 Solution

ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2009/a/2/c/solution

The solution is the continuous Doléans-Dade exponential:

S_{t} = exp {X_{t} - X_{0} - \frac{1}{2} [X]_{t}} .

Here

[X]

is the continuous quadratic variation starting from zero, denoted by the angle-bracket notation in the source. If

X = M + A

with

A

continuous of finite variation, then

[X] = [M]

U = X - X_{0} - \frac{1}{2} [X]

[X]

. Applying Itô's formula to

e^{U}

gives

d S = S d U + \frac{1}{2} S d [U] = S d X - \frac{1}{2} S d [X] + \frac{1}{2} S d [X] = S d X .

The initial value is

S_{0} = 1

. This verification applies to the full semimartingale, including its finite-variation part.

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