 Solution

ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2007/a/4/b/solution

The deterministic stochastic integral in the solution is centered Gaussian with variance

v (t) = \int_{0}^{t} a (s)^{- 2} σ (s)^{2} d s .

Therefore

X (t, x)

a (t) x

V (t) = a (t)^{2} v (t) = \int_{0}^{t} e^{2 \int_{s}^{t} μ (r) d r} σ (s)^{2} d s .

For

t > 0

this is strictly positive, since

σ > 0

. Its density at

y

\frac{1}{2 πV ( t )} exp [- \frac{( y - a ( t ) x ) ^{2}}{2 V ( t )}] .

The boundedness assumptions on

σ

and its reciprocal in particular ensure nondegeneracy; compact-interval continuity is already sufficient for the displayed finite-time construction.

 New to topics? Read the docs here!