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XV: 07, 118-141, LNM 850 (1981)
KUNITA, Hiroshi
Some extensions of Ito's formula (Stochastic calculus)
The standard Ito formula expresses the composition of a smooth function $f$ with a continuous semimartingale as a stochastic integral, thus implying that the composition itself is a semimartingale. The extensions of Ito formula considered here deal with more complicated composition problems. The first one concerns a composition Let $(F(t, X_t)$ where $F(t,x)$ is a continuous semimartingale depending on a parameter $x\in R^d$ and satisfying convenient regularity assumptions, and $X_t$ is a semimartingale. Typically $F(t,x)$ will be the flow of diffeomorphisms arising from a s.d.e. with the initial point $x$ as variable. Other examples concern the parallel transport of tensors along the paths of a flow of diffeomorphisms, or the pull-back of a tensor field by the flow itself. Such formulas (developed also by Bismut) are very useful tools of stochastic differential geometry
Keywords: Stochastic differential equations, Flow of a s.d.e., Change of variable formula, Stochastic parallel transport
Nature: Original
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XXVI: 11, 127-145, LNM 1526 (1992)
Relèvement horizontal d'une semimartingale càdlàg (Stochastic differential geometry, Stochastic calculus)
For filtering purposes, the lifting of a manifold-valued semimartingale $X$ to the tangent space at $X_0$ is extended here to the case when $X$ has jumps. The value of $L_t$ involves the inverse of the exponential at $X_{t-}$ applied to $X_t$, and a parallel transport from $X_0$ to $X_{t-}$
Comment: The same method is described in a more general setting by Kurtz-Pardoux-Protter Ann I.H.P. (1995). In turn, this is a particular instance of a very general scheme due to Cohen (Stochastics Stoch. Rep. (1996)
Keywords: Stochastic parallel transport, Stochastic differential equations, Jumps
Nature: Original
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