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Hello gals and guys. It´s almost the end of the semester, could someone perhaps give me a hand with the following problem:Suppose Y is an Ft measurable Random Variable such that E[|Y|^2] < infinite. Note that Mt = E(Y|Ft) is a square integrable martingale with respect to Ft and can be represented using martingale representation theorem as:0 <= t <= T, for ssome unique adapted process g. Find g in the following cases:a) Y = W^2(t)b) Y = W^3(t)c) Y= exp(sigma*Wt) for some positive constant sigma.

well,in the case of Y=(W(t))^2 then E[Y \ Ft] = t, so M(t)=tin the case of Y=(W(t))^3 then E[Y \ Ft] =0 , so M(t)=0in the case of Y=exp(σ W(t)) then E[Y\Ft]=exp{(1/2)(σ^2) t }so you can find g

You may also do it using Ito's lemma.

QuoteOriginally posted by: mindedwell,in the case of Y=(W(t))^2 then E[Y \ Ft] = t, so M(t)=tin the case of Y=(W(t))^3 then E[Y \ Ft] =0 , so M(t)=0in the case of Y=exp(σ W(t)) then E[Y\Ft]=exp{(1/2)(σ^2) t }so you can find gMinded, thanks for your reply.I have used the information that you gave me. I infer that the expected value of the second case is because the Brownian motion has an odd exponent and has mean zero. Can you explain how you got the other expected values please? For the rest of the problem, I think that the way to solve for g as you mention as the end is to put for each of the cases. Is this the correct way to approach it as you mentioned?Thanks in advance for your assistance!

Functional Itô Calculus Bruno Dupire Bloomberg L.P. July 17, 2009Abstract: Itô calculus deals with functions of the current state whilst we deal with functions of the current path to acknowledge the fact that often the impact of randomness is cumulative. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem, providing an alternative to the Clark-Ocone formula from Malliavin Calculus. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. http://papers.ssrn.com/sol3/papers.cfm? ... id=1435551

I think, you should post this in the Student's forum!Nevertheless, I have done tasks like this in the exercise section of Oksendal's Stochastic Differential Equation.First you probably got something wrong: Y is not a Variable as it depends on t, so you have a familiy of random variables. Unfordunately it is not clear to whether you Filtration is the one to which your Brownian Motion is adapted but I guess so, since the Integral is constructed according to the brownian motion requires it to be adapted. Your task should be fairly simple, just play around with Ito Formula, which can be found in varying degress in the Standard Literature (i.e. Oksendal, Karatzas Shreve) or just take a look at Wikipedia.