Solution Of Stochastic Integral Equation Research Articles

Numerical solution of stochastic integral equations using CAS wavelets

Numerical solution of stochastic integral equations using CAS wavelets

Numerical solution of stochastic integral equations by using Bernoulli operational matrix

In this paper, a new computational method based on stochastic operational matrix for integration of Bernoulli polynomials is proposed for solving nonlinear Volterra–Fredholm–Hammerstein stochastic integral equations. By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra–Fredholm–Hammerstein stochastic integral equation is reduced to nonlinear system of algebraic equations with unknown Bernoulli coefficients. This work is inspired by Bazm (2015), where the authors study the deterministic integral equations. In order to show the rate of convergence of the suggested approach, we present theorems on convergence analysis and error estimation. Some illustrative error estimations and examples are provided and included to demonstrate applicability and accuracy of the technique.

On tightness of solutions of stochastic integral equations driven by p-semimartingales

We obtain sufficient conditions for tightness of a sequence of solutions of stochastic integral equations driven by p-semimartingales.

On estimation of the Hurst index of solutions of stochastic integral equations

Let X be a solution of a stochasti Let X be a solution of a stochastic integral equation driven by a fractional Brownian motion BH and let Vn(X, 2) = \sumn k=1(\DeltakX)2, where \DeltakX = X( k+1/n ) - X(k/n ). We study the 
 ditions n2H-1Vn(X, 2) convergence almost surely as n → ∞ holds. This fact is used to obtain a strongly consistent estimator of the Hurst index H, 1/2 < H < 1. 

Catalytic Discrete State Branching Models and Related Limit Theorems

Catalytic discrete state branching processes with immigration are defined as strong solutions of stochastic integral equations. We provide main limit theorems of those processes using different scalings. The class of limit processes of the theorems includes essentially all continuous state catalytic branching processes and spectrally positive regular affine processes.

On the approximation of the solutions of stochastic equations with θ-integrals

The paper investigates the problem of approximation of stochastic θ-integrals and the solutions of stochastic differential equations. The complete classification of the methods of approximation of stochastic θ-integrals in the convolution algebra is proposed. It is proved that the solutions of stochastic integral equations with θ-integral can be approximated by the solutions of finite-difference equations with averaging.

Solution of stochastic integral equation for multiple scattering by method of Taylor series expansion

Using the solution for multidimensional integral equation obtained in the author's earlier paper by the Taylor expansion method, the stochastic integral equation for a multiply scattered field is solved in stochastic form. From this, the mean value and space‐time correlation function of the multiply scattered field are obtained, in terms of the shape, size, number density and distribution region of scatterers. Both the near and far field are obtained. The results show that, for the first‐order approximation, the results agree with those obtained from the Born approximation, for the second‐ and third‐order approximations, the results agree with those obtained from the Neumann series expansion method in [1]. Therefore, these results also show better agreement with experimental results than those obtained from Twersky's formula.

On existence,asymptotic behaviour and stability of solutions of stochastic integral equations

This paper deals with the equation where H is an adapted corlol process, Z is a semimartingale such that Zo=0 a.s.and satisfies some regularity conditions. There are given sufficient conditions for the existence of random solutions and their asymptotic behaviour. We base on the notion of a measure of noncompactness in Banach space and the fixed–point theorem of Darbo type Results of this paper generalize those of [5],[7]

Existence of a Strong Solution of the Itô Stochastic Differential Equation

Previous article Next article Existence of a Strong Solution of the Itô Stochastic Differential EquationI. V. FedorenkoI. V. Fedorenkohttps://doi.org/10.1137/1129012PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. K. Zvonkin and , N. V. Krylov, Strong solutions of stochastic differential equations, Proceedings of the School and Seminar on the Theory of Random Processes (Druskininkai, 1974), Part II (Russian), Inst. Fiz. i Mat. Akad. Nauk Litovsk. SSR, Vilnius, 1975, 9–88, (In Russian.) 54:14100 Google Scholar[2] A. Yu. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR Sbornik, 39 (1981), 387–404 0462.60063 CrossrefGoogle Scholar[3] A. V. Mel'nikov, On strong solutions of stochastic differential equations with nonsmooth coefficients, Theory Prob. Appl., 24 (1979), 147–150 0432.60076 LinkGoogle Scholar[4] A. V. Skorokhod, Studies in the theory of random processes, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965viii+199 32:3082b 0146.37701 Google Scholar[5] Toshio Yamada, On a comparison theorem for solutions of stochastic differential equations and its applications, J. Math. Kyoto Univ., 13 (1973), 497–512 49:4093 0277.60047 CrossrefGoogle Scholar[6] I. I. Gikman and , A. V. Skorokhod, Stochastic Differential Equations, Springer-Verlag, Berlin-New York, 1972 Google Scholar[7] Philip Hartman, Ordinary differential equations, John Wiley & Sons Inc., New York, 1964xiv+612 30:1270 0125.32102 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails On a class of stochastic differential equations arising from the stochastic approximation theory20 January 2011 | Stochastics and Stochastic Reports, Vol. 44, No. 3-4 Cross Ref Volume 29, Issue 1| 1985Theory of Probability & Its Applications History Submitted:06 July 1981Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1129012Article page range:pp. 121-123ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Successive approximations for solutions of stochastic integral equations of Volterra type

in an abstract Wiener space (i, H, B), where w(s) is a B-valued Wiener process. The first integral is in the sense of Ito (see Kuo [5]) and the second is in the sense of Bochner. The determinist case is discussed by Chen [ 1 ] and a finite dimensional stochastic case is discussed by Kravec (3,4]. Conditions which guarantee the convergence in maximal quadratic mean of Picard’s successive approximations to a solution of (1.1) are given. Our assumptions made on rp, Ki are like that of Chen [ 11.

