If I have two stochastic processes $X_t$ and $Y_t$ that are dependent however $dX_tdY_t=0$ where $dX_t=a(z,x,t)dt$ and $dY_t=b(z,x,t)dt$ and $dZ_t=dW_t$(Brownian motion) then from Ito formula $du(Xt,Yt,Zt)=u_tdt+u_XdX+u_Ydt+u_ZdZ+0.5u_{ZZ}dZdZ$, which is an SDE and if I want to solve it for all terms of dt be zero: it implies I have a pde: $u_t+a*u_x+b*u_y+u_z=0$. Here is the question: this pde in three space variables which is to be solved assumes x,z and y are independent but it came from considering dependent variables. Where is the paradox? So when we write a pde which follows from Ito we care only about the fact that they are uncorrelated? Because I don't see any dependence embedded into the pde...Please clarify that for me.
independent vs. uncorrelated
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stochastic-processes
pde
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0@ Medan : As far as I know yes. – 2012-04-17