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VI: 08, 105-108, LNM 258 (1972)

**KAZAMAKI, Norihiko**

Note on a stochastic integral equation (Stochastic calculus)

Though this paper has been completely superseded by the theory of stochastic differential equations with respect to semimartingales (see 1124), it has a great historical importance as the first step in this direction: the semimartingale involved is the sum of a locally square integrable martingale and a continuous increasing process

Comment: The author developed the subject further in*Tôhoku Math. J.* **26**, 1974

Keywords: Stochastic differential equations

Nature: Original

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X: 16, 240-244, LNM 511 (1976)

**YAMADA, Toshio**

On the uniqueness of solutions of stochastic differential equations with reflecting barrier conditions (Stochastic calculus, Diffusion theory)

A stochastic differential equation is considered on the positive half-line, driven by Brownian motion, with time-dependent coefficients and a reflecting barrier condition at $0$ (Skorohod style). Skorohod proved pathwise uniqueness under Lipschitz condition, and this is extended here to moduli of continuity satisfying integral conditions

Comment: This extends to the reflecting barrier case the now classical result in the ``free'' case due to Yamada-Watanabe,*J. Math. Kyoto Univ.*, **11**, 1971. Many of these theorems have now simpler proofs using local times, in the spirit of Revuz-Yor, *Continuous Martingales and Brownian Motion,* Chapter IX

Keywords: Stochastic differential equations, Boundary reflection

Nature: Original

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XI: 24, 376-382, LNM 581 (1977)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Équations différentielles stochastiques (Stochastic calculus)

This is an improved and simplified exposition of the existence and uniqueness theorem for solutions of stochastic differential equations with respect to semimartingales, as proved by the first author in*Zeit. für W-theorie,* **36**, 1976 and by Protter in *Ann. Prob.* **5**, 1977. The theory has become now so classical that the paper has only historical interest

Keywords: Stochastic differential equations, Semimartingales

Nature: Exposition, Original additions

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XI: 27, 411-414, LNM 581 (1977)

**KOSKAS, Maurice**

Images d'équations différentielles stochastiques (Stochastic calculus)

This paper answers a natural question: can one take computations performed on ``canonical'' versions of processes back to their original spaces? It is related to Stricker's work (*Zeit. für W-theorie,* **39**, 1977) on the restriction of filtrations

Keywords: Stochastic differential equations

Nature: Original

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XII: 13, 114-131, LNM 649 (1978)

**YAMADA, Toshio**

Sur une construction des solutions d'équations différentielles stochastiques dans le cas non-lipschitzien (Stochastic calculus)

The results of this paper improve on those of the author's paper (*Zeit. für W-theorie,* **36**, 1976) concerning a one-dimensional stochastic differential equations of the classical Ito type, whose coefficients satisfy a Hölder-like condition instead of the standard Lipschitz condition. The proofs are simplified, and strong convergence of the Cauchy method is shown

Comment: Such equations play an important role in the theory of Bessel processes (see chapter XI of Revuz-Yor,*Continuous Martingales and Brownian Motion,* Springer 1999

Keywords: Stochastic differential equations, Hölder conditions

Nature: Original

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XIII: 25, 281-293, LNM 721 (1979)

**ÉMERY, Michel**

Équations différentielles stochastiques lipschitziennes~: étude de la stabilité (Stochastic calculus)

This is the main application of the topologies on processes and semimartingales introduced in 1324. Using a very general definition of stochastic differential equations turns out to make the proof much simpler, and the existence and uniqueness of solutions of such equations is proved anew before the stability problem is discussed. Useful inequalities on stochastic integration are proved, and used as technical tools

Comment: For all of this subject, the book of Protter*Stochastic Integration and Differential Equations,* Springer 1989, is a useful reference

Keywords: Stochastic differential equations, Stability

Nature: Original

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XIII: 52, 611-613, LNM 721 (1979)

**MEYER, Paul-André**

Présentation de l'``inégalité de Doob'' de Métivier et Pellaumail (Martingale theory)

In the theory of stochastic differential equations with respect to discontinuous semimartingales, stopping processes ``just before'' a stopping time $T$ (at $T-$) is a basic technique, but since it does not preserve the martingale property, Doob's inequality cannot be used to control the stopped process. The inequality discussed here is an efficient substitute, used by Métivier-Pellaumail (*Ann. Prob.* **8**, 1980) to develop the whole theory of stochastic differential equations

