help in understanding Ito formula (Bjork text)

islandboy · March 31st, 2007, 5:46 pm

this is from the Bjork book, Arbitrage Theory in Continuous Time, pages 351 to 352I'm trying to understand how Bjork used the Ito Formula to solve the following:Given:and lettingBjork gave the P-dynamics of Z2 as follows (we ignore the terms in dt):I understand that the Ito Formula states dW(t) term to be: but given the complicated structure of Z2(t), I couldn't understand how to compute the dW(t) term.Can anyone help?Thanks!

islandboy · March 31st, 2007, 6:01 pm

As a follow-up question what if instead we are given:where and are scalar P-Weiner process with correlation rho, and Y1(t) and Y2(t) are independent weiner processes.According the Bjork, if we use the Ito formula again, instead of justas in the above.that is, if we define Z2(t) as in the above, we have:can anyone explain how this is so?thanks again.

Alan · March 31st, 2007, 6:54 pm

The first case you posted follows from the second with rho = 1.The second follows from Ito's formula for f(S1,S2). The trick is to recognize a particular linear combo of dW1 and dW2 as the "d" of a -1D- Brownian motion W* If you can't see it, I suggest you post what you think df is from Ito's formula.Then people can point out dW*.regards,

islandboy · March 31st, 2007, 7:08 pm

hi thanks for replying.QuoteThe second follows from Ito's formula for f(S1,S2).that's the one I couldn't understand. Are we supposed to expand Z2(t) and get its derivative with respect to S1 and S2 repectively...cause that kind of computation seems too unnecessarily complicated and messy.QuoteThe trick is to recognize a particular linear combo of dW1 and dW2 as the "d" of a -1D- Brownian motion W* What do you mean? Can you explain further? I apologize, I don't understand what you mean here.thanks again for the assist

Alan · April 1st, 2007, 12:59 am

Yes, expand out Z2 = Z2(S1,S2) using Ito. Yes, it's messy. Go do it.Since you know the answer, why don't you try to organize the termsso they look like the answer. What has to be true for it to work?

islandboy · April 1st, 2007, 6:00 am

I'll try that out.thanks again for the help Alan