Strong uniqueness of solution of stochastic integral equations for semimartingale components

Strong uniqueness of solution of stochastic integral equations for semimartingale components

On properties of strong solutions of stochastic equations with respect to semimartingales

The paper is devoted to strong solutions of stochastic integral equations with respect to semimartingales. We study existence (for non-Lipschitz coefficients) and asymptotic behaviour of strong solutions and obtain also a number of results on weak and square mean convergence of these solutions.

An Extension of Kazamaki's Results on BMO Differentials

Kazamaki has shown that if (M~)nz1 M are BMO martingales with continuous paths and lim M' = M in BMO, then e(Mf) converges in Al to &(M), where 6(M) denotes the stochastic exponential of M. While Kazamaki's result does not extend to the right continuous case, it does extend locally. It is shown here that if M', M are semimartingales and M' converges locally in A (a semimartingale BMO-type norm) to M then X' converges locally in AT (1 _ p < 0o) to X, where X', X are resepctively solutions of stochastic integral equations with Lipschitz-type coefficients and differentials dM', dM. (The coefficients are also allowed to vary.) This is a stronger stability than usually holds for solutions of stochastic integral equations, reflecting the strength of the Y norm.

On the existence and uniqueness of solutions of stochastic integral equations of the Volterra type

On the existence and uniqueness of solutions of stochastic integral equations of the Volterra type

On a class of stochastic integral equations of mixed type

In this paper, the concept of a random contractor is used to investigate the existence and uniqueness of solutions of stochastic integral equations of mixed type in the general form of x ( t ; ω ) = h ( t ; ω ) + ∫ t 0 q ( t, s, x ( s ; ω ); ω ) ds + ∫ ∞ 0 r ( t, s, x ( s ; ω ); ω ) ds, t ⩾ 0, where ( ( Ω , %plane1D;49C; , P ) ) is a complete probability space. Both of the integrals are assumed to be Bochner integrals.

Recursive integral equations for the detection of counting processes

A recursive stochastic integral equation for the detection of counting processes is derived from a previously known formula [5] of the likelihood ratio. This is done quite simply by using a result due to Doleans-Dade [4] on the solution of stochastic integral equations.

On the existence and unicity of solutions of stochastic integral equations

Let M be a local martingale, A be an adapted process with finite variation on each finite interval and H be an adapted cadlag process (i.e. H is continuous on the right and has finite left limits). We shall prove that the equation $$X_t = H_t + \int\limits_0^t {f(s,X_{s - } )dM_s + } \int\limits_0^t {g(s,X_{s - } )dA_s }$$ (1) has one and only one solution, provided the random functions f and g satisfy the properties (L) given below, i.e. a Lipschitz condition $$|g(s,\omega ,x) - g(s,\omega ,y)| + |f(s,\omega ,x) - f(s,\omega ,y)|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } K|x - y|,$$ and two less stringent properties.

Absolute Continuity of Measures Corresponding to Diffusion Processes in Banach Space

This paper is concerned with the stochastic processes in Banach space arising from the solutions of stochastic integral equations. It is shown that under certain assumptions the corresponding measures of two such stochastic processes are equivalent. The Radon-Nikodym derivative is also given.

Local Behaviour of Solutions of Stochastic Integral Equations

Let X denote the solution process of the stochastic equation $dX(t) = a(X(t))dt + \sigma (X(t))dW(t)$. In this paper, conditions on $a( \cdot )$ and $\sigma ( \cdot )$ are given under which the sample paths of X are differentiate at $t = 0$ with probability one. Variations of these results are obtained leading to a new uniqueness criterion for solutions of stochastic equations. If $\sigma ( \cdot )$ is Hölder continuous with exponent greater than $\tfrac {1}{2}$ and $a( \cdot )$ satisfies a Lipschitz condition, it is shown that in the one-dimensional case the above equation has only one continuous solution.

Local behaviour of solutions of stochastic integral equations

Let X denote the solution process of the stochastic equation d X ( t ) = a ( X ( t ) ) d t + σ ( X ( t ) ) d W ( t ) dX(t) = a(X(t))dt + \sigma (X(t))dW(t) . In this paper, conditions on a ( ⋅ ) a( \cdot ) and σ ( ⋅ ) \sigma ( \cdot ) are given under which the sample paths of X are differentiate at t = 0 t = 0 with probability one. Variations of these results are obtained leading to a new uniqueness criterion for solutions of stochastic equations. If σ ( ⋅ ) \sigma ( \cdot ) is Hölder continuous with exponent greater than 1 2 \tfrac {1}{2} and a ( ⋅ ) a( \cdot ) satisfies a Lipschitz condition, it is shown that in the one-dimensional case the above equation has only one continuous solution.