Stochastic Calculus Research Articles

MULTIPLE SOLITONS, BIFURCATIONS, CHAOTIC PATTERNS AND FISSION/FUSION, ROGUE WAVES SOLUTIONS OF TWO-COMPONENT EXTENDED (2+1)-D ITÔ CALCULUS SYSTEM

The exploration of nonlinear phenomena entails the representation of intricate systems with space-time variables, and across this line, Itô calculus, as the stochastic calculus version of the change pertaining to the variables formula and chain rules, involves the second derivative of f, coming from the property that Brownian motion has non-zero quadratic variation. To this end, the extended ([Formula: see text])-dimensional two-component Itô equation, as an applicable mathematical tool employed in this study for enhancing our understanding of the complex dynamics inherent in multidimensional physical systems, serves the purpose of modeling and understanding dynamic phenomena pervading various disciplines. For modeling complex phenomena, fractional differential equations (FDEs), ordinary differential equations (ODEs), partial differential equations (PDEs) as well as the other ones provide benefits in terms of accuracy and tractability. Accordingly, our study provides the analysis of the two-component nonlinear extended ([Formula: see text])-dimensional Itô equation using the Hirota bilinear method to derive multiple soliton solutions, including novel variations along with their dispersion coefficients, which shed light into the intriguing attributes of the Itô equation. The investigation further encompasses diverse soliton types, such as the general first-order soliton, second-order soliton with fission and bifurcation, third-order soliton, and fourth-order soliton with fission and bifurcation. Besides these, the study also explores the rogue wave and lump solutions by varying parameters across distinct planes. Consequently, these results validate the characteristics and utility of the two-component nonlinear extended ([Formula: see text])-dimensional Itô equation and its relevance to related systems. The novel findings based on the Itô calculus systems revealed, through the analyses, theoretical and experimental aspects in combination with the graphical presentation of the parameter effects on solitons in line with the analyses obtained. These have enhanced the understanding of the dynamics of intricate attributes governed by the two-component nonlinear extended ([Formula: see text])-dimensional Itô equation, particularly concerning chaotic patterns, fission and bifurcation soliton nonlinear complexities.

Hyers-Ulam stability of Hadamard fractional stochastic differential equations

The current article is used to investigate the Hyers-Ulam stability (HUS) of Hadamard stochastic fractional differential equations (HSFDE) by using a version of some fixed point theorem (FPT), a technical lemma and some classical stochastic calculus tools. To show the interest of our results, we present two examples. In this manner, we generalize some recent interesting results.

The local well-posedness for a stochastic system of nonlinear dispersive equations

<p style='text-indent:20px;'>This paper consider the Cauchy problem for the Schr<inline-formula><tex-math id="M1">\begin{document}$ {\rm \ddot{o}} $\end{document}</tex-math></inline-formula>dinger equation coupled with the stochastic Benjamin-Ono equation. A priori estimates for the stochastic integral and the nonlinear terms corresponding to the coupling system are achieved by using Fourier transform restriction method introduced by Bourgain. It is shown that the Cauchy problem is locally well-posed as the initial data in appropriate Sobolev spaces.</p>

A New Proof of the Expansion of Iterated Ito Stochastic Integrals with Respect to the Components of a Multidimensional Wiener Process Based on Generalized Multiple Fourier Series and Hermite Polynomials

The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality, namely, for the case of continuous or piecewise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.

Stochastic integration with respect to a cylindrical special semi martingale

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.

