Vague convergence of locally integrable martingale measures (Q1338745)

The author introduces a concept of vague convergence of locally integrable (orthogonal) martingale measures as follows. Let \(M^ n\), \(n \in \mathbb{N}\), and \(M\) be \(F_ t\)-martingale measures on a locally compact Hausdorff space \(\mathbf E\), given a filtered probability space \((\Omega, F, F_ t, P)\). \(M^ n\) vaguely converges to \(M\) in distribution \((M^ n @>v{\mathcal L}>> M)\) if, for all functions \(f\in C_ K( \mathbb{R}_ + \times {\mathbf E})\) (continuous, with a compact support), \[ \int^ \bullet _ 0 \int_ E f(s,x) M^ n (ds, dx) @>{\mathcal L}>> \int^ \bullet_ 0 \int_ E f(s,x) M(ds,dx),\quad n \to \infty. \] (\(@>{\mathcal L}>>\) denotes the convergence in distribution.) Conditions for vague convergence are provided in terms of convergence of characteristics of martingale measures. The convergence of stochastic integrals \(f^ n \cdot M^ n @>v{\mathcal L}>> f \cdot M\) and \[ \int^ \bullet _ 0 \int_ E f^ n (s,x) M^ n (ds,dx) @>{\mathcal L}>> \int^ \bullet_ 0 \int_ E f(s, x) M(ds, dx), \quad n \to \infty, \] is also studied when \(f^ n \to f\) in a certain sense.