Estimate for stochastic integrals

Lithuanian Mathematical Journal -     

Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Z_n, n?1, such that EK²(Z₁, Z₂)<∞, set A_n = ∑₁?i?j?nK(Z_i,Z_j), n?1. If the Z_n's are two-dimensional and K is the determinant function, A_n is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the A_n's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.     

Laplace approximation for stochastic line integrals

We prove a precision of large deviation principle for current-valued processes such as shown in Bolthausen et al. (Ann Probab 23(1):236–267, 1995) for mean empirical measures. The class of processes we consider is determined by the martingale part of stochastic line integrals of 1-forms on a compact Riemannian manifold. For the pair of the current-valued process and mean empirical measures, we give an asymptotic evaluation of a nonlinear Laplace transform under a nondegeneracy assumption on the Hessian of the exponent at equilibrium states. As a direct consequence, our result implies the Laplace approximation for stochastic line integrals or periodic diffusions. In particular, we recover a result in Bolthausen et al. (Ann Probab 23(1):236–267, 1995) in our framework.     

Approximation theorems for set-valued stochastic integrals

The article is devoted to new properties of Aumann, Lebesgue, and Itô set-valued stochastic integrals considered in papers [1 Kisielewicz, M. (2014). Properties of generalized set-valued stochastic integrals. Discuss. Math. (DICO) 34:131–147. [Google Scholar],2 Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449:1893–1910.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, it contains some approximation theorems for Aumann and Itô set-valued stochastic integrals. Hence, in particular, it follows that Aumann and Lebesgue set-valued stochastic integrals cover a.s., both for measurable and IF-nonanticipative integrably bounded set-valued stochastic processes.     

Sharp maximal inequality for stochastic integrals

Let be a nonnegative supermartingale and be a predictable process with values in . Let denote the stochastic integral of with respect to . The paper contains the proof of the sharp inequality

where . A discrete-time version of this inequality is also established.

     

Differentiation formulas for stochastic integrals in the plane

For a one-parameter process of the form X_t=X₀+∫^t₀φ_sdW_s+∫^t₀ψ_sds, where W is a Wiener process and ∫φdW is a stochastic integral, a twice continuously differentiable function f(X_t) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a two-parameter Wiener process.Let {W₂, z∈R²₊} be a Wiener process with a two-dimensional parameter. Ertwhile, we have defined stochastic integrals ∫ φdWand ∫ψdWdW, as well as mixed integrals ∫h dz dW and ∫gdW dz. Now let X_z be a two-parameter process defined by the sum of these four integrals and an ordinary Lebesgue integral. The objective of this paper is to represent a suitably differentiable function f(X_z) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(X_z) on increasing paths in $\text{R}{}^2{}_{\mathrm{+}}$.     

Frechet-valued martingales and stochastic integrals

Stochastic processes with values in a separable Frechet space whose a itinuous linear functional are real-valued square integrable martingales are investigated. The coordinate measures on the Fréchet space are obtained from cylinder set measures on a Hilbert space that is dense in the Fréchet space. Real-valued stochastic integrals are defined from the Fréchet-valued martingales using integrands from the topological dual of the aforementioned Hilbert space. An increasing process with values in the self adjoint operators on the Hilbert space plays a fundamental role in the definition of stochastic integrals. For Banach-valued Brownian motion the change of variables formula of K. Itô is generalized. A converse to the construction of the measures on the Fréchet space from cylinder set measures on a Hilbert space is also obtained.     

Generalized stochastic integrals and the malliavin calculus

Summary The paper first reviews the Skorohod generalized stochastic integral with respect to the Wiener process over some general parameter space T and it's relation to the Malliavin calculus as the adjoint of the Malliavin derivative. Some new results are derived and it is shown that every sufficiently smooth process {u_t, tT} can be decomposed into the sum of a Malliavin derivative of a Wiener functional, and a process whose generalized integral over T vanishes. Using the results on the generalized integral, the Bismut approach to the Malliavin calculus is generalized by allowing non adapted variations of the Wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions.Let e _i be a non-random complete orthonormal system on T, the Ogawa integral u W is defined as _i (e _i u) e _i dW where the integrals are Wiener integrals. Conditions are given for the existence of an intrinsic Ogawa integral i.e. independent of the choice of the orthonormal system and results on it's relation to the Skorohod integral are derived.The transformation of measures induced by (W + u d u non adapted is discussed and a Girsanov-type theorem under certain regularity conditions is derived.The work of M.Z. was supported by the Fund for Promotion of Research at the Technion     

Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations

Some inequalities concerning the Itô stochastic integral and solutions of stochastic different equations are obtained.     

Convergence of Riemann sums for stochastic integrals

Summary Let b be a Brownian motion and f a function in L ²[0,1]. If is a partition of [0,1], denote by f the step function obtained by replacing f by its mean values in each subinterval. As becomes fine, the martingale f db converges to fdb in L ² but not necessarily almost surely. We determine precisely which Lipschitz conditions on f imply a.s. convergence. A similar thing is done for non-anticipating random functions.