Mathematical model to assess energy consumption using water inflow-drainage system of iron-ore mines in terms of a stochastic process

Purpose is to develop a unified mathematical model to assess energy efficiency of a water inflow-drainage process as the real variant of stochastic method for water pumping from underground workings of iron-ore mines. Methods. The research process was based upon the methods of probability theory as well as stochastic modelling methods. The stochastic function integration has been reduced to summation of its ordinates and further transition to a proper boundary. Findings. A mathematical model of a water inflow-drainage system as a stochastic process has been developed in terms of input parameters of a standard operating iron-ore mine. The abovementioned has made it possible to assess realistically, substantiate, and obtain possibilities for a specific production facility as well as for generalization of the results involving determination of stochastic characteristics of drainage process. Originality. For the first time, a mathematical model of drainage from underground levels of iron-ore mines has been developed as a stochastic process. The process characteristics have been identified relying upon randomness of a water pumping technique. In contrast to the available settings, the new model parameters characterize their dispersion. Possibility to obtain complete characteristics of energy consumption has been obtained: for drainage; for water accumulation volume in underground water collectors; for water pumping from the specified mine depths over the specific period as random processes. A number of drainage features have been analyzed and differentiated being determined with the help of nor-mal law of water accumulation velocity in the underground water collectors in iron-ore mines. Practical implications. In terms of operating iron-ore mine, a generalized drainage mathematical model has been developed as a stochastic process using statistical data concerning water accumulation velocity in the underground water collectors. It has been proved that if the ordinates of water accumulation velocity in the underground water collectors obey the normal distribution law then it is expedient to characterize drainage as a stochastic process. The developed methods, studying drain-age as a stochastic process, help expand the research boundaries involving other auxiliary operations performed during underground mining of iron ore raw materials.

Selfsimilar Free Additive Processes and Freely Selfdecomposable Distributions

In the paper by Fan (Inf Dim Anal Quant Probab Rel Topics 9:451–469, 2006), he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved that the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan’s in general, of selfsimilarity via linear combinations of non-commutative stochastic processes, although the two definitions are equivalent for non-commutative stochastic processes with freely independent increments. We secondly prove the converse of Fan’s result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free Lévy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free Lévy processes in terms of their free cumulant transforms is also given, and several examples are discussed.

An analysis of approximation algorithms for iterated stochastic integrals and a Julia and Matlab simulation toolbox

For the approximation and simulation of twofold iterated stochastic integrals and the corresponding Lévy areas w.r.t. a multi-dimensional Wiener process, we review four algorithms based on a Fourier series approach. Especially, the very efficient algorithm due to Wiktorsson and a newly proposed algorithm due to Mrongowius and Rößler are considered. To put recent advances into context, we analyse the four Fourier-based algorithms in a unified framework to highlight differences and similarities in their derivation. A comparison of theoretical properties is complemented by a numerical simulation that reveals the order of convergence for each algorithm. Further, concrete instructions for the choice of the optimal algorithm and parameters for the simulation of solutions for stochastic (partial) differential equations are given. Additionally, we provide advice for an efficient implementation of the considered algorithms and incorporated these insights into an open source toolbox that is freely available for both Julia and Matlab programming languages. The performance of this toolbox is analysed by comparing it to some existing implementations, where we observe a significant speed-up.

A numerical solution for a stochastic beam equation exhibiting purely viscous behavior

AbstractThe deflection of Euler–Bernoulli beams under stochastic dynamic loading, exhibiting purely viscous behavior, is characterized by partial differential equations of the fourth order. This paper proposes a computational method to determine the approximate solution to such equations. The functions are approximated using two‐dimensional shifted Legendre polynomials. An operational matrix of integration and an operational matrix of stochastic integration are derived. The operational matrices assist in breaking down the problem under consideration into a set of algebraic equations that may be solved using any known numerical technique that leads to the solution of the stochastic beam equation. The well‐posedness of the problem is studied. The proposed methodology is demonstrated to be practical for addressing the novel stochastic dynamic loading problem by confirming the outcome using a few numerical examples. Thus the effectiveness and applicability of the technique are ensured. The solution quality is explored through diagrams. The accuracy of the method is substantiated by comparing it with the Runge–Kutta method of order 1.5 (R–K 1.5). The absolute error caused by the proposed technique is comparably much less than R–K 1.5. A simulation analysis is carried out with MATLAB, and an algorithm is developed.

LibRoadRunner 2.0: a high performance SBML simulation and analysis library.

