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XVII: 18, 179-184, LNM 986 (1983)

**HE, Sheng-Wu**; **YAN, Jia-An**; **ZHENG, Wei-An**

Sur la convergence des semimartingales continues dans ${\bf R}^n$ et des martingales dans une variété (Stochastic calculus, Stochastic differential geometry)

Say that a continuous semimartingale $X$ with canonical decomposition $X_0+M+A$ converges perfectly on an event $E$ if both $M_t$ and $\int_0^t|dA_s|$ have an a.s. limit on $E$ when $t\rightarrow \infty $. It is established that if $A_t$ has the form $\int_0^tH_sd[M,M]_s$, $X$ converges perfectly on the event $\{\sup_t|X_t|+\lim\sup_tH_t <\infty \}$. A similar (but less simple) statement is shown for multidimensional $X$; and an application is given to martingales in manifolds: every point of a manifold $V$ (with a connection) has a neighbourhood $U$ such that, given any $V$-valued martingale $X$, almost all paths of $X$ that eventually remain in $U$ are convergent

Comment: The latter statement (martingale convergence) is very useful; more recent proofs use convex functions instead of perfect convergence. The next talk 1719 is a small remark on perfect convergence

Keywords: Semimartingales, Martingales in manifolds

Nature: Original

Retrieve article from Numdam

Sur la convergence des semimartingales continues dans ${\bf R}^n$ et des martingales dans une variété (Stochastic calculus, Stochastic differential geometry)

Say that a continuous semimartingale $X$ with canonical decomposition $X_0+M+A$ converges perfectly on an event $E$ if both $M_t$ and $\int_0^t|dA_s|$ have an a.s. limit on $E$ when $t\rightarrow \infty $. It is established that if $A_t$ has the form $\int_0^tH_sd[M,M]_s$, $X$ converges perfectly on the event $\{\sup_t|X_t|+\lim\sup_tH_t <\infty \}$. A similar (but less simple) statement is shown for multidimensional $X$; and an application is given to martingales in manifolds: every point of a manifold $V$ (with a connection) has a neighbourhood $U$ such that, given any $V$-valued martingale $X$, almost all paths of $X$ that eventually remain in $U$ are convergent

Comment: The latter statement (martingale convergence) is very useful; more recent proofs use convex functions instead of perfect convergence. The next talk 1719 is a small remark on perfect convergence

Keywords: Semimartingales, Martingales in manifolds

Nature: Original

Retrieve article from Numdam