Qu, Dassios, and Zhao (2021) suggested an exact simulation method for tempered stable Ornstein–Uhlenbeck processes, but their algorithms contain some errors. This short note aims to correct their algorithms and conduct some numerical experiments.

We study 2-stage game-theoretic problem oriented 3-stage service policy computing, convolutional neural network (CNN) based algorithm design, and simulation for a blockchained buffering system with federated learning. More precisely, based on the game-theoretic problem consisting of both “win-lose” and “win-win” 2-stage competitions, we derive a 3-stage dynamical service policy via a saddle point to a zero-sum game problem and a Nash equilibrium point to a non-zero-sum game problem. This policy is concerning users-selection, dynamic pricing, and online rate resource allocation via stable digital currency for the system. The main focus is on the design and analysis of the joint 3-stage service policy for given queue/environment state dependent pricing and utility functions. The asymptotic optimality and fairness of this dynamic service policy is justified by diffusion modeling with approximation theory. A general CNN based policy computing algorithm flow chart along the line of the so-called big model framework is presented. Simulation case studies are conducted for the system with three users, where only two of the three users can be selected into the service by a zero-sum dual cost game competition policy at a time point. Then, the selected two users get into service and share the system rate service resource through a non-zero-sum dual cost game competition policy. Applications of our policy in the future blockchain based Internet (e.g., metaverse and web3.0) and supply chain finance are also briefly illustrated.

We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of order $\lambda \in (0,1)$. The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some mild assumptions on the noise, we prove that the solution has moments of all orders. In addition, we provide its connection to the solution of some Skorokhod reflection problem. As an illustration of our results and motivation for applications, we also suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes. We complete the study by providing a numerical scheme for the solution.

We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

In this paper an exact rejection algorithm for simulating paths of the coupled Wright–Fisher diffusion is introduced. The coupled Wright–Fisher diffusion is a family of multivariate Wright–Fisher diffusions that have drifts depending on each other through a coupling term and that find applications in the study of networks of interacting genes. The proposed rejection algorithm uses independent neutral Wright–Fisher diffusions as candidate proposals, which are only needed at a finite number of points. Once a candidate is accepted, the remainder of the path can be recovered by sampling from neutral multivariate Wright–Fisher bridges, for which an exact sampling strategy is also provided. Finally, the algorithm’s complexity is derived and its performance demonstrated in a simulation study.

Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.

There are two types of tempered stable (TS) based Ornstein–Uhlenbeck (OU) processes: (i) the OU-TS process, the OU process driven by a TS subordinator, and (ii) the TS-OU process, the OU process with TS marginal law. They have various applications in financial engineering and econometrics. In the literature, only the second type under the stationary assumption has an exact simulation algorithm. In this paper we develop a unified approach to exactly simulate both types without the stationary assumption. It is mainly based on the distributional decomposition of stochastic processes with the aid of an acceptance–rejection scheme. As the inverse Gaussian distribution is an important special case of TS distribution, we also provide tailored algorithms for the corresponding OU processes. Numerical experiments and tests are reported to demonstrate the accuracy and effectiveness of our algorithms, and some further extensions are also discussed.

We propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this by means of a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as giving bounds on the p-order moments, for all $p\geq1$. We show convergence of an Euler–Maruyama scheme for the process, and also give the rate of convergence, which is dependent on the self-exciting dynamics of the process. Moreover, we discuss various applications of this process, and give examples of different functions to model self-exciting behavior.

In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $ H \gt \frac{1}{2}$. The drift term of the equation is locally Lipschitz and unbounded in the neighbourhood of the origin. We show the existence, uniqueness and positivity of the solutions. The estimates of moments, including the negative power moments, are given. We also develop the implicit Euler scheme, proved that the scheme is positivity preserving and strong convergent, and obtain rate of convergence. Furthermore, by using Lamperti transformation, we show that our results can be applied to stochastic interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear Aït-Sahalia type model.

