such as, the derivative of the following process in the direction of (W1,W2)
ST=rStdt+VtSt[(1-rho^2)dWt1+rho dWt2].
Actually what I want to know is 'is it possible to define a directional derivative as in the calculus. (something like Gradient operator)'
Probably the functional Ito Calculus may help you. There are two main works, one by Bruno Dupire and the other by Cont and Fournie.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1435551
http://projecteuclid.org/download/pdfview_1/euclid.aop/1358951982
Hope these will be helpful!
Best
Logic tells us that if you can define ordinary derivative in stochastic calculus in 1D space you can easily generalized it into 3D in terms of Gradient operator without having any further restriction on the system. While doing that one should keep in mind that only the expectation values of the random variable or function are matters, which behave like:
Ordinary variables and functions in calculus.
Directional Derivative of: a = a. Grad = (a.Grad) ,
={ L pDx + M pDy + N pDz} . Where L,M,N are the direction cosines of a relative to the coordinate axes, pDx,yz partial differential derivative operators.Bold letters indicate vectorial quantities here.
In ITO calculus so-called independent variables Xt and Ys are implicit functions of time over the well defined paths. Therefore one has to be little careful in the new definitions pD and Grad operator. Dr. Maggis referred very valuable references in this respect. Where path dependence symbolical taking care of using different letters such as X. Y..Z with subscribe t, s etc. for the same variable, which are tracking different paths in time domain.
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