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In this work we present an easy and direct construction of a Stochastic integral respect a cylindrical martingale-Lévy process. To do this, we apply the radonification technique. The radonification technique has been very useful to define an genuine stochastic process starting from a cylindrical process combine with Hilbert–Schmidt operators. With this work we present a self-contained construction, easy to follow to understand this new integration theories that are getting strong nowadays. [ABSTRACT FROM AUTHOR]

The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter H > 1∕2. The integral is defined initially on the processes that are" piecewise" predictable on a short horizon. Then the integral is extended on a wide class of square integrable adapted random processes. This class is described via a mild restriction on the growth rate of the conditional mean square error for the forecast on an arbitrarily short horizon given current observations. On the other hand, a pathwise regularity, such as Hölder condition, etc., is not required for the integrand. The suggested integration can be interpreted as foresighted integration for integrands featuring certain restrictions on the forecasting error. This integration is based on Itô's integration and does not involve Malliavin calculus or Wick products. In addition, it is shown that these stochastic integrals depend right continuously on H at H = 1/2. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the propagation of the weak representation property (WRP) to an independently enlarged filtration. More precisely, we consider an F -semimartingale X possessing the WRP with respect to F and an H -semimartingale Y possessing the WRP with respect to H . Assuming that F and H are independent, we show that the G -semimartingale Z = (X , Y) has the WRP with respect to G , where G : = F ∨ H . In our setting, X and Y may have simultaneous jump-times. Furthermore, their jumps may charge the same predictable times. This generalizes all available results about the propagation of the WRP to independently enlarged filtrations. [ABSTRACT FROM AUTHOR]

In this research, we introduce the stochastic integration with respect to a cylindrical special semi-martingale, which is a specific case of general integration, with specific properties of special semi-martingales. [ABSTRACT FROM AUTHOR]

A standing assumption in the literature on proportional transaction costs is efficient friction. Together with robust no free lunch with vanishing risk, it rules out strategies of infinite variation as they usually appear in frictionless markets. In this paper, we show how the models with and without transaction costs can be unified. The bid and ask prices of a risky asset are given by càdlàg processes which are locally bounded from below and may coincide at some points. In a first step, we show that if the bid–ask model satisfies "no unbounded profit with bounded risk" for simple strategies, then there exists a semimartingale lying between the bid and ask price processes. In a second step, under the additional assumption that the zeros of the bid–ask spread are either starting points of an excursion away from zero or inner points from the right, we show that for every bounded predictable strategy specifying the amount of risky assets, the semimartingale can be used to construct the corresponding self-financing risk-free position in a consistent way. Finally, the set of most general strategies is introduced, which also provides a new view on the frictionless case. [ABSTRACT FROM AUTHOR]

We provide a detailed analysis of the Gelfand integral on Fréchet spaces, showing among other things a Vitali theorem, dominated convergence and a Fubini result. Furthermore, the Gelfand integral commutes with linear operators. The Skorohod integral is conveniently expressed in terms of a Gelfand integral on Hida distribution space, which forms our prime motivation and example. We extend several results of Skorohod integrals to a general class of pathwise Gelfand integrals. For example, we provide generalizations of the Hida–Malliavin derivative and extend the integration-by-parts formula in Malliavin Calculus. A Fubini-result is also shown, based on the commutative property of Gelfand integrals with linear operators. Finally, our studies give the motivation for two existing definitions of stochastic Volterra integration in Hida space. [ABSTRACT FROM AUTHOR]

Precise and detailed knowledge of the internal structure of hadrons is one of the most actual problems in elementary particle physics. In view of the planned high energy physics facilities, in particular the Electron–Ion Collider constructed in Brookhaven National Laboratory (National Academies of Sciences, Engineering and Medicine, 2018, [1]), the Chinese Electron–Ion Collider of China (Chen, 2018 [2]), or upgraded versions of CERN’s LHC experiments, it is important to prepare adequate theoretical tools to compare and correctly interpret experimental results. One of the model frameworks allowing to estimate hadron structure functions is the combination of the McLerran–Venugopalan initial condition model together with the JIMWLK equation which describes the evolution in rapidity of the initial distribution. In this package we present a parallel C++ implementation of both these ingredients. In order to allow a thorough assessment of systematic effects several discretizations of the JIMWLK kernel are implemented both in position and momentum spaces. The effects of the running coupling in three different definitions are provided. The main code is supplemented with test and check programs for all main functionalities. The clear structure of the code allows easy implementation of further improvements such as the collinear constraint (Hatta and Iancu, 2016).

