 Solution

ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2007/a/1/b/solution

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

independent of

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

(x, v)

is the product

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

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

(B_{s}, B_{t})

\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.

 New to topics? Read the docs here!