Stochastic Diff Eq SDE

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Consider the following SDE $$d\sigma = a(\sigma,t)dt + b(\sigma,t)dW$$

The Forward Equation (FKE) is given by $$\frac{\partial p}{\partial t} = \frac{1}{2}\frac{\partial ^2}{\partial \sigma^2}(b^2p) - \frac{\partial}{\partial \sigma}(ap)$$

a. The steady state solution is given by setting $\frac{\partial p}{\partial t} = 0.$ Considering the boundary conditions that as $\sigma \rightarrow \infty$, $p \rightarrow 0$ and $\frac{\partial p}{\partial t} \rightarrow 0$, show that the steady state solution is given by $$p (\sigma) = \frac{A}{b^2} e^{\int^\sigma \frac{2a}{b^2}d\sigma'}$$

b. Show that for $$d\sigma^2 = C_1(C_2 - \sigma^2) dt + C_3\sigma dW$$ The steady state solution for the FKE is given by $$p(\sigma) = \frac{4C}{C^2_3} \sigma \frac{4(C_1C_2 - \frac{C^2_3}{4})}{C^2_3}e^{\frac{-2C_1 \sigma^2}{C^2_3}}$$

Hint: You need to find the SDE for $d\sigma$ first

You can assume dW represents increment in the Wiener process W

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Please click on the link above to view the question –  Mike Feb 1 '13 at 14:23
No. Please make your question self-contained. –  Did Feb 1 '13 at 15:35
Did, if you mean by self contained to have the question uploaded into the box then the website doesnt allow me to do so yet as I am a new member. So instead of writing the question in the box I have provided a link for an image of the question which was written on a pdf document. Just to make the question read clearly and easily. –  Mike Feb 1 '13 at 15:43
What about writing down your question yourself? I know, this sounds crazy... :-) but plenty of people are doing it. –  Did Feb 1 '13 at 18:32
Now that the post is self-contained (thanks @John), the OP might want to add their own thoughts on the questions, which parts they know how to solve, etc., etc., etc. –  Did Feb 4 '13 at 16:46
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Stochastic Calculus can be daunting (even if you are only learning an elementary amount) when done in a short space of time.

However with a few hints I know you can make a good attempt at the question you posted.

1. The SDE started in the question is a random walk. The two parameters in the SDE are not constant - they are funtions of $\sigma$ and time. However you are told to solve for the Steady State, i.e. a situation where time is no longer a variable. This means you can change the partial derivatives for standard derivatives.

2. The key to the part a) is realising that the FKE can be reduced from a second order differential question to a first order differential equation. Once you've done this you can solve for $p$ (Remember to do what the question says with $\frac{dp}{dt}$.

3. Hint: Remember $b$ is not a constant in the FKE. What rule might you have to use to differentiate two variables?

4. Think of the SDE given here simply as a process. It could be a stock price, an interest rate - a process. We can see that this process is labelled $\sigma^2.$ And the change in this process $d\sigma$ is the SDE.

5. There are times when we wish to model the change in another process - which itself is a function of a different process - for example modelling the change in an option price based on the lying.

6. You need to look at the "one formula which you should know like the back of your hand" and derive this new process. Hint: Look at the other question you asked and how the solution for that involved a simple variable substitution (then follow that answer closely).

7. Once you've done that the second part is exactly like the first - just substitute the variables and manipulate the algebra.

Give it a go - you can do it.

And Mike, learn MathJax - it will be extremely useful for you. There are MathJax articles on this site and it can be learned very quickly.

Hope this helps.

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