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:

$\displaystyle dX_t= \frac{b-X_t}{1-t}dt + dW_t\newline X_0 = a \in \mathbb R$

Where b is a real constant.

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

Where alpha is a real constant.

Looking forward to hearing from you! :)

Re: Need help solving two stochastic differential equations

Hello,

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

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

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

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

FYI, this looks like what is called a brownian bridge... (it's in the Wikipedia article)

Re: Need help solving two stochastic differential equations