∣p,S^{(p)}_N = \sum_{k=1}^{N}\left|W(t_k)-W(t_{k-1})\right|^p, so that SN(1)S^{(1)}_N is the length of the sampled path and SN(2)S^{(2)}_N the sum of squared increments. ((a)) Write each increment as Δt Zk\sqrt{\Delta t}\,Z_k with ZkZ_k a standard Gaussian, and show that ⟨SN(p)⟩=cp Tp/2N1−p/2\langle S^{(p)}_N\rangle=c_p\,T^{p/2}N^{1-p/2} with cp=⟨∣Z∣p⟩c_p=\langle|Z|^p\rangle. Hence show that as N→∞N\to\infty the mean tends to zero for p>2p>2 and grows without limit for p<2p<2, and that it equals TT for every NN when p=2p=2. ((b)) For p=2p=2, using ⟨Z4⟩=3\langle Z^4\rangle=3 and the independence of the increments, find var(SN(2))\mathrm{var}(S^{(2)}_N) and show it vanishes as N→∞N\to\infty. Explain why p=2p=2 is the only power for which the sum tends to a finite, nonzero number that is not random. ((c)) A path is sampled at N=4.0×105N=4.0\times10^5 equally spaced points over T=2.0 sT=2.0\,\mathrm{s}. Compute the standard deviation of SN(2)S^{(2)}_N and express it as a fraction of TT.

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