Stochastic calculus with respect to continuous finite quadratic variation processes

The quadratic variation of a continuous process (when it exists) is defined through a regularization procedure. A large class of finite quadratic variation processes is provided, with a particular emphasis on Gaussian processes. For such processes a calculus is developed with application to the study of some stochastic differential equations.

[1]  C. S. G. David.  Stochastic Analysis , 2021, Nature.

[2]  L. Decreusefond,et al.  Stochastic Analysis of the Fractional Brownian Motion , 1999 .

[3]  M. Zähle.  Integration with respect to fractal functions and stochastic calculus. I , 1998 .

[4]  J. Wolf.  Transformations of semi-martingales and local dirichlet processes , 1997 .

[5]  S. Tindel.  Stochastic parabolic equations with anticipative initial condition , 1997 .

[6]  Wolf Jochen.  An Itô formula for local dirichlet processes , 1997 .

[7]  Francesco Russo,et al.  The generalized covariation process and Ito formula , 1995 .

[8]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[9]  A. Shiryayev,et al.  Quadratic covariation and an extension of Itô's formula , 1995 .

[10]  Francesco Russo,et al.  Forward, backward and symmetric stochastic integration , 1993 .

[11]  H. Kuo,et al.  White Noise: An Infinite Dimensional Calculus , 1993 .

[12]  A. Russek.  On an extension of the stochastic integral , 1991 .

[13]  J. Potthoff,et al.  Itô's lemma without non-anticipatory conditions , 1991 .

[14]  M. Zakai.  Stochastic integration, trace and the skeleton of Wiener functionals , 1990 .

[15]  D. Nualart,et al.  Stochastic processes possessing a skorohod integral representation , 1990 .

[16]  F. Utzet,et al.  Stratonovich integral and trace , 1990 .

[17]  J. Bertoin.  Sur Une Integrale Pour Les Processus A $\alpha$-Variation Bornee , 1989 .

[18]  David Nualart,et al.  Stochastic calculus with anticipating integrands , 1988 .

[19]  J. Bertoin.  Une extension d'une inégalité de Burkholder, Davis, Gundy pour les processus à α-variation bornée et applications , 1988 .

[20]  H. Kuo.  White noise approach to stochastic integration , 1988 .

[21]  J. Bertoin.  Les processus de dirichlet et tant qu'espace de banach , 1986 .

[22]  J. Colombeau.  Multiplication of distributions , 1983 .

[23]  Héctor J. Sussmann,et al.  An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point , 1977 .

[24]  Annika Lang.  Stochastic Partial Differential Equations , 2021, Computer Vision.

[25]  M. Zähle.  On the Link Between Fractional and Stochastic Calculus , 1999 .

[26]  J. Kerstan.  ORDINARY DIFFERENTIAL EQUATIONS WITH FRACTAL NOISE , 1997 .

[27]  A. S. ÜstünelÉcole.  Stochastic Analysis of the Fractional Brownian Motion , 1996 .

[28]  P. Vallois,et al.  Ito formula forC1-functions of semimartingales , 1996 .

[29]  C. Stricker,et al.  Componentwise and Vector Stochastic Integration with Respect to Certain Multi-Dimensional Continuous Local Martingales , 1995 .

[30]  F. Russo.  Colombeau Generalized Functions and Stochastic Analysis , 1994 .

[31]  Terry Lyons,et al.  Decomposition of Dirichlet Processes and its Application , 1994 .

[32]  P. Protter.  Stochastic integration and differential equations : a new approach , 1990 .

[33]  P. Protter.  Stochastic integration and differential equations , 1990 .

[34]  Terry Lyons,et al.  A crossing estimate for the canonical process on a Dirichlet space and tightness result , 1988 .

[35]  J. Colombeau,et al.  Elementary introduction to new generalized functions , 1985 .

[36]  渡辺 信三.  Lectures on stochastic differential equations and Malliavin calculus , 1984 .

[37]  S. Mohammed.  Stochastic functional differential equations , 1984 .

[38]  Hans Föllmer,et al.  Calcul d'ito sans probabilites , 1981 .

[39]  福島 正俊.  Dirichlet forms and Markov processes , 1980 .

[40]  Hani J. Doss,et al.  Liens entre equations di erentielles stochastiques et ordinaires , 1977 .