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The questions about finding (conditional) mathematical expectations, the joint and marginal distribution of different functionals from the trajectories of random processes, expressed through the process itself, the ordinary stochastic integral and the stochastic Ito integral (stochastic integrals are understood as integrals in the mean square sense) are among the important issues of both the theory itself random processes and its numerous applications. But it is not always possible to find (joint) distributions of the indicated functionals by direct computations, therefore, they usually resort to some methods of finding the required characteristics. One of such methods is the so-called method of differential equations, which reduces the problem of finding joint distributions of functionals from random processes to solving (connected with these functionals) partial differential equations. The aim of the present paper is to find a joint distribution of the above-mentioned types of functionals and the integrands present in the definitions of these functionals depend both on the time and on the spatial coordinates. To do this, we first derive an equation for the joint characteristic function of the functionals under consideration and show that in some special cases the determination of the Laplace transformation of the solution of this equation can be reduced to the solution of an ordinary differential equation with constant coefficients. As an application, explicit distributions of some functionals of the Wiener process are found and some of their possible applications are discussed.

We suggest a new approach to stochastic integration in infinite-dimensional spaces that is based on representing random variables on Banach spaces as real-valued processes on an interval. We prove stochastic integrability of operator-valued processes on general separable Banach spaces under the conditions that do not depend on the norm of the space and show how our methods can be applied to studying infinite-dimensional stochastic differential equations. In particular, our results provide a natural construction of the stochastic integral in abstract Wiener spaces. [ABSTRACT FROM AUTHOR]

In this paper we study the Assouad dimension of graphs of certain Lévy processes and functions defined by stochastic integrals. We do this by introducing a convenient condition which guarantees a graph to have full Assouad dimension and then show that graphs of our studied processes satisfy this condition. [ABSTRACT FROM AUTHOR]

We consider processes of the form μ(t) = μ((0, t]), where μ is a σ-additive in probability stochastic set function. Convergence of a random Fourier series to μ(t) is proved, and the approximation of integrals with respect to μ using Fejer sums is obtained. For this approximation, we prove the convergence of solutions of the heat equation driven by μ. [ABSTRACT FROM AUTHOR]

In this article, we provide predictable and chaotic representations for Itô-Markov additive processes X. Such a process is governed by a finite-state continuous time Markov chain J which allows one to modify the parameters of the Itô-jump process (in so-called regime switching manner). In addition, the transition of J triggers the jump of X distributed depending on the states of J just prior to the transition. This family of processes includes Markov modulated Itô-Lévy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stochastic integrals with respect to some explicitly constructed orthogonal martingales. We identify the predictable representation of a square-integrable martingale as a sum of stochastic integrals of predictable processes with respect to Brownian motion and power-jumps martingales related to all the jumps appearing in the model. This result generalizes the seminal result of Jacod-Yor and is of importance in financial mathematics. The derived representation then allows one to enlarge the incomplete market by a series of power-jump assets and to price all market-derivatives. [ABSTRACT FROM AUTHOR]

Random contractions and Bernoulli selections are recognized as strong analytical tools of probability distributions theory. The paper investigates the distribution of a Bernoulli selection incorporating a stochastic integral and a random contraction. Moreover, the paper establishes a practical interpretation of the formulated Bernoulli selection. [ABSTRACT FROM AUTHOR]

In this paper, we first study the martingale problem in a sublinear expectation space. The critical tool is the Evans–Krylov theorem on regularity properties for solutions of fully nonlinear PDEs. Based on the analysis for the martingale problem and inspired by the rough path theory, we then develop stochastic calculus with respect to a general stochastic process, and derive an Itô type formula and the integration-by-parts formula. Our framework is analytic in that it does not rely on the probabilistic concept of “independence” as in the G -expectation theory. [ABSTRACT FROM AUTHOR]

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a d -dimensional fractional Brownian motion (fBm) B t with Hurst parameter H > 1 2 , where the integrands are vector fields applied to B t . It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion. [ABSTRACT FROM AUTHOR]

In Karandikar-Rao [11], the quadratic variation [ M, M] of a (local) martingale was obtained directly using only Doob's maximal inequality and it was remarked that the stochastic integral can be defined using [ M, M], avoiding using the predictable quadratic variation 〈 M, M〉 (of a locally square integrable martingale) as is usually done. This is accomplished here- starting with the result proved in [11], we construct ∫ f dX where X is a semimartingale and f is predictable and prove dominated convergence theorem (DCT) for the stochastic integral. Indeed, we characterize the class of integrands f for this integral as the class L( X) of predictable processes f such that | f| serves as the dominating function in the DCT for the stochastic integral. This observation seems to be new. We then discuss the vector stochastic integral ∫ 〈 f, dY〉 where f is ℝ valued predictable process, Y is ℝ valued semimartingale. This was defined by Jacod [6] starting from vector valued simple functions. Memin [13] proved that for (local) martingales M, ... M: If N are martingales such that N → N for every t and if ∃ f such that N = ∫ 〈 f, dM〉, then ∃ f such that N = ∫ 〈 f, dM〉. Taking a cue from our characterization of L( X), we define the vector integral in terms of the scalar integral and then give a direct proof of the result due to Memin stated above. This completeness result is an important step in the proof of the Jacod-Yor [4] result on martingale representation property and uniqueness of equivalent martingale measure. This result is also known as the second fundamental theorem of asset pricing. [ABSTRACT FROM AUTHOR]