Stochastic Differential Equations

[courtrigrad]

Hello all

I am doing a project concerning volatility and drift structure of various markets. If we have $$dr = u(r,t)dt + w(r,t)dX$$ is this a partial differntial equation or just a differential equation? $$r$$ is the spot rate $$t$$ is time and $$X$$ is a random variable.

Thanks

 

[learningphysics]

Partial differential equation.

 

[courtrigrad]

ok so in other words $$dr = \frac{\partial u}{\partial t} dt + \frac{\partial w}{\partial t} dX$$?

Thanks

 

[learningphysics]

No.

$$dr = \frac{\partial r}{\partial t} dt + \frac{\partial r}{\partial X} dX$$

 

[saltydog]

Hello all

I am doing a project concerning volatility and drift structure of various markets. If we have $$dr = u(r,t)dt + w(r,t)dX$$ is this a partial differntial equation or just a differential equation? $$r$$ is the spot rate $$t$$ is time and $$X$$ is a random variable.

Thanks

Well, I must admit that looks confussing to me. Would you kindly explain what's the dependent variable and what are the independent variable?

As I see it, it looks like the following:

We wish to find the function r(t,X) such that:

$$\frac{\partial r}{\partial t}=\frac{\partial u}{\partial t}+\frac {\partial w}{\partial X}$$

such that u(r,t) and w(r,t) are given functions of the "dependent" variable r(t,X) and t.

I still think this isn't right but maybe an improvement you can correct.

 

[courtrigrad]

yes i think salty dog that is right. We have two functions u and t with parameters r and t. I am not sure, as I am just studying calculus!

 

[saltydog]

yes i think salty dog that is right. We have two functions u and t with parameters r and t. I am not sure, as I am just studying calculus!

Noooooo dude. That's not quite right what you said: need to precisely define what the function is, the dependent variable, independent variables and what partials are involved. I'm kind and won't tell you perhaps PDEs are not something for you to be looking at if you're just into Calculus.

 

[saltydog]

Noooooo dude. That's not quite right what you said: need to precisely define what the function is, the dependent variable, independent variables and what partials are involved. I'm kind and won't tell you perhaps PDEs are not something for you to be looking at if you're just into Calculus.

Wait a minute. I'm sorry. I mean it's ok to look at them and be in wonder about them but perhaps not expect to be able to solve them if you're just starting Calculus. I once had a Chem teacher who showed me a triple integral a long time ago before I knew what it was. He expressed utter wonder at the time and I didn't understand. I do now!

 

[courtrigrad]

just taking mathwonk's advice. i am reading Courant's calculus book in addition to studying finance. i am skipping around and I understand the basic concept that in a partial derivative you keep variables fixed.

 

[polyb]

courtrigrad,

I've been looking on the arxives and found this, it has a lot to do with what you have been asking about and specifically markets, options, and stochastics:

http://xxx.lanl.gov/PS_cache/physics/pdf/0001/0001040.pdf

There are some pretty good references in the bibliography that you may want to look into for further reading. This paper was also published in Physica A which carries a lot of the financial/physics papers.

I also found this 'elementary'(HA!) introduction to stochastic calculus. Scroll down towards the bottom of the page and the notes are in a pdf format.

http://www.statslab.cam.ac.uk/~afrb2/

Good luck on your project and I hope this helps a bit. There truly is a lot to the subject and you have only begun to scratch the surface, so have fun and keep digging.

BTW, have you had a chance to consult with a teacher on narrowing down the topic for your project?

 

[courtrigrad]

yea i am investigating the drift and volatility structure of current data/ Polyb, thanks al ot for your great help. Maybe we can discuss more about the project, and your ideas as well

Thanks