Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2007 / A / 1 / b

Table of contents
- Solution b

Solution

Independent increments give

B_{s} \sim N (0, s)

independent of

B_{t} - B_{s} \sim N (0, t - s)

. Their joint probability density function at

(x, v)

is the product

\frac{1}{2 π s ( t - s )} exp (- \frac{x ^{2}}{2 s} - \frac{v ^{2}}{2 ( t - s )}) .

The change of variables

(x, v) \mapsto (x, y) = (x, x + v)

has Jacobian determinant one. Therefore the joint probability density function of

(B_{s}, B_{t})

is

\frac{1}{2 π s ( t - s )} exp (- \frac{x ^{2}}{2 s} - \frac{( y - x ) ^{2}}{2 ( t - s )}) .

In particular these coordinates are correlated, with covariance

s

, although the first pair has independent coordinates.

 Ancestors (12)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version