Keywords: Doob's inequality, Stochastic differential equations

Nature: Exposition

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XIII: 53, 614-619, LNM 721 (1979)

**YOEURP, Chantha**

Solution explicite de l'équation $Z_t=1+\int_0^t |Z_{s-}|\,dX_s$ (Stochastic calculus)

The title describes completely the paper

Keywords: Stochastic differential equations

Nature: Original

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XIV: 13, 118-124, LNM 784 (1980)

**ÉMERY, Michel**

Équations différentielles stochastiques. La méthode de Métivier-Pellaumail (Stochastic calculus)

Métivier-Pellaumail introduced the idea of an increasing process $(A_t)$ controlling a semimartingale $X$ as the property $$E[\,(sup_{t<T} \int_0^t H_s dX_s)^2\,] \le E[\,A_{T-}\,\int_0^{T-} H_s^2 dA_s\,]$$ for all stopping times $T$ and bounded previsible processes $(H_t)$. For a proof see 1414. Métivier-Pellaumail used this inequality to develop the theory of stochastic differential equations (including stability) without localization and pasting together at jump times. Here their method is applied to the topology of semimartingales

Comment: See 1352. A general reference on the Métivier-Pellaumail method can be found in their book*Stochastic Integration,* Academic Press 1980. See also He-Wang-Yan, *Semimartingale Theory and Stochastic Calculus,* CRC Press 1992

Keywords: Semimartingales, Spaces of semimartingales, Stochastic differential equations, Doob's inequality, Métivier-Pellaumail inequality

Nature: Original

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XIV: 24, 209-219, LNM 784 (1980)

**PELLAUMAIL, Jean**

Remarques sur l'intégrale stochastique (Stochastic calculus)

This is an exposition of stochastic integrals and stochastic differential equations for Banach space valued processes along the lines of Métivier-Pellaumail*Stochastic Integration* (1980), the class of semimartingales being defined by the Métivier-Pellaumail inequality (1413)

Keywords: Stochastic integrals, Stochastic differential equations, Métivier-Pellaumail inequality

Nature: Exposition

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XIV: 32, 282-304, LNM 784 (1980)

**KUNITA, Hiroshi**

On the representation of solutions of stochastic differential equations (Stochastic calculus)

This paper concerns stochastic differential equations in the standard form $dY_t=\sum_i X_i(Y_t)\,dB^i(t)+X_0(Y_t)\,dt$ where the $B^i$ are independent Brownian motions, the stochastic integrals are in the Stratonovich sense, and $X_i,X_0$ have the geometric nature of vector fields. The problem is to find a deterministic (and smooth) machinery which, given the paths $B^i(.)$ will produce the path $Y(.)$. The complexity of this machinery reflects that of the Lie algebra generated by the vector fields. After a study of the commutative case, a paper of Yamato settled the case of a nilpotent Lie algebra, and the present paper deals with the solvable case. This line of thought led to the important and popular theory of flows of diffeomorphisms associated with a stochastic differential equation (see for instance Kunita's paper in*Stochastic Integrals,* Lecture Notes in M. 851)

Comment: On a closely related subject, see the paper of Fliess and Norman-Cyrot, 1623

Keywords: Stochastic differential equations, Lie algebras, Campbell-Hausdorff formula

Nature: Original

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XIV: 33, 305-315, LNM 784 (1980)

**YAN, Jia-An**

Sur une équation différentielle stochastique générale (Stochastic calculus)

The differential equation considered is of the form $X_t= \Phi(X)_t+\int_0^tF(X)_s\,dM_s$, where $M$ is a semimartingale, $\Phi$ maps adapted cadlag processes into themselves, and $F$ maps adapted cadlag process into previsible processes---not locally bounded, this is the main technical point. Some kind of Lipschitz condition being assumed, existence, uniqueness and stability are proved

Keywords: Stochastic differential equations

Nature: Original

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XV: 05, 44-102, LNM 850 (1981)

**MEYER, Paul-André**

Géométrie stochastique sans larmes (Stochastic differential geometry)

Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is*continuous semimartingales in manifolds,* following L.~Schwartz (LN **780**, 1980), but with additional features: an indication of J.M.~Bismut hinting to a definition of continuous *martingales * in a manifold, and the author's own interest on the forgotten intrinsic definition of the second differential $d^2f$ of a function. All this fits together into a geometric approach to semimartingales, and a probabilistic approach to such geometric topics as torsion-free connexions

