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u418936
Topic Author
Posts: 121
Joined: October 28th, 2003, 9:31 pm

### need help solving two sde's

hii need to solve the following system of sde's: dx(t) = k*[y(t) - x(t)]dt + v*dW1(t) dy(t) = a*[b - x(t)]dt + w*dW2(t)where k, v, a, and w are constants, and W1() and W2() are standard BM's. I can figure out the second one, but i'm having a lot of trouble solving the first one. i'd greatly appreciate any help.thanks

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

### need help solving two sde's

remember that they are linked ... so write them out in matrix notation and it will be clear
knowledge comes, wisdom lingers

DrEvil
Posts: 36
Joined: April 22nd, 2004, 10:17 pm

### need help solving two sde's

Looks like a modified Vasicek model where the instantaneous reversion mean follows another (possibly correlated) Vasicek process. Any reason you're doing this other than simulate a rate process so chaotic it gives Oldrich a headache?

u418936
Topic Author
Posts: 121
Joined: October 28th, 2003, 9:31 pm

### need help solving two sde's

i'm using the kalman filter to get x() and y() from zero coupon treasury yields. it's working pretty (ie my measurement errors are usually less than 10 bps using daily spline data from 1990 to today) well when i discretize the two equations and run the KF, but I want to find the exact solutions to see if i can make it work better.i see in principle how you ought to be able to solve the system using matrices, but i don't know matrix differentiation very well. if anybody could start me in the right direction, i'd really appreciate it.

spacemonkey
Posts: 443
Joined: August 14th, 2002, 3:17 am

### need help solving two sde's

Since you know how to solve an SDE of the form,G(t)=G(0)+int_0^t (a[b - G(u)]du)+int_0^t c dW,then you should try to find constants R (there should be two) so that G(t)=R*x(t)+y(t) is of this form. Then you can solve for G(t) to get a system of linear equations that you can solve for x(t) and y(t).
Last edited by spacemonkey on April 26th, 2004, 10:00 pm, edited 1 time in total.

hazerider
Posts: 174
Joined: July 24th, 2003, 3:45 pm

### need help solving two sde's

I seem to remember something similar from school. I think you need to come up with the Markov representation of the two processes, where you introduce some transformation to decouple the relationship between x and y, and come up with new processes.

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