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Though this paper has been completely superseded by the theory of stochastic differential equations with respect to semimartingales (see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 equationsNature: Original
Retrieve article from Numdam
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 reflectionNature: Original Retrieve article from Numdam
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,
SemimartingalesNature: Exposition,
Original additions Retrieve article from Numdam
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 equationsNature: Original Retrieve article from Numdam
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 conditionsNature: Original Retrieve article from Numdam
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,
StabilityNature: Original Retrieve article from Numdam
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 equationsNature: Exposition Retrieve article from Numdam
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 equationsNature: Original Retrieve article from Numdam
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 inequalityNature: Original Retrieve article from Numdam
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 inequalityNature: Exposition Retrieve article from Numdam
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,
1623Keywords: Stochastic differential equations,
Lie algebras,
Campbell-Hausdorff formulaNature: Original Retrieve article from Numdam
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 equationsNature: Original Retrieve article from Numdam
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 integralsNature: Original Retrieve article from Numdam
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
1922Keywords: Stochastic differential equations,
Flow of a s.d.e.Nature: Exposition,
Original additions Retrieve article from Numdam
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 transportNature: Original Retrieve article from Numdam
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 variablesNature: Original Retrieve article from Numdam
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
1624Keywords: Stochastic differential equations,
Flow of a s.d.e.Nature: Original Retrieve article from Numdam
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
1432Keywords: Stochastic differential equations,
Lie algebras,
Chen's iterated integrals,
Campbell-Hausdorff formulaNature: Original Retrieve article from Numdam
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.,
InjectivityNature: Exposition,
Original additions Retrieve article from Numdam
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
3214Keywords: Semimartingales in manifolds,
Stochastic differential equations,
Local characteristics,
Nelson's stochastic mechanics,
Transfer principleNature: Original Retrieve article from Numdam
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
1657Keywords: Semimartingales in manifolds,
Stochastic differential equationsNature: Original Retrieve article from Numdam
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 equationsNature: Original Retrieve article from Numdam
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
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
1505Keywords: Stochastic differential equations,
Semimartingales in manifolds,
Transfer principleNature: Original Retrieve article from Numdam
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 integralsNature: Original Retrieve article from Numdam
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,
JumpsNature: Original Retrieve article from Numdam
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 manifoldsNature: Correction Retrieve article from Numdam
XLIII: 07, 191-214, LNM 2006 (2011)
RIEDLE, Markus
Cylindrical Wiener ProcessesKeywords: Cylindrical Wiener process,
Cylindrical process,
Cylindrical measure,
Stochastic integrals,
Stochastic differential equations,
Radonifying operator,
Reproducing kernel Hilbert spaceNature: 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,
$p$-variationNature: 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 equationsNature: 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,
$h$-transformNature: Original