Comment: A short introduction by the same author can be found in*Stochastic Integrals,* Springer LNM 851. The same ideas are expanded and presented in the supplement to Volume XVI and the book by Émery, *Stochastic Calculus on Manifolds *

Keywords: Semimartingales in manifolds, Martingales in manifolds, Transfer principle, Stochastic differential equations, Stochastic integrals, Stratonovich integrals

Nature: Original

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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: 38, 561-586, LNM 850 (1981)

**PELLAUMAIL, Jean**

Solutions faibles et semi-martingales (Stochastic calculus, General theory of processes)

From the author's summary: ``we consider a stochastic differential equation $dX=a(X)\,dZ$ where $Z$ is a semimartingale and $a$ is a previsible functional which is continuous for the uniform norm. We prove the existence of a weak solution for such an equation''. The important point is the definition of a weak solution: it turns out to be a ``fuzzy process'' in the sense of 1536, i.e., a fuzzy r.v. taking values in the Polish space of cadlag sample functions

Keywords: Stochastic differential equations, Weak solutions, Fuzzy random variables

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: 23, 257-267, LNM 920 (1982)

**FLIESS, Michel**; **NORMAND-CYROT, Dorothée**

Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T.~Chen (Stochastic calculus)

Consider a s.d.e. in a manifold, $dX_t=\sum_i A_i(X)\,dM^i_t$ (Stratonovich differentials), driven by continuous real semimartingales $M^i_t$, and where the $A_i$ have the geometrical nature of vector fields. Such an equation has a counterpart in which the $M^i(t)$ are arbitrary deterministic piecewise smooth functions, and if this equation can be solved by some deterministic machinery, then the s.d.e. can be solved too, just by making the input random. Thus a bridge is drawn between s.d.e.'s and problems of deterministic control theory. From this point of view, the complexity of the problem reflects that of the Lie algebra generated by the vector fields $A_i$. Assuming these fields are complete (i.e., generate true one-parameter groups on the manifold) and generate a finite dimensional Lie algebra (which then is the Lie algebra of a matrix group), the problem can be linearized. If the Lie algebra is nilpotent, the solution then can be expressed explicitly as a function of a finite number of iterated integrals of the driving processes (Chen integrals), and this provides the required ``deterministic machine''. It thus appears that results like those of Yamato (*Zeit. für W-Theorie,* **47**, 1979) do not really belong to probability theory

Comment: See Kunita's paper 1432

Keywords: Stochastic differential equations, Lie algebras, Chen's iterated integrals, Campbell-Hausdorff formula

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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XVI-S: 57, 165-207, LNM 921 (1982)

**MEYER, Paul-André**

Géométrie différentielle stochastique (bis) (Stochastic differential geometry)

A sequel to 1505. The main theme is that an ordinary differential equation has a non unique extension as a stochastic differential equation: besides the Stratonovich one, given by the ``transfer principle'', there are other possibilities: choosing among them requires some additional, connection-like, structure. The most striking application is the Dohrn-Guerra correction to the parallel transport along a semimartingale

Comment: For complements, see Émery 1658, Hakim-Dowek-Lépingle 2023, Émery's monography*Stochastic Calculus in Manifolds* (Springer, 1989) and article 2428, and Arnaudon-Thalmaier 3214

Keywords: Semimartingales in manifolds, Stochastic differential equations, Local characteristics, Nelson's stochastic mechanics, Transfer principle

Nature: Original

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XVI-S: 58, 208-216, LNM 921 (1982)

**ÉMERY, Michel**

En marge de l'exposé de Meyer : ``Géométrie différentielle stochastique'' (Stochastic differential geometry)

Marginal remarks to Meyer 1657

Keywords: Semimartingales in manifolds, Stochastic differential equations

Nature: Original

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XIX: 07, 91-112, LNM 1123 (1985)

**SCHWARTZ, Laurent**

Construction directe d'une diffusion sur une variété (Stochastic differential geometry)

This seems to be the first use of Witney's embedding theorem to construct a process (a Brownian motion, a diffusion, a solution to some s.d.e.) in a manifold $M$ by embedding $M$ into some $**R**^d$. Very general existence and uniqueness results are obtained

