Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
    
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
add comment

1 Answer

up vote 2 down vote accepted

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: $$ \mathrm{d}\left( Y_t^2 \exp\left( \lambda t + \mu W_t \right) \right) $$

share|improve this answer
    
Thanks a million! :D –  L1meta Nov 19 '11 at 12:53
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.