This article presents libRoadRunner 2.0, an extensible, high-performance, cross-platform, open-source software library for the simulation and analysis of models expressed using the systems biology markup language (SBML). libRoadRunner is a self-contained library, able to run either as a component inside other tools via its C++, C and Python APIs, or interactively through its Python or Julia interface. libRoadRunner uses a custom just-in-time (JIT) compiler built on the widely used LLVM JIT compiler framework. It compiles SBML-specified models directly into native machine code for a large variety of processors, making it fast enough to simulate extremely large models or repeated runs in reasonable timeframes. libRoadRunner is flexible, supporting the bulk of the SBML specification (except for delay and non-linear algebraic equations) as well as several SBML extensions such as hierarchical composition and probability distributions. It offers multiple deterministic and stochastic integrators, as well as tools for steady-state, sensitivity, stability and structural analyses. libRoadRunner binary distributions for Windows, Mac OS and Linux, Julia and Python bindings, source code and documentation are all available at https://github.com/sys-bio/roadrunner, and Python bindings are also available via pip. The source code can be compiled for the supported systems as well as in principle any system supported by LLVM-13, such as ARM-based computers like the Raspberry Pi. The library is licensed under the Apache License Version 2.0.

Construction of minimal mass blow-up solutions to rough nonlinear Schrödinger equations

We study the focusing mass-critical rough nonlinear Schrödinger equations, where the stochastic integration is taken in the sense of controlled rough path. In both dimensions one and two, the minimal mass blow-up solutions are constructed, which behave asymptotically like the pseudo-conformal blow-up solutions near the blow-up time. Furthermore, the global well-posedness is obtained if the mass of initial data is below that of the ground state. These results yield that the mass of ground state is exactly the threshold of global well-posedness and blow-up in the stochastic focusing mass-critical case. Similar results are also obtained for a class of nonlinear Schrödinger equations with lower order perturbations, particularly in the absence of the standard pseudo-conformal symmetry and the conservation law of energy.

Design of a Stochastic Traffic Regulator for End-to-End Network Delay Guarantees

Providing end-to-end network delay guarantees in packet-switched networks such as the Internet is highly desirable for mission-critical and delay-sensitive data transmission, yet it remains a challenging open problem. Since deterministic bounds are based on the worst-case traffic behavior, various frameworks for stochastic network calculus have been proposed to provide less conservative, probabilistic bounds on network delay, at least in theory. However, little attention has been devoted to the problem of regulating traffic according to stochastic burstiness bounds, which is necessary in order to guarantee the delay bounds in practice. We design and analyze a stochastic traffic regulator that can be used in conjunction with results from stochastic network calculus to provide probabilistic guarantees on end-to-end network delay. Two alternative implementations of the stochastic regulator are developed and compared. Numerical results are provided to demonstrate the performance of the proposed stochastic traffic regulator.

SNC BASED NETWORK LAYER DESIGN FOR UNDERWATER COMMUNICATION USED IN OPEN OCEAN FISH FARMING

Aquaculture is gradually moving toward Open Ocean Fish Farming (OOFF) in recent times because of the huge environmental impacts caused by onshore fish farming. Although Offshore fish farming has a lot of benefits over onshore fish farming it has its threats as well. Harmful Algae Blooms (HAB) are one such threat to Offshore fish farming. HAB is the formation of algae in groups that are toxic. The HAB can significantly affect the health of the fish that are being cultured. To prevent and act on HAB, monitoring of different environmental parameters is done in the field that is of concern. Traditionally for these kinds of monitoring human drivers are employed. But involving human drivers for monitoring in offshore conditions is cost expensive and difficult. Technology-enabled autonomous monitoring is needed in an offshore environment. Here comes underwater Wireless Sensor Networks (UWSN) which can be used for inspection of underwater fish farms for the detection of HAB. In UWSN loss is one of the main parameters that affect the Quality of Service. So this research focuses on the development of a Stochastic Network Calculus (SNC) based mathematical model of the network layer design for underwater communication that can be used in OOFF. In this work, we introduced the loss factor parameter into SNC and derived the loss bound for underwater communication, and the same is evaluated using discrete event simulation.