Comment: This method has since become standard in stochastic differential geometry; see for instance Émery's book*Stochastic Calculus in Manifolds* (Springer, 1989)

Keywords: Diffusions in manifolds, Stochastic differential equations

Nature: Original

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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

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XXIV: 28, 407-441, LNM 1426 (1990)

**ÉMERY, Michel**

On two transfer principles in stochastic differential geometry (Stochastic differential geometry)

Second-order stochastic calculus gives two intrinsic methods to transform an ordinary differential equation into a stochastic one (see Meyer 1657, Schwartz 1655 or Emery*Stochastic calculus in manifolds*). The first one gives a Stratonovich SDE and needs coefficients regular enough; the second one gives an Ito equation and needs a connection on the manifold. Discretizing time and smoothly interpolating the driving semimartingale is known to give an approximation to the Stratonovich transfer; it is shown here that another discretized-time procedure converges to the Ito transfer. As an application, if the ODE makes geodesics to geodesics, then the Ito and Stratonovich SDE's are the same

Comment: An error is corrected in 2649. The term ``transfer principle'' was coined by Malliavin,*Géométrie Différentielle Stochastique,* Presses de l'Université de Montréal (1978); see also Bismut, *Principes de Mécanique Aléatoire* (1981) and 1505

Keywords: Stochastic differential equations, Semimartingales in manifolds, Transfer principle

Nature: Original

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XXVI: 10, 113-126, LNM 1526 (1992)

**TAYLOR, John C.**

Skew products, regular conditional probabilities and stochastic differential equations: a technical remark (Stochastic calculus, Stochastic differential geometry)

This is a detailed study of the transfer principle (the solution to a Stratonovich stochastic differential equations can be pathwise obtained from the driving semimartingale by solving the corresponding ordinary differential equation) in the case of an equation where the solution of another equation plays the role of a parameter

Comment: The term ``transfer principle'' was coined by Malliavin,*Géométrie Différentielle Stochastique,* Presses de l'Université de Montréal (1978); see also Bismut, *Principes de Mécanique Aléatoire* (1981)

Keywords: Transfer principle, Stochastic differential equations, Stratonovich integrals

Nature: Original

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XXVI: 11, 127-145, LNM 1526 (1992)

**ESTRADE, Anne**; **PONTIER, Monique**

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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XXVI: 49, 633-633, LNM 1526 (1992)

**ÉMERY, Michel**

Correction au Séminaire~XXIV (Stochastic differential geometry)

An error in 2428 is pointed out; it is corrected by Cohen (*Stochastics Stochastics Rep.* **56**, 1996)

Keywords: Stochastic differential equations, Semimartingales in manifolds

Nature: Correction

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XLIII: 07, 191-214, LNM 2006 (2011)

**RIEDLE, Markus**

Cylindrical Wiener Processes

Keywords: Cylindrical Wiener process, Cylindrical process, Cylindrical measure, Stochastic integrals, Stochastic differential equations, Radonifying operator, Reproducing kernel Hilbert space

Nature: Original

XLIII: 11, 269-307, LNM 2006 (2011)

**PAGÈS, Gilles**; **SELLAMI, Afef**

Convergence of multi-dimensional quantized SDE's (Integration theory, Theory of processes)

Keywords: Functional quantization, Stochastic differential equations, Stratonovich integrals, Stationary quantizers, Rough paths, Itô map, Hölder semi-norm, $p$-variation

Nature: Original

XLIV: 04, 75-103, LNM 2046 (2012)

**QIAN, Zhongmin**; **YING, Jiangang**

Martingale representations for diffusion processes and backward stochastic differential equations (Stochastic calculus)

Keywords: Backward Stochastic Differential equations, Dirichlet forms, Hunt processes, Martingales, Natural filtration, Non-linear equations

Nature: Original

XLV: 09, 245-275, LNM 2078 (2013)

**JACOB, Emmanuel**

Langevin Process Reflected on a Partially Elastic Boundary II (Theory of processes)

Keywords: Langevin process, second order reflection, recurrent extension, excursion measure, stochastic differential equations, $h$-transform

Nature: Original

Note on a stochastic integral equation (Stochastic calculus)

Though this paper has been completely superseded by the theory of stochastic differential equations with respect to semimartingales (see 1124), it has a great historical importance as the first step in this direction: the semimartingale involved is the sum of a locally square integrable martingale and a continuous increasing process

