The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type

We consider the integral equation driven by a standard Brownian motion and fractional Brownian motion (fBm). Since fBm is not a semimartingale, we cannot use the semimartingale theory to define an integral with respect to the fBm. Furthermore, a well-developed theory of stochastic differential equations is not applicable to solve it. Existence and uniqueness conditions are obtained for a solution in the space of continuous functions with q-bounded variation, q>2.     

On the convergence to the multiple Wiener-Itô integral

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in C₀([0,T]). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function f∈L²(n[0,T]). We prove also the weak convergence in the space C₀([0,T]) to the second-order integral for two important families of processes that converge to a standard Brownian motion.     

Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1. For 0 < β < ? we show that a function Φ: (0, T) → ℒ(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H-cylindrical fractional Brownian motion.     

Wiener integrals, Malliavin calculus and covariance measure structure

We introduce the notion of covariance measure structure for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratic variation processes with stationary increments and the bifractional Brownian motion.     

A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4

We prove a change of variable formula for the 2D fractional Brownian motion of index H bigger or equal to 1/4. For H strictly bigger than 1/4, our formula coincides with that obtained by using the rough paths theory. For H=1/4 (the more interesting case), there is an additional term that is a classical Wiener integral against an independent standard Brownian motion.     

Approximation of the finite dimensional distributions of multiple fractional integrals

We construct a family Inεt(f) of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener-Itô integral with respect to the fractional Brownian motion. We assume that and we prove our approximation result for the integrands f in a rather general class.     

The weak McShane integral

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ-finite outer regular quasi Radon measure space (S,Σ, T, µ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if it is Pettis and weakly McShane integrable on S. On the other hand, we prove that if an X-valued function is weakly McShane integrable on S, then it is Pettis integrable on each member of an increasing sequence (S _l ) _l?1 of measurable sets of finite measure with union S. For weakly sequentially complete spaces or for spaces that do not contain a copy of c ₀, a weakly McShane integrable function on S is always Pettis integrable. A class of functions that are weakly McShane integrable on S but not Pettis integrable is included.     

CONVERGENCE RATE OF AN APPROXIMATION TO MULTIPLE INTEGRAL OF FBM

In this article, we study the rate of convergence of the polygonal approximation to multiple stochastic integral Sp （f） of fractional Brownian motion of Hurst parameter H 〈 1/2 when the fractional Brownian motion is replaced by its polygonal approximation. Under different conditions on f and for different p, we obtain different rates.     

Integration with respect to fractal functions and stochastic calculus. I

The classical Lebesgue–Stieltjes integral ∫ ^b _a fdg of real or complex-valued functions on a finite interval (a,b) is extended to a large class of integrands f and integrators g of unbounded variation. The key is to use composition formulas and integration-by-part rules for fractional integrals and Weyl derivatives. In the special case of H?lder continuous functions f and g of summed order greater than 1 convergence of the corresponding Riemann–Stieltjes sums is proved. The results are applied to stochastic integrals where g is replaced by the Wiener process and f by adapted as well as anticipating random functions. In the anticipating case we work within Slobodeckij spaces and introduce a stochastic integral for which the classical It? formula remains valid. Moreover, this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional Brownian motion. Received: 14 January 1998 / Revised version: 9 April 1998     

Absolutely summing operators and integration of vector-valued functions

Let (Ω,Σ,μ) be a complete probability space and an absolutely summing operator between Banach spaces. We prove that for each Dunford integrable (i.e., scalarly integrable) function the composition u○f is scalarly equivalent to a Bochner integrable function. Such a composition is shown to be Bochner integrable in several cases, for instance, when f is properly measurable, Birkhoff integrable or McShane integrable, as well as when X is a subspace of an Asplund generated space or a subspace of a weakly Lindelöf space of the form C(K). We also study the continuity of the composition operator f?u○f. Some other applications are given.     

Normal convergence using Malliavin calculus with applications and examples

We prove the chain rule in the more general framework of the Wiener–Poisson space, allowing us to obtain the so-called Nourdin–Peccati bound. From this bound, we obtain a second-order Poincaré-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener–Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein–Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to infinitely many “small” jumps (particularly fractional Lévy processes), and the product of two Ornstein–Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). We also obtain bounds for their rate of convergence to normality.     

