Question regarding Wiener Process

[imfe]

Can someone explain me how to get a new Weiner process based on the ratio of two processes

Form example, if I have one process dx/x = r1*t + sigmaX * dW and dy/y = r2*t + sigmaY*dW

How can I get a new Weiner process for X/Y such that the drift of X/Y = r3 in new measure W-hat

How do I write the SDE for both X and Y in this new measure W-hat?

To apply this: how can I get a new Weiner process for following?


Can you provide an example of the problem where dx/x and dy/y are given and we are required to find a process for X/Y with a specified drift.
For Example, if dx/x = 0.8*t + 0.6*dW and dy/y = 0.4t + 0.3*dw, how can I find a new process for X/Y such that the drift is 0.7?

Also how do I write SDE for X and Y in the new W-hat measure?


Please provide me steps so that I can understand the whole process of solving the problem.

I really appreciate this help.

 

[Vic_Siqiao]

using ito lemma to get the process for X/Y, i think this should be the first step

 

[stelios]

try to apply itô formula to X/Y after by identification you might find something interesting

 

[imfe]

try to apply itô formula to X/Y after by identification you might find something interesting

Can some one get me started on this ?

WHen you try to apply ito lemma to X/Y - do you mean first I find X = Xo*exp((0.8- 0.5* (0.6)^2)t + 0.6W) and Y in similar fashion, find X/Y and then proceed?

What do I do with X/Y that I get in some form such as X/Y = (Xo/Yo)* exp(some terms from above)?

Please help with some more details.

Thx.

 

[Vic_Siqiao]

dX/Y=Xd(1/Y)+ 1/Y*dX+dXd(1/Y)

of course first u need to find out the process for d(1/Y),still using Ito lemma.