Comment: The author developed the subject further in

Keywords: Stochastic differential equations

Nature: Original

Retrieve article from Numdam

X: 16, 240-244, LNM 511 (1976)

On the uniqueness of solutions of stochastic differential equations with reflecting barrier conditions (Stochastic calculus, Diffusion theory)

A stochastic differential equation is considered on the positive half-line, driven by Brownian motion, with time-dependent coefficients and a reflecting barrier condition at $0$ (Skorohod style). Skorohod proved pathwise uniqueness under Lipschitz condition, and this is extended here to moduli of continuity satisfying integral conditions

Comment: This extends to the reflecting barrier case the now classical result in the ``free'' case due to Yamada-Watanabe,

Keywords: Stochastic differential equations, Boundary reflection

Nature: Original

Retrieve article from Numdam

XI: 24, 376-382, LNM 581 (1977)

Équations différentielles stochastiques (Stochastic calculus)

This is an improved and simplified exposition of the existence and uniqueness theorem for solutions of stochastic differential equations with respect to semimartingales, as proved by the first author in

Keywords: Stochastic differential equations, Semimartingales

Nature: Exposition, Original additions

Retrieve article from Numdam

XI: 27, 411-414, LNM 581 (1977)

Images d'équations différentielles stochastiques (Stochastic calculus)

This paper answers a natural question: can one take computations performed on ``canonical'' versions of processes back to their original spaces? It is related to Stricker's work (

Keywords: Stochastic differential equations

Nature: Original

Retrieve article from Numdam

XII: 13, 114-131, LNM 649 (1978)

Sur une construction des solutions d'équations différentielles stochastiques dans le cas non-lipschitzien (Stochastic calculus)

The results of this paper improve on those of the author's paper (

Comment: Such equations play an important role in the theory of Bessel processes (see chapter XI of Revuz-Yor,

Keywords: Stochastic differential equations, Hölder conditions

Nature: Original

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XIII: 25, 281-293, LNM 721 (1979)

Équations différentielles stochastiques lipschitziennes~: étude de la stabilité (Stochastic calculus)

This is the main application of the topologies on processes and semimartingales introduced in 1324. Using a very general definition of stochastic differential equations turns out to make the proof much simpler, and the existence and uniqueness of solutions of such equations is proved anew before the stability problem is discussed. Useful inequalities on stochastic integration are proved, and used as technical tools

Comment: For all of this subject, the book of Protter

Keywords: Stochastic differential equations, Stability

Nature: Original

Retrieve article from Numdam

XIII: 52, 611-613, LNM 721 (1979)

Présentation de l'``inégalité de Doob'' de Métivier et Pellaumail (Martingale theory)

In the theory of stochastic differential equations with respect to discontinuous semimartingales, stopping processes ``just before'' a stopping time $T$ (at $T-$) is a basic technique, but since it does not preserve the martingale property, Doob's inequality cannot be used to control the stopped process. The inequality discussed here is an efficient substitute, used by Métivier-Pellaumail (

Keywords: Doob's inequality, Stochastic differential equations

Nature: Exposition

Retrieve article from Numdam

XIII: 53, 614-619, LNM 721 (1979)

Solution explicite de l'équation $Z_t=1+\int_0^t |Z_{s-}|\,dX_s$ (Stochastic calculus)

The title describes completely the paper

Keywords: Stochastic differential equations

Nature: Original

Retrieve article from Numdam

XIV: 13, 118-124, LNM 784 (1980)

Équations différentielles stochastiques. La méthode de Métivier-Pellaumail (Stochastic calculus)

Métivier-Pellaumail introduced the idea of an increasing process $(A_t)$ controlling a semimartingale $X$ as the property $$E[\,(sup_{t<T} \int_0^t H_s dX_s)^2\,] \le E[\,A_{T-}\,\int_0^{T-} H_s^2 dA_s\,]$$ for all stopping times $T$ and bounded previsible processes $(H_t)$. For a proof see 1414. Métivier-Pellaumail used this inequality to develop the theory of stochastic differential equations (including stability) without localization and pasting together at jump times. Here their method is applied to the topology of semimartingales

Comment: See 1352. A general reference on the Métivier-Pellaumail method can be found in their book

Keywords: Semimartingales, Spaces of semimartingales, Stochastic differential equations, Doob's inequality, Métivier-Pellaumail inequality

