Stochastic Calculus Research Articles

Homogenization, Linearization, and Large‐Scale Regularity for Nonlinear Elliptic Equations

AbstractWe consider nonlinear, uniformly elliptic equations with random, highly oscillating coefficients satisfying a finite range of dependence. We prove that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized equation). We also obtain a quantitative estimate on the rate of this homogenization. These results lead to a better understanding of differences of solutions to the nonlinear equation, which is of fundamental importance in quantitative homogenization. In particular, we obtain a large‐scale C0, 1 estimate for differences of solutions—with optimal stochastic integrability. Using this estimate, we prove a large‐scale C1, 1 estimate for solutions, also with optimal stochastic integrability. Each of these regularity estimates are new even in the periodic setting. As a second consequence of the large‐scale regularity for differences, we improve the smoothness of the homogenized Lagrangian by showing that it has the same regularity as the heterogeneous Lagrangian, up to C2, 1. © 2020 Wiley Periodicals LLC

Theorem of Furstenberg type for multiplicative stochastic integrals

Abstract This paper contains the proof of the positivity of the top Lyapunov exponent for the multiplicative stochastic integrals related to the diffusion processes on the Lee algebra of the matrices with zero trace.

Parameter Estimation of the Stochastic Primary Nucleation Kinetics by Stochastic Integrals Using Focused-Beam Reflectance Measurements

The kinetic parameters of stochastic primary nucleation were estimated for the batch-cooling crystallization of L-arginine. It is difficult for process analytical tools to detect the first nucleus. In this study, the latent period for the total number of crystals to be increased to a predetermined threshold was repeatedly measured with focused-beam reflectance measurements. Consequently, the latent periods were different in each measurement due to the stochastic behavior of both primary and secondary nucleation. Therefore, at first, the distribution of the latent periods was estimated by a Monte Carlo simulation for some combinations of the kinetic parameters of primary nucleation. In the simulation, stochastic integrals of the population and mass balance equations were solved. Then, the parameters of the distribution of latent periods were estimated and correlated with the kinetic parameters of primary nucleation. The resulting correlation was represented by a mapping. Finally, the parameters of the actual distribution were input into the inverse mapping, and the kinetic parameters were estimated as the outputs. The estimated kinetic parameters were validated using statistical techniques, which implied that the observed distribution function of the latent periods for the thresholds used in the estimation coincided reasonably with the simulated one based on the estimated parameters.

Equilibrium Theory in Continuous Time

This paper contains my (still incomplete) lecture notes on equilibrium theory in continuous time. It is intended for PhD students with a reasonably solid background in stochastic calculus.

Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

This work focuses on the dynamics of particles in a confined geometry with position-dependent diffusivity, where the confinement is modelled by a periodic channel consisting of unit cells connected by narrow passage ways. We consider three functional forms for the diffusivity, corresponding to the scenarios of a constant (D0), as well as a low (Dm) and a high (Dd) mobility diffusion in cell centre of the longitudinally symmetric cells. Due to the interaction among the diffusivity, channel shape and external force, the system exhibits complex and interesting phenomena. By calculating the probability density function, mean velocity and mean first exit time with the Itô calculus form, we find that in the absence of external forces the diffusivity Dd will redistribute particles near the channel wall, while the diffusivity Dm will trap them near the cell centre. The superposition of external forces will break their static distributions. Besides, our results demonstrate that for the diffusivity Dd, a high dependence on the x coordinate (parallel with the central channel line) will improve the mean velocity of the particles. In contrast, for the diffusivity Dm, a weak dependence on the x coordinate will dramatically accelerate the moving speed. In addition, it shows that a large external force can weaken the influences of different diffusivities; inversely, for a small external force, the types of diffusivity affect significantly the particle dynamics. In practice, one can apply these results to achieve a prominent enhancement of the particle transport in two- or three-dimensional channels by modulating the local tracer diffusivity via an engineered gel of varying porosity or by adding a cold tube to cool down the diffusivity along the central line, which may be a relevant effect in engineering applications. Effects of different stochastic calculi in the evaluation of the underlying multiplicative stochastic equation for different physical scenarios are discussed.

