Convergence of stochastic integrals to a continuous local martingale with conditionally independent increments

For each $T>0$, let a tensor-valued stochastic process $Y_T$ be defined by \[ Y_T(t)=\int _0^tD Z_T(s)\otimes \vartheta _T(s), \] where $Z_T$ is an $\mathbf {R}^d$-valued locally square integrable martingale with respect to some filtration $\mathbb {F}_T$ and where $\vartheta _T$ is an $\mathbf {R}^d$-valued $\mathbb {F}_T$-predictable stochastic process such that $\int _0^t|\vartheta _T(s)|^2D\operatorname {tr}\langle Z_T\rangle (s)<\infty$ for all $t$. In this paper, conditions are found for the convergence $(Y_T, \langle Y_T\rangle )\stackrel {\textrm {law}}\longrightarrow (Y, \langle Y\rangle )$, where $Y$ is a continuous local martingale with conditionally independent increments given $\langle Y\rangle$.