Nature: Original

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XIV: 24, 209-219, LNM 784 (1980)

Remarques sur l'intégrale stochastique (Stochastic calculus)

This is an exposition of stochastic integrals and stochastic differential equations for Banach space valued processes along the lines of Métivier-Pellaumail

Keywords: Stochastic integrals, Stochastic differential equations, Métivier-Pellaumail inequality

Nature: Exposition

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XIV: 32, 282-304, LNM 784 (1980)

On the representation of solutions of stochastic differential equations (Stochastic calculus)

This paper concerns stochastic differential equations in the standard form $dY_t=\sum_i X_i(Y_t)\,dB^i(t)+X_0(Y_t)\,dt$ where the $B^i$ are independent Brownian motions, the stochastic integrals are in the Stratonovich sense, and $X_i,X_0$ have the geometric nature of vector fields. The problem is to find a deterministic (and smooth) machinery which, given the paths $B^i(.)$ will produce the path $Y(.)$. The complexity of this machinery reflects that of the Lie algebra generated by the vector fields. After a study of the commutative case, a paper of Yamato settled the case of a nilpotent Lie algebra, and the present paper deals with the solvable case. This line of thought led to the important and popular theory of flows of diffeomorphisms associated with a stochastic differential equation (see for instance Kunita's paper in

Comment: On a closely related subject, see the paper of Fliess and Norman-Cyrot, 1623

Keywords: Stochastic differential equations, Lie algebras, Campbell-Hausdorff formula

Nature: Original

Retrieve article from Numdam

XIV: 33, 305-315, LNM 784 (1980)

Sur une équation différentielle stochastique générale (Stochastic calculus)

The differential equation considered is of the form $X_t= \Phi(X)_t+\int_0^tF(X)_s\,dM_s$, where $M$ is a semimartingale, $\Phi$ maps adapted cadlag processes into themselves, and $F$ maps adapted cadlag process into previsible processes---not locally bounded, this is the main technical point. Some kind of Lipschitz condition being assumed, existence, uniqueness and stability are proved

Keywords: Stochastic differential equations

Nature: Original

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XV: 05, 44-102, LNM 850 (1981)

Géométrie stochastique sans larmes (Stochastic differential geometry)

Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is

Comment: A short introduction by the same author can be found in

Keywords: Semimartingales in manifolds, Martingales in manifolds, Transfer principle, Stochastic differential equations, Stochastic integrals, Stratonovich integrals

Nature: Original

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

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

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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

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XV: 38, 561-586, LNM 850 (1981)

Solutions faibles et semi-martingales (Stochastic calculus, General theory of processes)

From the author's summary: ``we consider a stochastic differential equation $dX=a(X)\,dZ$ where $Z$ is a semimartingale and $a$ is a previsible functional which is continuous for the uniform norm. We prove the existence of a weak solution for such an equation''. The important point is the definition of a weak solution: it turns out to be a ``fuzzy process'' in the sense of 1536, i.e., a fuzzy r.v. taking values in the Polish space of cadlag sample functions

Keywords: Stochastic differential equations, Weak solutions, Fuzzy random variables

Nature: Original

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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

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XVI: 23, 257-267, LNM 920 (1982)

Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T.~Chen (Stochastic calculus)

Consider a s.d.e. in a manifold, $dX_t=\sum_i A_i(X)\,dM^i_t$ (Stratonovich differentials), driven by continuous real semimartingales $M^i_t$, and where the $A_i$ have the geometrical nature of vector fields. Such an equation has a counterpart in which the $M^i(t)$ are arbitrary deterministic piecewise smooth functions, and if this equation can be solved by some deterministic machinery, then the s.d.e. can be solved too, just by making the input random. Thus a bridge is drawn between s.d.e.'s and problems of deterministic control theory. From this point of view, the complexity of the problem reflects that of the Lie algebra generated by the vector fields $A_i$. Assuming these fields are complete (i.e., generate true one-parameter groups on the manifold) and generate a finite dimensional Lie algebra (which then is the Lie algebra of a matrix group), the problem can be linearized. If the Lie algebra is nilpotent, the solution then can be expressed explicitly as a function of a finite number of iterated integrals of the driving processes (Chen integrals), and this provides the required ``deterministic machine''. It thus appears that results like those of Yamato (

