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

Given a finite set of observed data {_Xtk(_ω0),_Ytk(_ω0)}$\{X_{t_k}({\omega }_0)\text{,}{Y}_{t_k}({\omega }_0)\}$ of just one sample path at n$n$ regularly spaced time of the processes _Xt$X_t$ and _Yt$Y_t$ satisfying d_Xt=_a0(t)dt+_a1(t)d_W1(t)+_a2(t)d_W2(t)+d_J1(t),d_Yt=_b0(t)dt+_b1(t)d_W1(t)+_b2(t)d_W2(t)+d_J2(t),t∈0,T]$d{X}_t=a_0(t)dt+a_1(t)d{W}_1(t)+a_2(t)d{W}_2(t)+d{J}_1(t)\text{,}d{Y}_t=b_0(t)dt+b_1(t)d{W}_1(t)+b_2(t)d{W}_2(t)+d{J}_2(t)\text{,}t\in 0\text{,}T]$, where _J1,_J2$J_1\text{,}{J}_2$ 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)${\nu }_{X\text{,}Y}(t)=a_1(t)b_1(t)+a_2(t)b_2(t)$ 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 _Jt$J_t$.