Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports 78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242.     

Continuous solutions of volterra integral equations with curvilinear stochastic integrals

The existence and uniqueness of a continuous solution of a stochastic integral equation with curvilinear integrals is proved.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 38–48, 1988.     

On an approach to the definition of a stochastic integral with respect to a Gaussian measure

We define a stochastic Riemann integral with respect to a Gaussian measure. The class of integrable functions is introduced in which there exists a solution of a stochastic Fredholm integral equation. It is shown by examples how to pass from the integral defined here to the Itô and Stratonovich integrals.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 100–108, 1986.     

Convergence of solutions of a class of two-parameter stochastic integral equations

The theorem on the continuous dependence on a parameter of the solutions of a class of stochastic integral equations with random coefficients containing as summands along with a Lebesgue integral, two-parameter stochastic integrals with respect to a Wiener and a centered Poisson measure is proved.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 97–105, 1988.     

Max-Plus Stochastic Processes

This paper is concerned with processes which are max-plus counterparts of Markov diffusion processes governed by Ito sense stochastic differential equations. Concepts of max-plus martingale and max-plus stochastic differential equation are introduced. The max-plus counterparts of backward and forward PDEs for Markov diffusions turn out to be first-order PDEs of Hamilton–Jacobi–Bellman type. Max-plus additive integrals and a max-plus additive dynamic programming principle are considered. This leads to variational inequalities of Hamilton–Jacobi–Bellman type.     

Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces

We define stochastic integrals of Banach valued random functions w.r.t. compensated Poisson random measures. Different notions of stochastic integrals are introduced and sufficient conditions for their existence are established. These generalize, for the case where integration is performed w.r.t. compensated Poisson random measures, the notion of stochastic integrals of real valued random functions introduced in Ikeda and Watanabe (1989) [Stochastic Differential Equations and Diffusion Processes (second edition), North-Holland Mathematical Library, Vol. 24, North Holland Publishing Company, Amsterdam/Oxford/New York.], (in a different way) in Bensoussan and Lions (1982) [Contróle impulsionnel et inquations quasi variationnelles. (French) [Impulse control and quasivariational inequalities] Méthodes Mathématiques de l'Informatique [Mathematical Methods of Information Science], Vol. 11. (Gauthier-Villars, Paris), and Skorohod, A.V. (1965) [Studies in the theory of random processes (Addison-Wesley Publishing Company, Inc, Reading, MA), Translated from the Russian by Scripta Technica, Inc. ], to the case of Banach valued random functions. The relation between these two different notions of stochastic integrals is also discussed here.     

Multiple stochastic integrals constructed by special expansions of products of the integrating stochastic processes

The paper deals with problems of constructing multiple stochastic integrals in the case when the product of increments of the integrating stochastic process admits an expansion as a finite sum of series with random coefficients. This expansion was obtained for a sufficiently wide class including centered Gaussian processes. In the paper, some necessary and sufficient conditions are obtained for the existence of multiple stochastic integrals defined by an expansion of the product of Wiener processes. It was obtained a recurrent representation for the Wiener stochastic integral as an analog of the Hu–Meyer formula.     

Finite-dimensional distributions of invariant measure in nonlinear stochastic differential systems

The article studies stochastic processes defined by Ito stochastic differential equations with Wiener white noise. Methods are considered that find exact expressions for finite-dimensional densities of random stationary and nonstationary (in the narrow sense) processes based on construction of integral invariants of specially chosen systems of ordinary differential equations. Translated from Algoritmy Upravleniya i Identifikatsii, pp. 129–140, 1997.     

Explicit One-Step Numerical Method with the Strong Convergence Order of 2.5 for Ito Stochastic Differential Equations with a Multi-Dimensional Nonadditive Noise Based on the Taylor–Stratonovich Expansion

Computational Mathematics and Mathematical Physics - A strongly converging method of order 2.5 for Ito stochastic differential equations with multidimensional nonadditive noise based on the unified...     