Asymptotic Behavior of Weighted Power Variations of Fractional Brownian Motion in Brownian Time

We study the asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time \(Z_t:= X_{Y_t},t \geqslant 0\), where X is a fractional Brownian motion and Y is an independent Brownian motion.     

Birkhoff integral for multi-valued functions

The aim of this paper is to study Birkhoff integrability for multi-valued maps , where (Ω,Σ,μ) is a complete finite measure space, X is a Banach space and cwk(X) is the family of all non-empty convex weakly compact subsets of X. It is shown that the Birkhoff integral of F can be computed as the limit for the Hausdorff distance in cwk(X) of a net of Riemann sums ∑_nμ(A_n)F(t_n). We link Birkhoff integrability with Debreu integrability, a notion introduced to replace sums associated to correspondences when studying certain models in Mathematical Economics. We show that each Debreu integrable multi-valued function is Birkhoff integrable and that each Birkhoff integrable multi-valued function is Pettis integrable. The three previous notions coincide for finite dimensional Banach spaces and they are different even for bounded multi-valued functions when X is infinite dimensional and X∗ is assumed to be separable. We show that when F takes values in the family of all non-empty convex norm compact sets of a separable Banach space X, then F is Pettis integrable if, and only if, F is Birkhoff integrable; in particular, these Pettis integrable F's can be seen as single-valued Pettis integrable functions with values in some other adequate Banach space. Incidentally, to handle some of the constructions needed we prove that if X is an Asplund Banach space, then cwk(X) is separable for the Hausdorff distance if, and only if, X is finite dimensional.     

Integral Representation with Adapted Continuous Integrand with Respect to Fractional Brownian Motion

We show that if a random variable is a final value of an adapted Hölder continuous process, then it can be represented as a stochastic integral with respect to fractional Brownian motion, and the integrand is an adapted process, continuous up to the final point.     

The moment problem on the Wiener space

Consider an L¹-continuous functional ? on the vector space of polynomials of Brownian motion at given times, suppose ? commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, , mapping the Wiener space to R.In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which ? can be written in the form ∫⋅dμ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables are nonnegative.     

Central limit theorem for an iterated integral with respect to fBm with H>1/2

We construct an iterated stochastic integral with respect to fractional Brownian motion (fBm) with H>1/2. The first integrand is a deterministic function, and each successive integral is with respect to an independent fBm. We show that this symmetric stochastic integral is equal to the Malliavin divergence integral. By a version of the Fourth Moment Theorem of Nualart and Peccati [10], we show that a family of such integrals converges in distribution to a scaled Brownian motion. An application is an approximation to the windings for a planar fBm, previously studied by Baudoin and Nualart [2].     

On Double Stratonovich Fractional Integrals and Some Strong and Weak Approximations

Abstract

Double Stratonovich integrals with respect to the odd part and even part of the fractional Brownian motion are constructed. The first and the second moments of such integrals are explicitly identified. As application of double Stratonovich integrals a strong law of large numbers for efBm and ofBm is derived.

Riemann–Stieltjes integral approximations to double Stratonovich fractional integrals are also considered. The strong convergence (almost surely and mean square) is obtained for approximations based on explicit series expansions of the fractional Brownian processes. The weak convergence is derived for approximations by processes with absolutely continuous paths which converge weakly to the considered fractional Brownian processes. The above-mentioned convergences are obtained for deterministic integrands which are given by bimeasures.     

Continuity in Law with Respect to the Hurst Parameter of the Local Time of the Fractional Brownian Motion

We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter H converges weakly to that of the local time of , when H tends to H ₀.      

A Set-indexed Fractional Brownian Motion

We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no “really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.      

Regularity of Skorohod integral processes based on integrands in a finite Wiener chaos

Summary Letf be a square integrable kernel on them-dimensional unit cube,U the Skorohod integral process in them ^th Wiener chaos associated with it. Isoperimetric inequalities for functions on Wiener space yield the exponential integrability of the increments ofU. To this result we apply the majorizing measure technique to show thatU possesses a continuous version and give an upper bound of its modulus of continuity.