Question

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:

Answer 1

If X and Y are jointly normal random variables with parameters $\mu_x, \sigma_x^2, \mu_y, \sigma_y^2, \rho$ , then Y | X is normally distributed with

$E[Y | X = x] = \mu_y &plus; \rho \sigma_y (x - \mu_x)/\sigma_x$

$Var[Y | X= x] = (1 - \rho^2) \sigma_y^2$

Let X = W_t and Y = W_s , so we have X ~ N(0, t) and Y ~ N(0, s) and

$\rho = \frac{Cov(X,Y)}{\sigma_X \sigma_Y} = \frac{min(t, s)}{\sqrt{t}\sqrt{s}} = \frac{s}{\sqrt{t}\sqrt{s}} = \frac{\sqrt{s}}{\sqrt{t}}$ (s < t)

$E[Y | X = x] = 0 &plus; \frac{\sqrt{s}}{\sqrt{t}}. \sqrt{s} \left ( (x - 0)/\sqrt{t} \right ) = \frac{sx}{t}$

$Var[X | Y = y] = \left ( 1 - \frac{s}{t} \right )* s = \frac{s}{t} \left ( t - s\right )$

Thus, the distribution of W_s | W_t = x is,

$W_s | W_t = x \sim N \left ( \frac{s}{t}x, \frac{s}{t} \left ( t - s\right ) \right )$