Mathematical Modeling in Finance with Stochastic Processes

206 CHAPTER 6. STOCHASTIC CALCULUS

Sources

This discussion is adapted from Financial Calculus: An introduction to
derivative pricingby M Baxter, and A. Rennie, Cambridge University Press,
1996, pages 52–62 and “An Algorithmic Introduction to the Numerical Sim-
ulation of Stochastic Differential Equations”, by Desmond J. Higham, in
SIAM Review, Vol. 43, No. 3, pages 525–546, 2001.

Problems to Work for Understanding

1. Find the solution of the stochastic differential equation

dY(t) =Y(t)dt+ 2Y(t)dW

2. Find the solution of the stochastic differential equation

dY(t) =tY(t)dt+ 2Y(t)dW

Note the difference with the previous problem, now the multiplier of

thedtterm is a function of time.

3. Find the solution of the stochastic differential equation

dY(t) =μY(t)dt+σY(t)dW

4. Find the solution of the stochastic differential equation

dY(t) =μtY(t)dt+σY(t)dW

Note the difference with the previous problem, now the multiplier of

thedtterm is a function of time.

5. Find the solution of the stochastic differential equation

dY(t) =μ(t)Y(t)dt+σY(t)dX

Note the difference with the previous problem, now the multiplier of the

dtterm is a general (technically, a locally bounded integrable) function

of time.