Stratonovich type integration with respect to fractional Brownian motion with Hurst parameter less than $1/2$

Let $B^{H}$ be a fractional Brownian motion with Hurst parameter $H\in (0,1/2)$ and $p:\mathbb{R}\rightarrow \mathbb{R}$ a polynomial function. The main purpose of this paper is to introduce a Stratonovich type stochastic integral with respect to $B^{H}$, whose domain includes the process $p(B^{H})$. That is, an integral that allows us to integrate $p(B^{H})$ with respect to $B^{H}$, which does not happen with the symmetric integral given by Russo and Vallois (Probab. Theory Related Fields 97 (1993) 403–421) in general. Towards this end, we combine the approaches utilized by León and Nualart (Stochastic Process. Appl. 115 (2005) 481–492), and Russo and Vallois (Probab. Theory Related Fields 97 (1993) 403–421), whose aims are to extend the domain of the divergence operator for Gaussian processes and to define some stochastic integrals, respectively. Then, we study the relation between this Stratonovich integral and the extension of the divergence operator (see León and Nualart (Stochastic Process. Appl. 115 (2005) 481–492)), an Itô formula and the existence of a unique solution of some Stratonovich stochastic differential equations. These last results have been analyzed by Alòs, León and Nualart (Taiwanese J. Math. 5 (2001) 609–632), where the Hurst paramert $H$ belongs to the interval $(1/4,1/2)$.

A functional non-central limit theorem for multiple-stable processes with long-range dependence

A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets.

On inverse initial value problems for the stochastic strongly damped wave equation

We consider two final value problem of a stochastic strongly damped wave equation driven by white noise (Section 3) and fractional noise (Section 4). We show that a stochastic integral in the solution is not stable and the problem is not well posed. To regularize the problem in two cases of noise, we apply the Fourier truncated method to control the frequency in such a way that it depends on the noisy level, which is the upper bound of the errors appearing in the input data. Furthermore, the convergence rate of the regularized solution is investigated.

Stochastic Fubini Theorem for Jump Noises in Banach Spaces

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.

Lindeberg’s central limit theorems for martingale like sequences under sub-linear expectations

The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes, especially stochastic integrals and differential equations. In this paper, general central limit theorems and functional central limit theorems are obtained for martingale like random variables under the sub-linear expectation. As applications, the Lindeberg central limit theorem and functional central limit theorem are obtained for independent but not necessarily identically distributed random variables, and a new proof of the L\'evy characterization of a G-Brownian motion without using stochastic calculus is given. For proving the results, we have also established Rosenthal's inequality and the exceptional inequality for the martingale like random variables.

On the dynamics of two photons interacting with a two-qubit coherent feedback network

The purpose of this paper is to study the dynamics of a quantum coherent feedback network composed of two two-level systems (qubits) driven by two counter-propagating photons, one in each input channel. The coherent feedback network enhances the nonlinear photon–photon interaction inside the feedback loop. By means of quantum stochastic calculus and the input–output framework, the analytic form of the steady-state output two-photon state is derived. Based on the analytic form, the applications on the Hong–Ou–Mandel (HOM) interferometer and marginally stable single-photon devices using this coherent feedback structure have been demonstrated.

Uniform contraction of a stochastic integral in strategic operations

Random contractions are generally recognized as strong analytical tools of probability theory with significant practical applications in a wide variety of disciplines. The paper formulates a uniform random contraction of a stochastic integral. Properties and applications in strategic decision making and other practical disciplines of the formulated uniform random contraction are also established by the paper.

Stochastic Network Calculus for Delay Variation in UWAN Multichannel Communication

The behavior of underwater acoustic wireless communication is challenging one compared to terrestrial communication due to the unpredictable changes in underwater. The multichannel communication needs to consider a lot of parameters at the time of modeling and investigation such as losses and noise. Thus, communication channel separation is curiously inflexible to investigate all properties of underwater. Compared with other modes of communication, acoustics waves afford the sturdy data delivery and receiving capability in underwater. The channel allocation and detachment appearing quite difficult in acoustic with respect to techniques that induce the transmission of data digitally, which compared with the major properties of attenuation variation in underwater. In this proposal, we concentrate on the numerical results for the transmission and receiver end delay variation with the help of stochastic network calculus to measure the deviation between the simulation and numerical results (SNC) with respect to multichannel(MC).

A Regularity Theory for Random Elliptic Operators

Since the seminal results by Avellaneda & Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring work, Armstrong & Smart proved large-scale Lipschitz estimates for such operators with random coefficients satisfying a finite-range of dependence assumption. In the present contribution, we extend the intrinsic large-scale regularity of Avellaneda & Lin (namely, intrinsic large-scale Schauder and Calderón-Zygmund estimates) to elliptic systems with random coefficients. The scale at which this improved regularity kicks in is characterized by a stationary field r* which we call the minimal radius. This regularity theory is qualitative in the sense that r* is almost surely finite (which yields a new Liouville theorem) under mere ergodicity, and it is quantifiable in the sense thatr* has high stochastic integrability provided the coefficients satisfy quantitative mixing assumptions. We illustrate this by establishing optimal moment bounds on r* for a class of coefficient fields satisfying a multiscale functional inequality, and in particular for Gaussian-type coefficient fields with arbitrary slow-decaying correlations.

