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In this paper, a new combination method is being proposed for the numerical solution of nonlinear Ito–Volterra stochastic integral equations based on Adomian decomposition method (ADM), Triangular function (TF) approximation, quadrature methods and Ito stochastic integration formula. The presented method is developed in two steps. First, we apply ADM to the main equation then for calculating the components, we used TF-approximation together with quadrature and Ito stochastic integration formula. Some theorems related to error and convergence analysis of the suggested method are also stated. Finally, several examples confirm the applicability, efficiency and accuracy of the method, along with comparisons.

In this paper, we use stochastic integration in the framework of signed measures, together with the technique of time changes. Let Q be a bounded non-null signed measure on $$(\varOmega ,{\mathcal {F}}, {{P}}),$$ such that $$\left| Q\right| $$ and P are equivalent. In the first part of the paper, we generalize the results of stochastic calculus in Beghdadi-Sakrani (Seminaire de probabilites XXXVI, Springer, 2003) to Q-local martingales and we give some examples. In the second part, we prove that the class of Q-semimartingales is invariant under time changes. We establish the famous formulas of time-changed local martingales as well as the representation of a Q-local martingale as a time-changed Brownian motion.

A new integral with respect to an integer-valued random measure is introduced. In contrast to the finite variation integral ubiquitous in semimartingale theory (Jacod and Shiryaev, 2003, II.1.5), the new integral is closed under stochastic integration, composition, and smooth transformations. The new integral gives rise to a previously unstudied class of pure-jump processes: the sigma-locally finite variation pure-jump processes. As an application, it is shown that every semimartingale $X$ has a unique decomposition $$X = X_0 + X^{\mathrm{qc}}+X^{\mathrm{dp}},$$ where $X^{\mathrm{qc}}$ is quasi-left-continuous and $X^{\mathrm{dp}}$ is a sigma-locally finite variation pure-jump process that jumps only at predictable times, both starting at zero. The decomposition mirrors the classical result for local martingales (Yoeurp, 1976, Theoreme~1.4) and gives a rigorous meaning to the notions of continuous-time and discrete-time components of a semimartingale. Against this backdrop, the paper investigates a wider class of processes that are equal to the sum of their jumps in the semimartingale topology and constructs a taxonomic hierarchy of pure-jump semimartingales.

We study the least squares estimator for the drift parameter of the Langevin stochastic equation driven by the Rosenblatt process. Using the techniques of the Malliavin calculus and the stochastic integration with respect to the Rosenblatt process, we analyze the consistency and the asymptotic distribution of this estimator. We also introduce alternative estimators, which can be simulated, and we study their asymptotic properties.

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.

ESIAS is an atmospheric modeling system including the ensemble version of the Weather Forecasting and Research Model (WRF V3.7.1) and the ensemble version of the EURopean Air pollution Dispersion-Inverse Model (EURAD-IM), the latter uses the output of the WRF model to calculate, amongst others, the transportation of aerosols. To capture extreme weather events causing the uncertainty in the solar radiation and wind speed for the renewable energy industry, we employ ESIAS by using stochastic schemes, such as Stochastically Perturbed Parameterization Tendency (SPPT) and Stochastic Kinetic Energy Backscatter (SKEBS) schemes, to generate the random fields for ensembles of up to 4096 members. Our first goal is to produce 48 hourly weather predictions for the European domain with a 20 KM horizontal resolution to capture extreme weather events affecting wind, solar radiation, and cloud cover forecasts. We use the ensemble capability of ESIAS to optimize the physics configuration of WRF to have a more precise weather prediction. A total of 672 ensemble members are generated to study the effect of different microphysical schemes, cumulus schemes, and planetary boundary layer parameterization schemes. We examine our simulation outputs with 288 simulation hours in 2015 using model input from the Global Ensemble Forecast System (GEFS). Our results are validated by the cloud cover data from EUMETSAT CMSAF. Besides the precision of weather forecasting, we also determine the greatest spread by generating total 768 ensemble members: 16 stochastic members for each different configurations of physical parameterizations (48 combinations). The optimization of WRF will help for improving the air quality prediction by EURAD-IM, which will be demonstrated on a test case basis. Our results show that for the performed analysis the Community Atmosphere Model (CAM) 5.1, WRF Single-Moment 6-class scheme (WSM6), and the Goddard microphysics outstand the other 11 microphysics parameterizations, where the highest daily average matching rate is 64.2%. The Mellor–Yamada Nakanishi Niino (MYNN) 2 and MYNN3 schemes give better results compared to the other 8 planetary boundary layer schemes, and Grell 3D (Grell-3) works generally well with the above mentioned physical schemes. Overall, the combination of Goddard and MYNN3 produces the greatest spread comparing to the lowest spread (Morrison 2-moment & GFS) by 40%.

