# Need help solving two stochastic differential equations

• November 22nd 2011, 02:00 AM
Eloqent
Need help solving two stochastic differential equations
Hello, I'm new here, so I guess I should introduce myself. I'm a student at a math. finance course. We have some homework where we have to solve two stochastic differential equations, and I'm pretty stuck. Would love it if someone could show me how this is done! :)

The equations are the following:
$dX_t= \frac{b-X_t}{1-t}dt + dW_t\newline X_0 = a \in \mathbb R$

Where b is a real constant.

$dY_t=\frac{1}{Y_t}dt + \alpha Y_tdW_t \newline Y_0=y \in \mathbb R^+^+$

Where alpha is a real constant.

Looking forward to hearing from you! :)
• November 22nd 2011, 10:52 AM
Moo
Re: Need help solving two stochastic differential equations
Hello,

For the first one, apply Ito's lemma to $Y_t=\frac{X_t}{1-t}$

And you should get $dY_t=\frac{b}{(1-t)^2}\cdot dt+\frac{1}{1-t}\cdot dW_t$

And hence $\frac{X_t}{1-t}-X_0=\int_0^t \frac{b}{(1-s)^2}\ ds+\int_0^t \frac{1}{1-s}\ dW_s$

I must admit I kind of struggled when it came to finding $Y_t$. You know you have to apply Ito's lemma to some transformed $X_t$, so just try out simple functions of $X_t$ that may cancel out the $X_t$ in the right hand side of the equation.

FYI, this looks like what is called a brownian bridge... (it's in the Wikipedia article)
• November 22nd 2011, 02:06 PM
Eloqent
Re: Need help solving two stochastic differential equations
Thanks alot! :D