Real-time estimation scheme for the spot cross volatility of jump diffusion processes

Given a finite set of observed data {X(tk) (omega(0)), Y(tk) (omega(0))} of just one sample path at n regularly spaced time of the processes X(t) and Y(t) satisfying dX(t) = a(o)(t)dt + a(1)(t)dW(1)(t) + a(2)(t)dW(2)(t) + dJ(1)(t), dY(t) = b(0)(t)dt + b(1)(t)dW(2)(t) + b(2)(t)dW(2)(t) + dJ(2)(t), t is an element of [0, T], where J(1), J(2) are jump process, we are to investigate a numerical scheme for the estimation of the value nu(X,Y)(t) = a(1)(t)b(1)(t) + a(2)(t)b(2)(t) called cross volatility. Our framework also contains the volatility estimation problem as a special case. We will show that our scheme works under mild assumptions on the activity of the jump process J(t). (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.