Comment: See Kunita's paper 1432

Keywords: Stochastic differential equations, Lie algebras, Chen's iterated integrals, Campbell-Hausdorff formula

Nature: Original

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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

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XVI-S: 57, 165-207, LNM 921 (1982)

Géométrie différentielle stochastique (bis) (Stochastic differential geometry)

A sequel to 1505. The main theme is that an ordinary differential equation has a non unique extension as a stochastic differential equation: besides the Stratonovich one, given by the ``transfer principle'', there are other possibilities: choosing among them requires some additional, connection-like, structure. The most striking application is the Dohrn-Guerra correction to the parallel transport along a semimartingale

Comment: For complements, see Émery 1658, Hakim-Dowek-Lépingle 2023, Émery's monography

Keywords: Semimartingales in manifolds, Stochastic differential equations, Local characteristics, Nelson's stochastic mechanics, Transfer principle

Nature: Original

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XVI-S: 58, 208-216, LNM 921 (1982)

En marge de l'exposé de Meyer : ``Géométrie différentielle stochastique'' (Stochastic differential geometry)

Marginal remarks to Meyer 1657

Keywords: Semimartingales in manifolds, Stochastic differential equations

Nature: Original

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XIX: 07, 91-112, LNM 1123 (1985)

Construction directe d'une diffusion sur une variété (Stochastic differential geometry)

This seems to be the first use of Witney's embedding theorem to construct a process (a Brownian motion, a diffusion, a solution to some s.d.e.) in a manifold $M$ by embedding $M$ into some $

Comment: This method has since become standard in stochastic differential geometry; see for instance Émery's book

Keywords: Diffusions in manifolds, Stochastic differential equations

Nature: Original

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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

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XXIV: 28, 407-441, LNM 1426 (1990)

On two transfer principles in stochastic differential geometry (Stochastic differential geometry)

Second-order stochastic calculus gives two intrinsic methods to transform an ordinary differential equation into a stochastic one (see Meyer 1657, Schwartz 1655 or Emery

Comment: An error is corrected in 2649. The term ``transfer principle'' was coined by Malliavin,

Keywords: Stochastic differential equations, Semimartingales in manifolds, Transfer principle

Nature: Original

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XXVI: 10, 113-126, LNM 1526 (1992)

Skew products, regular conditional probabilities and stochastic differential equations: a technical remark (Stochastic calculus, Stochastic differential geometry)

This is a detailed study of the transfer principle (the solution to a Stratonovich stochastic differential equations can be pathwise obtained from the driving semimartingale by solving the corresponding ordinary differential equation) in the case of an equation where the solution of another equation plays the role of a parameter

Comment: The term ``transfer principle'' was coined by Malliavin,

Keywords: Transfer principle, Stochastic differential equations, Stratonovich integrals

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

Keywords: Stochastic parallel transport, Stochastic differential equations, Jumps

Nature: Original

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XXVI: 49, 633-633, LNM 1526 (1992)

Correction au Séminaire~XXIV (Stochastic differential geometry)

An error in 2428 is pointed out; it is corrected by Cohen (

Keywords: Stochastic differential equations, Semimartingales in manifolds

Nature: Correction

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XLIII: 07, 191-214, LNM 2006 (2011)

Cylindrical Wiener Processes

Keywords: Cylindrical Wiener process, Cylindrical process, Cylindrical measure, Stochastic integrals, Stochastic differential equations, Radonifying operator, Reproducing kernel Hilbert space

Nature: Original

XLIII: 11, 269-307, LNM 2006 (2011)

Convergence of multi-dimensional quantized SDE's (Integration theory, Theory of processes)

Keywords: Functional quantization, Stochastic differential equations, Stratonovich integrals, Stationary quantizers, Rough paths, Itô map, Hölder semi-norm, $p$-variation

Nature: Original

XLIV: 04, 75-103, LNM 2046 (2012)

Martingale representations for diffusion processes and backward stochastic differential equations (Stochastic calculus)

Keywords: Backward Stochastic Differential equations, Dirichlet forms, Hunt processes, Martingales, Natural filtration, Non-linear equations

Nature: Original

XLV: 09, 245-275, LNM 2078 (2013)

Langevin Process Reflected on a Partially Elastic Boundary II (Theory of processes)

Keywords: Langevin process, second order reflection, recurrent extension, excursion measure, stochastic differential equations, $h$-transform

Nature: Original