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

摘要

Given a finite set ol observed data {X_(tk)(ω_0), Y_(tk)(ω_0)} of just one sample path at n regularly spaced time of the processes X_t and Y_t satisfying dX_t = a_0(t)dt +a_1(t)dW(t) + a_2(t)dW_2(t) + dJ_t(t), dY_t = h_0(t)dt + b_t(t)dW_1(t) + h_2(t)dW_2(t) + dJ_2(t), t ∈ [0, T], where J_1. J_2 are jump process, we are to investigate a numerical scheme for the estimation of the value v_(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.