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

Given a finite set of observed data {Xtk(ω0),Ytk(ω0)} of just one sample path at n regularly spaced time of the processes Xt and Yt satisfying dXt=a0(t)dt+a1(t)dW1(t)+a2(t)dW2(t)+dJ1(t),dYt=b0(t)dt+b1(t)dW1(t)+b2(t)dW2(t)+dJ2(t),t∈[0,T], where J1,J2 are jump process, we are to investigate a numerical scheme for the estimation of the value νX,Y(t)=a1(t)b1(t)+a2(t)b2(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 Jt.