3. Let W(t be standard Brownian motion and let to > 0. Consider the random variable Min(to) min{W(s) 0 s< to}. Co...
Let W - {Wi,0< t < ) represent a standard Brownian motion Show that the process Z(s)-(zt-W f.0 < t-1) is a standard Brownian motion, where s > 0 is fixed
We consider a Standard Brownian Motion W={Wt,t>=o}, show that
for s<t, Ws|Wt=x the conditional distribution of the process
given a future valueWt=x
We consider a standard Brownian motion W W,t20) Show that for s < t, W /Wt-x the conditional distribution of the process given a future value Wi is given by the following Normal distribution:
Consider the standard Brownian motion{W(t),t≥0}. Find P(W(1)≥0, W(2)≥0)
8.2. Let W()-X(at)la for a >0. Verify that W(t is also Brownian motion
8.2. Let W()-X(at)la for a >0. Verify that W(t is also Brownian motion
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be Brownian motion t 0, let T be the first time Z(t) hits a. a) Show that T, is a random time for Z(t) and for B(t). b) Show P(T, 0o)1. c) Use martingale methods to compute E [e-m] for any θ > 0 r>
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be...
3. Brounian motion f(O)eR+ is a special case of a Gaussian process with mean zero and covariance C(s, t) = min(s, t) (a) What is the distribution of f(1), the Brownian motion at time t = 4? (Hint: it may be useful to function recall that for any random variable X, var(X)-(x, X) (b) Fix tE R. What is the distribution of f(t)? (c) What is the distribution of f(4)-f(2)? (Hint: it may be useful to utilize var(X-Y) = var(X)...
let {X(t), 1 2 0} denote a Brownian motion 8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion (d) Let Using (c) present an argument that
let {X(t), 1 2 0} denote a Brownian motion
8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion...
{ W, : t > 0} be a Brownian motion. Find E(W, (W2t --We), where 0 < t < 1: Let W Select one: t (1 -t) 0
{ W, : t > 0} be a Brownian motion. Find E(W, (W2t --We), where 0
8.2. Let W()-X(at)la for a >0. Verify that W(t is also Brownian motion
X(t).12 0 is a standard Brownian motion. Find the distribution of X(t) . 2. Assume that
X(t).12 0 is a standard Brownian motion. Find the distribution of X(t) . 2. Assume that