# Solutions to stochastic differential equations

I'm in a bit of trouble with my homework and was wondering if anyone could help me find the solutions to these two stochastic differential equations. Would really appreciate it! Thanks in advance! :)

1.


\begin{cases} dX_t= \frac{b-X_t}{1-t}dt + dW_t\newline X_0 = a \in \mathbb R \end{cases} Where $b$ is a real constant.

2.


\begin{cases} dY_t=\frac{1}{Y_t}dt + \alpha Y_tdW_t \newline Y_0=y \in \mathbb R^++ \end{cases} Where $a$ is a real constant.

3.


Verify which of the processes are affine

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the first one is Brownian bridge and I didn't get what 3. refer to – Ilya Nov 15 '11 at 11:42
Thanks! Will check that out :) – L1meta Nov 15 '11 at 13:00
Still haven't found out how to solve it. :( Can anyone point me in the right direction? – L1meta Nov 17 '11 at 13:15

HINT For item 1: Use Ito lemma to verify that $$\mathrm{d} \left( \frac{X_t-b}{1-t} \right) = \frac{1}{1-t} \mathrm{d} W_t$$

HINT for item 2: See if this answer of mine helps. But also think if you could match the constants so that the following expression has no diffusion component: $$\mathrm{d}\left( Y_t^2 \exp\left( \lambda t + \mu W_t \right) \right)$$

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Thanks a million! :D – L1meta Nov 19 '11 at 12:53