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
Consider the standard Brownian motion{W(t),t≥0}. Find P(W(1)≥0, W(2)≥0)
B(t) is a brownian motion. Find the distribution of B(t)=x | B(t+s) = x+k
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:
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...
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
4. [20 points] Let {B(t):t0 be a standard Brownian motion. Define a stochastic process (X (t):t20) by the formulas X (t) = tB(1 + t-1)-tB(1), x(0) = 0, t > 0, You may take for granted the fact that imt-«HX(t) = 0, with probability 1 (b) Explain why [X():t20 is a standard Brownian motion
4. [20 points] Let {B(t):t0 be a standard Brownian motion. Define a stochastic process (X (t):t20) by the formulas X (t) = tB(1 + t-1)-tB(1), x(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
3. Let W(t be standard Brownian motion and let to > 0. Consider the random variable Min(to) min{W(s) 0 s< to}. Compute the cumulative distribution function of Min(to)
3. Let W(t be standard Brownian motion and let to > 0. Consider the random variable Min(to) min{W(s) 0 s
If X(t), t>=0 is a Brownian motion process with drift mu and variance sigma squared for which X(0)=0, show that -X(t), t>=0 is a Brownian Motion process with drift negative mu and variance sigma squared.
{ 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