Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 4 / e

Table of contents
- Solution e

Solution

For

Y = X^{α}

, the Itô formula and the Bessel equation give

d Y_{t} = α X_{t}^{α - 1} d B_{t} + \frac{α ( d + α - 2 )}{2} X_{t}^{α - 2} d t .

Use the clock

C_{t} = α^{2} \int_{0}^{t} X_{s}^{2 α - 2} d s

and its inverse. The Dambis-Dubins-Schwarz theorem turns the first term into Brownian motion, while division of the drift by the clock rate gives

d Y_{u} = d W_{u} + \frac{d + α - 2}{2 α} \frac{d u}{Y _{u}} .

Hence

Y

is a Bessel process of dimension

d^{'} = 1 + \frac{d + α - 2}{α} = 2 + \frac{d - 2}{α} .

This is the Power time change of a Bessel process.

 Ancestors (10)

 View article source

 Discussion (0)

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