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(Solved): Suppose we divide the time interval [0, t] into n equally spaced periods so that t = t/n. Conside ...



Suppose we divide the time interval [0, t] into n equally spaced periods so that ∆t = t/n. Consider a particle doing a random walk, starting from X0 = 0, and within each period i = 1, 2, ..., n, the particle moves by Zi, so that Xt = nX i=1 Zi, and Zi = μ∆t + σ√∆tεi, where E(εi) = 0 and Var(εi) = 1, and εi are independent. μ and σ are constants. (1) Calculate E(Zi) and Var(Zi). (2) Calculate E(Xt) and Var(Xt). Do they depend on n? (3) For large n, what is the approximate distribution of Xt? 2



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