We consider decoupling inequalities for random variables taking values in a Banach space X. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar-type expansion in which only the pre-specified conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not affect the decoupling properties (in particular, it does not affect the constants involved). As a special case, we deal with one-sided moment inequalities for decoupled dyadic (i.e., Paley–Walsh) martingales and show that Burkholder–Davis–Gundy-type inequalities for stochastic integrals of X-valued processes can be obtained from decoupling inequalities for X-valued dyadic martingales. [ABSTRACT FROM AUTHOR]

This paper initiates the study of stochastic differential equations in the context of variable Lebesgue spaces. More precisely, we establish the existence and uniqueness of solutions of stochastic Burgers equations in variable Lebesgue spaces. To that end, we are forced to firstly develop a stochastic integration theory with respect to Brownian motions in this framework. The main ingredients of the stochastic integrability are a conditional expectation version of the Lenglart-Lépingle-Pratelli's inequality and Burkholder-Davis-Gundy's inequalities for continuous martingales in variable Lebesgue spaces. As by-products, we obtain similar results for stochastic differential equations of Itô type and stochastic wave equations. [ABSTRACT FROM AUTHOR]

We prove a general version of the stochastic Fubini theorem for stochastic integrals of Banach space valued processes with respect to compensated Poisson random measures under weak integrability assumptions, which extends this classical result from Hilbert space setting to Banach space setting. [ABSTRACT FROM AUTHOR]

In this work we introduce a theory of stochastic integration for operator-valued integrands with respect to some classes of cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed via the radonification of cylindrical martingales by a Hilbert-Schmidt operator theorem and unifies several other theories of stochastic integration in Hilbert spaces. In particular, our theory covers the theory of stochastic integration with respect to a Hilbert space valued Lévy process with second moments, with respect to a cylindrical Lévy processes with (weak) second moments and with respect to a Lévy-valued random martingale measures with finite second moment. As an application of our theory of integration we prove existence and uniqueness of solutions for stochastic stochastic partial differential equations driven by multiplicative cylindrical martingale-valued measure noise with rather general coefficients. Existence and uniqueness of solutions in the presence of multiplicative Lévy noise (with no moments assumptions) is also proved. [ABSTRACT FROM AUTHOR]

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We study the complexity of stochastic integration with respect to an isonormal process defined on a bounded Lipschitz domain Q ⊂ R d. We consider integration of functions from Sobolev spaces W p r (Q) and analyze the complexity in the deterministic and randomized setting. Matching upper and lower bounds for the n -th minimal error are established, this way determining the complexity of the problem. It turns out that the stochastic integration problem is closely related to approximation of the embedding of W p r (Q) into L 2 (Q). [ABSTRACT FROM AUTHOR]

Abstract Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of càdlàg functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation of "typical price paths" in the space of non-negative càdlàg paths and implies the existence of quadratic variation in the sense of Föllmer quasi surely under all martingale measures. Based on the robust existence of the quadratic variation, a model-free Itô integration is developed. [ABSTRACT FROM AUTHOR]

The paper concerns itself with establishing large deviation principles for a sequence of stochastic integrals and stochastic differential equations driven by general semimartingales in infinite-dimensional settings. The class of semimartingales considered is broad enough to cover Banach space-valued semimartingales and the martingale random measures. Simple usable expressions for the associated rate functions are given in this abstract setup. As illustrated through several concrete examples, the results presented here provide a new systematic approach to the study of large deviation principles for a sequence of Markov processes. [ABSTRACT FROM AUTHOR]

It is often of interest to model the incidence and duration of threshold exceedance events for an environmental variable over a set of monitoring locations. Such data arrive over continuous time and can be considered as observations of a two-state process yielding, sequentially, a length of time in the below threshold state followed by a length of time in the above threshold state, then returning to the below threshold state, etc. We have a two-state continuous time Markov process, often referred to as an alternating renewal process. The process is observed over a truncated time window and, within this window, duration in each state is modeled using a distinct cumulative intensity specification. Initially, we model each intensity over the window using a parametric regression specification. We extend the regression specification adding temporal random effects to enrich the model using a realization of a log Gaussian process over time. With only one type of renewal, this specification is referred to as a Gaussian process modulated renewal process. Here, we introduce Gaussian process modulation to the intensity for each state. Model fitting is done within a Bayesian framework. We clarify that fitting with a customary log Gaussian process specification over a lengthy time window is computationally infeasible. The nearest neighbor Gaussian process, which supplies sparse covariance structure, is adopted to enable tractable computation. We propose methods for both generating data under our models and for conducting model comparison. The model is applied to hourly ozone data for four monitoring sites at different locations across the United States for the ozone season of 2014. For each site, we obtain estimated profiles of up-crossing and down-crossing intensity functions through time. In addition, we obtain inference regarding the number of exceedances, the distribution of the duration of exceedance events, and the proportion of time in the above and below threshold state for any time interval. [ABSTRACT FROM AUTHOR]

Bayesian filtering solutions that are developed under the assumption of heavy-tailed uncertainties are more robust to outliers than the standard Gaussian ones. In this work, we consider robust nonlinear Bayesian filtering in the presence of multivariate t -distributed process and measurement noises. We develop a robust stochastic integration filter (RSIF) based on stochastic spherical-radial integration rule that achieves asymptotically exact evaluations of multivariate t -weighted integrals of nonlinear functions that arise in nonlinear Bayesian filtering framework. The superiority of the proposed scheme is demonstrated by comparing its performance against the cubature Kalman filter (CKF), a robust CKF, and the standard SIF in a representative example concerning bearings-only target tracking. [ABSTRACT FROM AUTHOR]

In light of a recent paper, we revisit the Compton scale region and point out that in this region there is stochastic behavior which can be modeled by Ito's method. All these lead to a QCD and quark-like behavior completely consistent with known empirical facts. [ABSTRACT FROM AUTHOR]