Diese Arbeit beschäftigt sich mit der Theorie stochastischer Integration und versucht einige Resultate über übliche Annahmen hinaus zu verallgemeinern.Ein entscheidender Schritt der Definition eines stochastischen Integrals ist es, dieses für Martingale zu definieren. Folgt man dem funktional-analytischen Zugang von Philip E. Protter, dann ist eine Ungleichung von Burkholder die entscheidende Zutat. Diese Arbeit verallgemeinert diese Ungleichung für stochastische Prozesse mit Werten in bestimmten Banachräumen, speziell wenn der Integrand Werte in solchen Banachräumen annimmt und der Integrator reellwertig sind. Für einen solchen Banachraum können wir ein stochastisches Integral für alle cáglád Prozesse mit Werten in diesem Banachraum definieren, wenn als Integrator ein reellwertiges Semimartingal verwendet wird. Ein weiterer großer Teil dieser Arbeit fokussiert sich auf das Bichteler-Dellacherie Theorem, das eine Charakterisierung jener stochastischen Prozesse im reellen Fall liefert, die als Integrator verwendet werden können. Es besagt, dass ein stochastisches Integral genau für cádlág Semimartingale definiert werden kann und, dass sich diese als Summe eines lokalen Martingals und eines Prozesses mit endlicher Variation schreiben lassen. Dieses Resultat verallgemeinern wir, indem wir die cádlág Annahme der Pfade fallen lassen, die analogen Voraussetzungen verwenden und den Prozess auf dem Level von Versionen noch immer als Summe eines lokalen Martingals und eines Prozesses mit endlicher Variation schreiben können. Genauer finden wir ein lokales Martingal mit cádlág Pfaden und einen Prozess von endlicher Variation, sodass für jeden Zeitpunkt der ursprüngliche Prozess fast sicher als Summe der beiden Prozesse geschrieben werden kann.Die gleichen Ideen können auf die Doob-Meyer Zerlegung angewandt werden, wodurch wir diese ebenso auf Supermartingale ohne stetige Pfade erweitern können., This thesis deals with the theory of stochastic integration and tries to generalize some results beyond standard assumptions.One crucial part of defining a stochastic integral is the step to define it for martingales. If one follows a functional analytic approach introduced by Philip E. Protter, an inequality due to Burkholder is the necessary ingredient. This thesis generalizes this inequality to a setting in which stochastic processes with values in certain Banach spaces are considered, in particular when the integrand is Banach space-valued and the integrator real-valued. For such a Banach space we can define a stochastic integral of all cáglád processes with values in that Banach space against a real-valued semimartingale.Another major part of this thesis focuses on the Bichteler-Dellacherie theorem which is the characterization of stochastic processes which can be taken as integrator in the real-valued case. It tells that a stochastic integral can be defined precisely for cádlág semimartingales and that those decompose into a local martingale and a finite variation process. We can extend this result by dropping the cádlág assumption on the paths, using the analogue assumptions of the theorem and still decomposing the process on a level of versions. In particular we can find a cádlág local martingale and a finite variation process such that for each time point the original process is the sum of those two, almost surely.The same ideas can be applied to the Doob-Meyer decomposition and we can again generalize this to supermartingales without continuity assumptions on its paths.

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. UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigaciones en Matemáticas Puras y Aplicadas (CIMPA)

Predictable covariation, L2-martingale and semi-martingale integrals, quadratic variation and covariation, substitution rule, Doleans exponential, change of measure, BDG inequalities, martingale integral, purely discontinuous semi-martingales, semi-martingale and martingale decompositions, exponential super-martingales and inequalities, quasi-martingales, stochastic integrators.

This thesis consists of four papers that broadly concerns two dierent topics. The rsttopic is so-called barycentric Markov processes. By a barycentric Markov process wemean a process that consists of a point/particle system evolving in (discrete) time,whose evolution depends in some way on the mean value of the current points in thesystem. In common for the three rst papers which are on this topic is that we studyhow all the points of the so-called core (a certain subset of points in the system) of thesystem converge to the same point.The rst article concerns how an N-point system behaves when we reject the K < N=2points that minimize the sample variance of the remaining N 􀀀K points (the core). Wethen replace the rejected points with K new points which follow some xed distributionand which are all independent from the past points. When K = 1 this is equivalent torejecting the point which is furthest from the center of mass. We prove that under ratherweak assumptions on the sampling distribution, the points of the core converge to thesame point as well as that regardless of any assumptions on our sampling distribution,the sampling variance of the core converges to zero or the core "drifts o to innity".The second article concerns a similar problem as the rst one. We once again consideran N-point system but at each time step we reject the point furthest from the centerof mass multiplied by a positive number p and replace it with a point from a xeddistribution with full support on [0; 1], which is independent from all past points. Ifp = 1 we obtain a special case of the previous article. If p 6= 1 it turns out thatthis process behaves very dierently from the process in the rst article, the stationarydistribution to which the core points converge turns out to be a Bernoulli distribution.The third article studies yet another N-point system but now on a discrete circle.During each time step we compute the distances for each point from the mean of it'stwo neighbours and reject the one with largest such distance (thereby obtaining ourcore) and replace it with a new point independent from past points. Two dierentcases are considered, the rst is with uniformly distributed points in [0; 1] and the otheris with a discrete uniform distribution (i.e. uniformly distributed on an equally spacedgrid).The fourth and last article is on the topic of stochastic calculus. The main objective isto study "stability" of integrators for stochastic integrals. We examine how converging sequences of processes in the role of integrators retain their convergence propertiesfor their corresponding integrals when the integrators are transformed under certainclasses of functions. The convergence is on one hand in the uniform (over compact timeintervals) in Lp-sense and on the other hand in the UCP-sense (uniform convergence inprobability on compact time intervals). We examine processes with quadratic variations(along some rening sequence) transformed by absolutely continuous functions as wellas Dirichlet processes transformed by C1 functions.

