B(t) is a brownian motion.
Find the distribution of B(t)=x | B(t+s) = x+k
Now, if X and Y are jointly normal random variables with
parameters
, then X | Y is normally distributed with
Let X = B(t) and Y = B(t + s), so we have X ~ N(0, t) and Y ~ N(0, t + s) and
Thus, the distribution of B(t)=x | B(t+s) = x+k is,
X = x | Y = x+k ~ N[ = t(x+k)/(s+t) ,
st /
(s+t)]
B(t) is a brownian motion. Find the distribution of B(t)=x | B(t+s) = x+k
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
Find the distribution of B(s) +B(t) when we have a Brownian motion with drift u and variance sigma square.
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:
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...
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
Let S(t), t >=0 be a Geometric Brownian motion process with drift mu = 0.1 and volatility sigma = 0.2. Find P(S(2) >S(1) > S(0))
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.