A New Stochastic Fubini-Type Theorem

When a stochastic process is given through an Itō integral, i.e. a stochastic integral, or a stochastic differential equation (SDE), an analytical solution does not have to exist—and even if there is a closed-form solution, the derivation of this solution can be very complex. When the solution of the stochastic process is not needed but only the expected value as a function of time, the question arises whether it is possible to use the expectation operator directly on the stochastic integral or on the SDE and to somehow calculate the expectation of the process as a Riemann integral over the expectation of the integrands and integrators. In this paper, we show that if the integrator is linear in expectation, the expectation operator and an Itō integral can be interchanged. Additionally, we state how this can be used on SDEs and provide an application from the field of technical trading, i.e. from the field of mathematical finance.

Stochastic optimal integration of decentralized heat pumps in a smart thermal and electric micro-grid

Heat pumps represent a link between different energy vectors and their application in thermal and electrical grids can improve the overall operational flexibility of the system. In this work, the optimal integration of electrically driven heat pumps within a hybrid distributed energy system is investigated. A multi-objective stochastic optimization methodology is proposed to evaluate the integrated optimal sizing and operation of the energy systems under uncertainties in climate, space occupancy, energy loads, and fuel costs. A case study is considered, namely a University campus, and two different configurations, with and without heat pumps, are compared. Both configurations include a cogeneration system, photovoltaic and solar thermal panels, and a wind turbine. The results show how the integration of heat pumps can reduce the operational cost of the system, increase the renewables share, provide a more robust design of the system, and moderate the risk of the investment. Indeed, the configuration with heat pumps entails a 50% higher expected value of the energy savings, a 28% increase of the renewable energy production, and higher energy savings in the worst-case scenario (13% vs. 5%).

Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review

We present a stochastic Lagrangian view of fluid dynamics. The velocity solving the deterministic Navier–Stokes equation is regarded as a mean time derivative taken over stochastic Lagrangian paths and the equations of motion are critical points of an associated stochastic action functional involving the kinetic energy computed over random paths. Thus the deterministic Navier–Stokes equation is obtained via a variational principle. The pressure can be regarded as a Lagrange multiplier. The approach is based on Itô’s stochastic calculus. Different related probabilistic methods to study the Navier–Stokes equation are discussed. We also consider Navier–Stokes equations perturbed by random terms, which we derive by means of a variational principle.

Fractional Brownian motion inference of multivariate stochastic differential equations

Recently, the financial mathematics has been emerged to interpret and predict the underlying mechanism that generates an incident of concern. A system of differential equations can reveal a dynamical development of financial mechanism across time. Multivariate wiener process represents the stochastic term in a system of stochastic differential equations (SDE). The standard wiener process follows a Markov chain, and hence it is a martingale (kind of Markov chain), which is a good integrator. Though, the fractional Wiener process does not follow a Markov chain, hence it is not a good integrator. This problem will produce an Arbitrage (non-equilibrium in the market) in the predicted series. It is undesired property that leads to erroneous conclusion, as it is not possible to build a mathematical model, which represents the financial phenomenon. If there is Arbitrage (unbalance) in the market, this can be solved by Wick-Ito-Skorohod stochastic integral (renormalized integral). This paper considers the estimation of a system of fractional stochastic differential equations (FSDE) using maximum likelihood method, although it is time consuming. However, it provides estimates with desirable characteristic with the most important consistency. Langevin method can be used to find the mathematical form of the functions of stochastic differential equations. This includes drift and diffusion by estimating conditional mean and variance from the data and finding the suitable function achieves the least error, and then estimating the parameters of the model by numerical optimal solution search method. Data used in this paper consist of three banking sector stock prices including Baghdad Bank (BBOB), the Commercial Bank (BCOI), and the National Bank (BNOI).

Time-Changed Local Martingales Under Signed Measures

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.

Online Drift Estimation for Jump-Diffusion Processes

We show the convergence of an online stochastic gradient descent estimator to obtain the drift parameter of a continuous-time jump-diffusion process. The stochastic gradient descent follows a stochastic path in the gradient direction of a function to find a minimum, which in our case determines the estimate of the unknown drift parameter. We decompose the deviation of the stochastic descent direction from the deterministic descent direction into four terms: the weak solution of the non-local Poisson equation, a Riemann integral, a stochastic integral, and a covariation term. This decomposition is employed to prove the convergence of the online estimator and we use simulations to illustrate the performance of the online estimator.