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XV: 06, 103-117, LNM 850 (1981)

**MEYER, Paul-André**

Flot d'une équation différentielle stochastique (Stochastic calculus)

Malliavin showed very neatly how an (Ito) stochastic differential equation on $**R**^n$ with $C^{\infty}$ coefficients, driven by Brownian motion, generates a flow of diffeomorphisms. This consists of three results: smoothness of the solution as a function of its initial point, showing that the mapping is 1--1, and showing that it is onto. The last point is the most delicate. Here the results are extended to stochastic differential equations on $**R**^n$ driven by continuous semimartingales, and only partially to the case of semimartingales with jumps. The essential argument is borrowed from Kunita and Varadhan (see Kunita's talk in the Proceedings of the Durham Symposium on SDE's, LN 851)

Comment: The results on semimartingales with jumps have been proved independently by Uppman. Some dust has been swept under the rugs about the non-explosion of the solution, and the results should be considered valid only in the globally Lipschitz case. See also Uppman 1624 and Léandre 1922

Keywords: Stochastic differential equations, Flow of a s.d.e.

Nature: Exposition, Original additions

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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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XV: 39, 587-589, LNM 850 (1981)

**ÉMERY, Michel**

Non-confluence des solutions d'une équation stochastique lipschitzienne (Stochastic calculus)

This paper proves that the solutions of a stochastic differential equation $dX_t=f(., t,X_t)\,dM_t$ driven by a continuous semimartingale $M$, where $f(\omega,t,x)$ is as usual previsible in $\omega$ and Lipschitz in $x$, are non-confluent, i.e., the solutions starting at different points never meet

Comment: See also 1506, 1507 (for less general s.d.e.'s), and 1624

Keywords: Stochastic differential equations, Flow of a s.d.e.

Nature: Original

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XVI: 24, 268-284, LNM 920 (1982)

**UPPMAN, Are**

Sur le flot d'une équation différentielle stochastique (Stochastic calculus)

This paper is a companion to 1506, devoted to the main results on the flow of a (Lipschitz) stochastic differential equation driven by continous semimartingales: non-confluence of solutions from different initial points, surjectivity of the mapping, smooth dependence on the initial conditions. The proofs have been greatly simplified

Keywords: Stochastic differential equations, Flow of a s.d.e., Injectivity

Nature: Exposition, Original additions

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XIX: 22, 271-274, LNM 1123 (1985)

**LÉANDRE, Rémi**

Flot d'une équation différentielle stochastique avec semimartingale directrice discontinue (Stochastic calculus)

Given a good s.d.e. of the form $dX=F\circ X_- dZ$, $X_{t-}$ is obtained from $X_t$ by computing $H_z(x) = x+F(x)z$, where $z$ stands for the jump of $Z$. Call $D$ (resp. $I$ the set of all $z$ such that $H_z$ is a diffeomorphism (resp. injective). It is shown that the flow associated to the s.d.e. is made of diffeomorphisms (respectively is one-to-one) iff all jumps of $Z$ belong to $D$ (resp. $I$)

Keywords: Stochastic differential equations, Flow of a s.d.e.

Nature: Original

Retrieve article from Numdam

Flot d'une équation différentielle stochastique (Stochastic calculus)

Malliavin showed very neatly how an (Ito) stochastic differential equation on $

Comment: The results on semimartingales with jumps have been proved independently by Uppman. Some dust has been swept under the rugs about the non-explosion of the solution, and the results should be considered valid only in the globally Lipschitz case. See also Uppman 1624 and Léandre 1922

Keywords: Stochastic differential equations, Flow of a s.d.e.

Nature: Exposition, Original additions

Retrieve article from Numdam

XV: 07, 118-141, LNM 850 (1981)

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

Keywords: Stochastic differential equations, Flow of a s.d.e., Change of variable formula, Stochastic parallel transport

Nature: Original

Retrieve article from Numdam

XV: 39, 587-589, LNM 850 (1981)

Non-confluence des solutions d'une équation stochastique lipschitzienne (Stochastic calculus)

This paper proves that the solutions of a stochastic differential equation $dX_t=f(., t,X_t)\,dM_t$ driven by a continuous semimartingale $M$, where $f(\omega,t,x)$ is as usual previsible in $\omega$ and Lipschitz in $x$, are non-confluent, i.e., the solutions starting at different points never meet

Comment: See also 1506, 1507 (for less general s.d.e.'s), and 1624

Keywords: Stochastic differential equations, Flow of a s.d.e.

Nature: Original

Retrieve article from Numdam

XVI: 24, 268-284, LNM 920 (1982)

Sur le flot d'une équation différentielle stochastique (Stochastic calculus)

This paper is a companion to 1506, devoted to the main results on the flow of a (Lipschitz) stochastic differential equation driven by continous semimartingales: non-confluence of solutions from different initial points, surjectivity of the mapping, smooth dependence on the initial conditions. The proofs have been greatly simplified

Keywords: Stochastic differential equations, Flow of a s.d.e., Injectivity

Nature: Exposition, Original additions

Retrieve article from Numdam

XIX: 22, 271-274, LNM 1123 (1985)

Flot d'une équation différentielle stochastique avec semimartingale directrice discontinue (Stochastic calculus)

Given a good s.d.e. of the form $dX=F\circ X_- dZ$, $X_{t-}$ is obtained from $X_t$ by computing $H_z(x) = x+F(x)z$, where $z$ stands for the jump of $Z$. Call $D$ (resp. $I$ the set of all $z$ such that $H_z$ is a diffeomorphism (resp. injective). It is shown that the flow associated to the s.d.e. is made of diffeomorphisms (respectively is one-to-one) iff all jumps of $Z$ belong to $D$ (resp. $I$)

Keywords: Stochastic differential equations, Flow of a s.d.e.

Nature: Original

Retrieve article from Numdam