The article analyzes the production loss, also referred to as reject, in the largest footwear producer of the World, the Alpargatas company. The article presents a discussion of the probabilistic modelling, the development of a stochastic model, that uses It\^{o} stochastic integration, based on temporal series provided by the company, and the identification of eventual solutions to the loss of production.

In this paper, we prove that the solution constructed in \cite{BR16} satisfies the stochastic vorticity equations with the stochastic integration being understood in the sense of the integration of controlled rough path introduced in \cite{G04}. As a result, we obtain the existence and uniqueness of the global solutions to the stochastic vorticity equations in 3D case for the small initial data independent of time, which can be viewed as a stochastic version of the Kato-Fujita result (see \cite{KF62}).

We develop a novel theory of weak and strong stochastic integration for cylindrical martingale-valued measures taking values in the dual of a nuclear space. This is applied to develop a theory of SPDEs with rather general coefficients. In particular, we can then study SPDEs driven by general L\'{e}vy processes in this context.

In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb E \|N_{\infty}\|^p \leq C_p\mathbb E \|M_{\infty}\|^p$. As a corollary we derive the sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms., Minor revision

In this paper we obtain a simple representation of the classical local times of fractional Brownian motion. We explore the properties of the quadratic variation of fractional Brownian motion { X ( t ) : t ≥ 0 } of index 0 H ≤ 1 ∕ 2 and obtain a generalisation of Tanaka’s formula from which we deduce that the local times at any point a are given by L ( t , a ) = lim n → ∞ 2 n ( 2 H − 1 ) ∑ k ∈ S l 2 | X ( k 2 − n ) − a | with S l = { k ∈ { 1 , 2 , … , l } : ( X ( k 2 − n ) − a ) ( X ( ( k − 1 ) 2 − n ) − a ) 0 } and l = ⌊ t 2 n ⌋ . This is the simplest expression of local times known to the author and it is amenable to numerical computations. Our arguments are elementary and do not use stochastic integration with respect to fractional Brownian motion.

We present a complexity analysis for strong approximation of Banach space valued and parameter dependent scalar stochastic It integration, driven by a Wiener process. Both definite and indefinite integration are considered. We analyze the Banach space valued version of the EulerMaruyama scheme. Based on these results, we define a multilevel algorithm for the parameter dependent stochastic integration problem and show its order optimality for various input classes.

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, the Chinese Electron-Ion Collider of China, 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., Comment: 12 pages, 2 figures, updated figures and references

This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic $L^p$- and $\phi$-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapie\'{n}, McConnell, and Woyczy\'{n}ski. The vast majority of the assertions presented in the paper is done under the sufficient and necessary UMD assumption on the corresponding Banach space., Comment: Final version, to appear in Probability Surveys

In this paper we consider Meyer-Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that $X$ is a UMD Banach space if and only if for any fixed $p\in (1,\infty)$, any $X$-valued $L^p$-martingale $M$ has a unique decomposition $M = M^d + M^c$ such that $M^d$ is a purely discontinuous martingale, $M^c$ is a continuous martingale, $M^c_0=0$ and \[ \mathbb E \|M^d_{\infty}\|^p + \mathbb E \|M^c_{\infty}\|^p\leq c_{p,X} \mathbb E \|M_{\infty}\|^p. \] An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-left continuous martingale and a martingale with accessible jumps. As an application we show that $X$ is a UMD Banach space if and only if for any fixed $p\in (1,\infty)$ and for all $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $$ \mathbb E \|N_{\infty}\|^p \leq C_{p,X}\mathbb E \|M_{\infty}\|^p. $$, Minor revision

In this thesis we study martingales and stochastic integration of processes withvalues in UMD Banach spaces.

In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation \begin{document}$ \, \mathrm{d}X(t) = F(X(t)) \, \mathrm{d}t + G(X(t)) \, \mathrm{d}L(t) $\end{document} driven by a cylindrical Levy process \begin{document}$ L $\end{document} is established. The coefficients \begin{document}$ F $\end{document} and \begin{document}$ G $\end{document} are assumed to satisfy the usual monotonicity and coercivity conditions. The noise is modelled by a cylindrical Levy processes which is assumed to belong to a certain subclass of cylindrical Levy processes and may not have finite moments.