Infinite-dimensional Integration Research Articles

Malliavin–Stein method: a survey of some recent developments

Initiated around the year 2007, the Malliavin–Stein approach to probabilistic approximations combines Stein’s method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin–Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin–Stein method, with specific emphasis on extensions and generalizations in the framework of Markov semigroups and of random point measures.

Numerical Approximation for Fractional Diffusion Equation Forced by a Tempered Fractional Gaussian Noise

This paper discusses the fractional diffusion equation forced by a tempered fractional Gaussian noise. The fractional diffusion equation governs the probability density function of the subordinated killed Brownian motion. The tempered fractional Gaussian noise plays the role of fluctuating external source with the property of localization. We first establish the regularity of the infinite dimensional stochastic integration of the tempered fractional Brownian motion and then build the regularity of the mild solution of the fractional stochastic diffusion equation. The spectral Galerkin method is used for space approximation; after that the system is transformed into an equivalent form having better regularity than the original one in time. Then we use the semi-implicit Euler scheme to discretize the time derivative. In terms of the temporal-spatial error splitting technique, we obtain the error estimates of the fully discrete scheme in the sense of mean-squared $L^2$-norm. Extensive numerical experiments confirm the theoretical estimates.

Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

Smolyak’s method, also known as sparse grid method, is a powerful tool to tackle multivariate tensor product problems solely with the help of efficient algorithms for the corresponding univariate problem.In this paper we study the randomized setting, i.e., we randomize Smolyak’s method. We provide upper and lower error bounds for randomized Smolyak algorithms with explicitly given dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worst case root mean square error (the typical error criterion for randomized algorithms, sometimes referred to as “randomized error”, ) and the root mean square worst case error (sometimes referred to as “worst-case error”).Randomized Smolyak algorithms can be used as building blocks for efficient methods such as multilevel algorithms, multivariate decomposition methods or dimension-wise quadrature methods to tackle successfully high-dimensional or even infinite-dimensional problems. As an example, we provide a very general and sharp result on the convergence rate of Nth minimal errors of infinite-dimensional integration on weighted reproducing kernel Hilbert spaces. Moreover, we are able to characterize the spaces for which randomized algorithms for infinite-dimensional integration are superior to deterministic ones. We illustrate our findings for the special instance of weighted Korobov spaces. We indicate how these results can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases (“spaces of increasing smoothness”) and to the problem of L2-approximation (function recovery).

Embeddings for infinite-dimensional integration and [formula omitted]-approximation with increasing smoothness

We study integration and L2-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness.

Integration by parts on the law of the modulus of the Brownian bridge

We prove an infinite dimensional integration by parts formula on the law of the modulus of the Brownian bridge $$BB=(BB_t)_{0 \le t \le 1}$$ from 0 to 0 in use of methods from white noise analysis and Dirichlet form theory. Additionally to the usual drift term, this formula contains a distribution which is constructed in the space of Hida distributions by means of a Wick product with Donsker’s delta (which correlates with the local time of |BB| at zero). This additional distribution corresponds to the reflection at zero caused by the modulus.

Infinite-dimensional integration and the multivariate decomposition method

We further develop the Multivariate Decomposition Method (MDM) for the Lebesgue integration of functions of infinitely many variables x 1 , x 2 , x 3 , … with respect to a corresponding product of a one dimensional probability measure. The method is designed for functions that admit a dominantly convergent decomposition f = ∑ u f u , where u runs over all finite subsets of positive integers, and for each u = { i 1 , … , i k } the function f u depends only on x i 1 , … , x i k . Although a number of concepts of infinite-dimensional integrals have been used in the literature, questions of uniqueness and compatibility have mostly not been studied. We show that, under appropriate convergence conditions, the Lebesgue integral equals the ‘anchored’ integral, independently of the anchor. For approximating the integral, the MDM assumes that point values of f u are available for important subsets u , at some known cost. In this paper, we introduce a new setting, in which it is assumed that each f u belongs to a normed space F u , and that bounds B u on ‖ f u ‖ F u are known. This contrasts with the assumption in many papers that weights γ u , appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights γ u were determined by minimizing an error bound depending on the B u , the γ u and the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper, only the bounds B u are assumed to be known. We give two examples in which we specialize the MDM: in the first case, F u is the | u | -fold tensor product of an anchored reproducing kernel Hilbert space; in the second case, it is a particular non-Hilbert space for integration over an unbounded domain.

Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration

We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration and approximation, and to their infinite-dimensional counterparts. In an application we consider weighted tensor product Sobolev spaces of mixed smoothness of any integer order, equipped with the classical, the anchored, or the ANOVA norm. Here we derive new results for multivariate and infinite-dimensional integration.

A unified approach to infinite-dimensional integration

An approach to infinite-dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite-dimensional construction of integrals as linear functionals, as much as possible independent of the underlying topological and measure theoretical structure. Various applications are given, including, next to Feynman path integrals, Schrödinger and diffusion equations, as well as higher order hyperbolic and parabolic equations.

Infinite dimensional oscillatory integrals as projective systems of functionals

The theory of infinite dimensional oscillatory integrals and some of its applications are discussed, with special attention to the relations with the original work of K. Ito in this area. A recent general approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented, together with some new developments.

Optimal Randomized Multilevel Algorithms for Infinite-Dimensional Integration on Function Spaces with ANOVA-Type Decomposition

In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first non-trivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or non-linear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. M\"uller-Gronbach, B. Niu, K. Ritter, J. Complexity 26 (2010), 229-254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.

Linearization of isometric embeddings between Banach spaces: An elementary approach

Mazur-Ulam’s classical theorem states that any isometric onto map between normed spaces is linear. This result has been generalized by T. Figiel [F] who showed that ifΦ is an isometric embedding from a Banach space X to a Banach space Y such that Φ(0) = 0 and vect [φ(X )] = Y , there exists a linear quotient map Q such that ‖Q‖ = 1 and Q ◦Φ= I dX . The third chapter of this short story is [GK] where it is shown that if a quotient map Q from a Banach space Y onto a separable Banach space X has a Lipschitz lifting, then it actually has a continuous linear lifting. Combining this statement with Figiel’s theorem provides another result from [GK] : if a separable Banach space X isometrically embeds into a Banach space Y , there exists an isometric linear embedding from X into Y . Nigel Kalton had an outstanding ability to set theorems and proofs in their proper frame. His articles are therefore fountains of ideas, irrigating each of the many fields to which he contributed : a non exhaustive survey on these contributions is [G]. The paper [GK] is no exception to this rule, and among other things it prepared the ground for far-reaching extensions, where e.g. the Lipschitz assumption is weakened to the Holder condition, leading to very different conclusions ( [K1], [K2], [K3] ). However, some readers could find daunting some arguments from [GK], whichwould probably turn down undergraduate students. Since the extension from [GK] of Mazur-Ulam’s theorem which is recalled above has a statement which is understandable to any student in mathematics, it seems appropriate to provide an elementary proof. This is the purpose of the present short note, where only basic functional analysis (elementary duality theory) and calculus (culminating at Fubini’s theorem for continuous functions on Rn) are used. Moreover this note is fully self-contained : even the generic smoothness of convex fuctions on Rn is shown, and a detailed proof of Figiel’s theorem is provided (following [BL], although Lemma 2 is not stated there). Themain result of the note is Theorem 5, whose proof follows the strategy of the proof of ( [GK], Corollary 3.2) but in such an elementary way that diagrams, free spaces and infinitedimensional integration or differentiation arguments are avoided. Thus our approach makes it clear that the core of the proof consists of finite-dimensional considerations. We can therefore teach at an undergraduate level Mazur-Ulam’s theorem and its natural extensions from [F] and [GK].

Infinite-dimensional integration on weighted Hilbert spaces

We study the numerical integration problem for functions with infinitely many variables which stem from a weighted reproducing kernel Hilbert space. We study the worst case ε \varepsilon -complexity which is defined as the minimal cost among all algorithms whose worst case error over the Hilbert space unit ball is at most ε \varepsilon . Here we assume that the cost of evaluating a function depends polynomially on the number of active variables. The infinite-dimensional integration problem is tractable if the ε \varepsilon -complexity is bounded by a constant times a power of 1 / ε 1/\varepsilon . The smallest such power is called the exponent of tractability. We provide improved lower bounds for the exponent of tractability for general finite-order weights and, with the help of multilevel algorithms, improved upper bounds for three newly defined classes of finite-order weights. The newly defined finite-intersection weights model the situation where each group of variables interacts with at most ρ \rho other groups of variables, ρ \rho some fixed number. For these weights we obtain sharp upper bounds for any decay of the weights and any polynomial degree of the cost function. For the other two classes of finite-order weights our upper bounds are sharp if, e.g., the decay of the weights is fast or slow enough. Furthermore, we deduce a lower bound for the exponent of tractability for arbitrary weights and a constructive upper bound for product weights.