On the calculation of certain singular integrals

A method for the approximate calculation of Cauchy-type integrals with logarithmic singularities is proposed. It is based on the expansion of a functionf(t) in a series in the Chebyshev polynomials.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 8, pp. 1174–1176, August, 1993.     

Meyer-Emery inequalities for norms of stochastic integrals with a parameter

We consider stochastic integrals with a parameter in the components. Inequalities for the functional norms of stochastic integrals are derived, generalizing the known inequalities of M. Emery [1] and P.-A. Meyer [2] to triples of spaces connected by the multiplication operation.Translated from Issledovaniya po Prikladnoi Matematike, Kazan', No. 14, pp. 134–143, 1987.     

Lévy processes and quasi-shuffle algebras

We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.     

Almost Sure Convergence of the Numerical Discretization of Stochastic Jump Diffusions

Based on the shuffle product expansion of exponential Lie series in terms of a Philip Hall basis for the stochastic differential equations of jump-diffusion type, we can establish Stratonovich–Taylor–Hall (STH) schemes. However, the STHr scheme converges only at order r in the mean-square sense. In order to have the almost sure Stratonovich–Taylor–Hall (ASTH) schemes, we have to include all the terms related to multiple Poissonian integrals as the moments of multiple Poissonian integrals always have lower orders of magnitudes as compared with those of multiple Brownian integrals.     

Absolute Continuity in Infinite Dimensions and Anticipating Stochastic Calculus

We generalize the change of variables formula for infinite dimensional integrals with respect to the Gaussian and exponential densities to the case of the uniform measure. The presentation of the result and its interpretation in terms of stochastic processes and anticipating stochastic calculus is unified. The expression of the Radon–Nykodim density function uses a Carleman–Fredholm determinant and a divergence operator.     

Two mathematical problems of canonical quantization. III. Stochastic vacuum mechanics

The problem of recovery of the measure from its logarithmic derivative is investigated. The role of this problem in stochastic mechanics, canonical quantization, and the theory of integration of functionals is discussed. It is shown that a measure that possesses logarithmic derivativeA is a stationary distribution of a diffusion process with drift coefficientA. This makes it possible to calculate integrals with respect to the measure by means of Monte Carlo methods.Power Institute, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 377–395, June, 1992.     

Elements of stochastic analysis for the case of Grassmann variables. III. Correlation functions

The present paper continues earlier studies [1,2], in which analogs were proposed in the case of Grassmann variables for concepts such as classical stochastic analysis, stochastic integrals, random processes, and stochastic partial differential equations and their solutions. This was done for the special case when the classical objects are functionals of a so-called smoothed Wiener process onR ₊×R ^v . In the present paper, the correlation functions of the solution of a stochastic partial differential equation are studied together with some applications.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 323–335, December, 1993.     

A property of stable systems of linear stochastic equations

The relationship between exponential mean square stability of systems of linear ordinary differential equations with Gaussian coefficients and the same stability of the corresponding linear stochastic Ito differential equations is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 147–152, February, 1990.     

随机扰动下统一混沌系统的有限时间同步

本文研究了具有随机扰动的统一混沌系统的有限时间同步问题,其中随机扰动是一维标准的维纳随机过程.利用了有限时间随机李雅普诺夫稳定性理论、伊藤公式,本文分三个步骤设立了三个控制器获得了驱动–响应系统在有限时间内的均方渐近同步.最后进行的数值模拟验证了理论结果的正确性和方法的有效性.     

Third-order asymptotic expansion of <Emphasis Type="Italic">M</Emphasis>-estimators for diffusion processes

For an unknown parameter in the drift function of a diffusion process, we consider an M-estimator based on continuously observed data, and obtain its distributional asymptotic expansion up to the third order. Our setting covers the misspecified cases. To represent the coefficients in the asymptotic expansion, we derive some formulas for asymptotic cumulants of stochastic integrals, which are widely applicable to many other problems. Furthermore, asymptotic properties of cumulants of mixing processes will be also studied in a general setting.