A Monte Carlo algorithm for multiple stochastic integrals of stable processes

In this paper, we provide a Monte Carlo method to calculate multiple stochastic integrals which is based on a.s. distributional convergence.

A sparse Bayesian framework for discovering interpretable nonlinear stochastic dynamical systems with Gaussian white noise

Extracting governing physics from data is a key challenge in many areas of science and technology. The existing techniques for equation discovery are mostly applicable to deterministic systems and require both input and state measurements. We here propose a novel data-driven framework for discovering nonlinear stochastic dynamical systems with Gaussian white noise. The proposed framework blends concepts of stochastic calculus, sparse learning algorithms, and Bayesian statistics to learn the governing physics from data. In particular, we combine sparsity–promoting spike and slab prior, Bayes law, and the Kramers–Moyal formula to identify stochastic differential equations from data. The proposed framework is highly efficient and works with sparse, noisy, and incomplete output measurements. The efficacy and robustness of the proposed approach are illustrated in several numerical examples involving both complete and partial state measurements. The results obtained indicate the potential of the proposed approach in discovering nonlinear stochastic dynamical systems subjected to Gaussian white noise excitation.

Stochastic evolution equations with Lévy noise in the dual of a nuclear space

In this article we give sufficient and necessary conditions for the existence of a weak and mild solution to stochastic evolution equations with (general) L\'{e}vy noise taking values in the dual of a nuclear space. As part of our approach we develop a theory of stochastic integration with respect to a L\'{e}vy process taking values in the dual of a nuclear space. We also derive further properties of the solution such as the existence of a solution with square moments, the Markov property and path regularity of the solution. In the final part of the paper we give sufficient conditions for the weak convergence of the solutions to a sequence of stochastic evolution equations with L\'{e}vy noises.

Infinitesimal invariance of completely Random Measures for 2D Euler Equations

We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields falls outside of the well-posedness regime of the PDE under consideration, so it is necessary to resort to stochastic integrals with respect to the candidate invariant measure in order to give a definition of the dynamics. Our findings generalize and unify previous results on Gaussian stationary solutions of Euler equations and point vortices dynamics. We also discuss difficulties arising when attempting to produce a solution flow for Euler’s equations preserving independently scattered random measures.

Before thy throne I slip in gently, hoping nobody will notice: A memorial to Robin Hudson (1940–2021)

We dedicate this to the life and work of Robin Hudson -- a mathematical physicist who developed the peerless quantum stochastic calculus, but who also inspired generations of researchers with both his intellect and wit.

Nash equilibria for relative investors via no-arbitrage arguments

Within a common arbitrage-free semimartingale financial market we consider the problem of determining all Nash equilibrium investment strategies for n agents who try to maximize the expected utility of their relative wealth. The utility function can be rather general here. Exploiting the linearity of the stochastic integral and making use of the classical pricing theory we are able to express all Nash equilibrium investment strategies in terms of the optimal strategies for the classical one agent expected utility problems. The corresponding mean field problem is solved in the same way. We give four applications of specific financial markets and compare our results with those given in the literature.

Variation of constants formulae for forward and backward stochastic Volterra integral equations

In this paper, we provide variation of constants formulae for linear (forward) stochastic Volterra integral equations (SVIEs, for short) and linear Type-II backward stochastic Volterra integral equations (BSVIEs, for short) in the usual Itô's framework. For these purposes, we define suitable classes of stochastic Volterra kernels and introduce new notions of the products of adapted L2-processes. Observing the algebraic properties of the products, we obtain the variation of constants formulae by means of the corresponding resolvent. Our framework includes SVIEs with singular kernels such as fractional stochastic differential equations. Also, our results can be applied to general classes of SVIEs and BSVIEs with infinitely many iterated stochastic integrals. The duality principle between generalized SVIEs and generalized BSVIEs is also proved.