4
$\begingroup$

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

  • 0
    Still haven't found out how to solve it. :( Can anyone point me in the right direction?2011-11-17

1 Answers 1

2

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) $