A Random String with Reflection in a Convex Domain

We study the motion of a random string in a convex domain O in ℝ d , namely the solution of a vector-valued stochastic heat equation, confined in the closure of O and reflected at the boundary of O. We study the structure of the reflection measure by computing its Revuz measure in terms of an infinite-dimensional integration by parts formula. Our method exploits recent results on weak convergence of Markov processes with log-concave invariant measures.

Tractability of infinite-dimensional integration in the worst case and randomized settings

We consider approximation of weighted integrals of functions with infinitely many variables in the worst case deterministic and randomized settings. We assume that the integrands f belong to a weighted quasi-reproducing kernel Hilbert space, where the weights have product form and satisfy γ j = O ( j − β ) for β > 1 . The cost of computing f ( x ) depends on the number Act ( x ) of active coordinates in x and is equal to $ ( Act ( x ) ) , where $ is a given cost function. We prove, in particular, that if the corresponding univariate problem admits algorithms with errors O ( n − κ / 2 ) , where n is the number of function evaluations, then the ∞ -variate problem is polynomially tractable with the tractability exponent bounded from above by max ( 2 / κ , 2 / ( β − 1 ) ) for all cost functions satisfying $ ( d ) = O ( e k ⋅ d ) , for any k ≥ 0 . This bound is sharp in the worst case setting if β and κ are chosen as large as possible and $ ( d ) is at least linear in d . The problem is weakly tractable even for a larger class of cost functions including $ ( d ) = O ( e e k ⋅ d ) . Moreover, our proofs are constructive.

Deterministic multi-level algorithms for infinite-dimensional integration on [formula omitted

Pricing a path-dependent financial derivative, such as an Asian option, requires the computation of E ( g ( B ) ) , the expectation of a payoff function g , that depends on a Brownian motion B . Employing a standard series expansion of B the latter problem is equivalent to the computation of the expectation of a function of the corresponding i.i.d. sequence of random coefficients. This motivates the construction and the analysis of algorithms for numerical integration with respect to a product probability measure on the sequence space R N . The class of integrands studied in this paper is the unit ball in a reproducing kernel Hilbert space obtained by superposition of weighted tensor product spaces of functions of finitely many variables. Combining tractability results for high-dimensional integration with the multi-level technique we obtain new algorithms for infinite-dimensional integration. These deterministic multi-level algorithms use variable subspace sampling and they are superior to any deterministic algorithm based on fixed subspace sampling with respect to the respective worst case error.

Multi-level Monte Carlo algorithms for infinite-dimensional integration on [formula omitted

We study randomized algorithms for numerical integration with respect to a product probability measure on the sequence space R N . We consider integrands from reproducing kernel Hilbert spaces, whose kernels are superpositions of weighted tensor products. We combine tractability results for finite-dimensional integration with the multi-level technique to construct new algorithms for infinite-dimensional integration. These algorithms use variable subspace sampling, and we compare the power of variable and fixed subspace sampling by an analysis of minimal errors.

Cumulants on the Wiener space

We combine infinite-dimensional integration by parts procedures with a recursive relation on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field. These findings yield a compact formula for cumulants on a fixed Wiener chaos, virtually replacing the usual “graph/diagram computations” adopted in most of the probabilistic literature.

Conservative stochastic Cahn–Hilliard equation with reflection

We consider a stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space–time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution. We propose a method based on infinite dimensional integration by parts formulae, obtaining existence and uniqueness of a strong solution for all continuous nonnegative initial conditions and detailed information on the associated invariant measure and Dirichlet form.

Integration by parts on the Brownian Meander

We prove infinite-dimensional integration by parts formulae for the laws of the Brownian Meander, of the Bessel Bridge of dimension 3 between z , z ′ ≥ 0 z,z’\geq 0 and of the Brownian Motion on the set of all paths taking values greater than or equal to a nonpositive constant. We give applications to SPDEs with reflection.

Irving Segal's Work on Infinite Dimensional Integration Theory

Irving Segal's Work on Infinite Dimensional Integration Theory