Let (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

This work investigates analytically, the use of piezoelectric tiles placed on stairways for vibrational energy harvesting – harnessing electrical power from natural vibrational phenomena – from pedestrian footfalls. While energy harvesting from pedestrian traffic along flat pathways has been studied in the linear regime and realised in practical applications, the greater amounts of energy naturally expended in traversing stairways suggest better prospects for harvesting. Considering the characteristics of two types of commercially available piezoelectric tiles – Navy Type III and Navy Type V – analytical models for the coupled electromechanical system are formulated. The harvesting potential of the tiles is then studied under conditions of both deterministic and carefully developed random excitation profiles for three distinct cases: linear, monostable nonlinear and an array of monostable nonlinear tiles on adjacent steps with linear coupling between them. The results indicate enhanced power output when the tiles are: (1) placed on stairways, (2) uncoupled and (3) subjected to excitation profiles with stochastic frequency. In addition, the Navy Type V tiles are seen to outperform the Navy Type III tiles. Finally, the strongly nonlinear regime outperforms the linear one suggesting that the realisation of commercially available piezoelectric tiles with appropriately tailored nonlinear characteristics will likely have a significant impact on energy harvesting from pedestrian traffic.

Vibration energy harvesting aims to harness the energy of ambient random vibrations for power generation, particularly in small-scale devices. Typically, stochastic excitation driving the harvester is modelled as a Brownian process and the dynamics are studied in the equilibrium state. However, non-Brownian excitations are of interest, particularly in the nonequilibrium regime of the dynamics. In this work we study the nonequilibrium dynamics of a generic piezoelectric harvester driven by Brownian as well as (non-Brownian) Lévy flight excitation, both in the linear and the Duffing regimes. Both the monostable and the bistable cases of the Duffing regime are studied. The first set of results demonstrate that Lévy flight excitation results in higher expectation values of harvested power. In particular, it is shown that increasing the noise intensity leads to a significant increase in power output. It is also shown that a linearly coupled array of nonlinear harvesters yields improved power output for tailored values of coupling coefficients. The second set of results show that Lévy flight excitation fundamentally alters the bifurcation characteristics of the dynamics. Together, the results underscore the importance of non-Brownian excitation characterised by Lévy flight in vibration energy harvesting, both from a theoretical viewpoint and from the perspective of practical applications.

An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.

This study investigates the phenomenon of targeted energy transfer (TET) from a linear oscillator to a nonlinear attachment behaving as a nonlinear energy sink for both transient and stochastic excitations. First, the dynamics of the underlying Hamiltonian system under deterministic transient loading is studied. Assuming that the transient dynamics can be partitioned into slow and fast components, the governing equations of motion corresponding to the slow flow dynamics are derived and the behaviour of the system is analysed. Subsequently, the effect of noise on the slow flow dynamics of the system is investigated. The Itô stochastic differential equations for the noisy system are derived and the corresponding Fokker–Planck equations are numerically solved to gain insights into the behaviour of the system on TET. The effects of the system parameters as well as noise intensity on the optimal regime of TET are studied. The analysis reveals that the interaction of nonlinearities and noise enhances the optimal TET regime as predicted in deterministic analysis.

New directions in Markov processes and research on master equations are showcased by example. The utility of Magnus expansions for handling time-varying rates is demonstrated. The useful notion in applied mathematics often turns out to be the pseudospectra and not simply the eigenvalues. We highlight that general principle with our own examples of Markov processes where exact eigenvalues are found and contrasted with the large errors produced by standard numerical methods. As a motivating application, isomerisation provides a running example and an illustration of our approaches to chemical kinetics. We also present a brief example of a totally asymmetric exclusion process.

This paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.

In error estimates of various numerical approaches for solving decoupled forward backward stochastic differential equations (FBSDEs), the rate of convergence for one variable is usually less than for the other. Under slightly strengthened smoothness assumptions, we show that the fully discrete Euler scheme admits a first-order rate of convergence for both variables.

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for non-linear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully non-linear case and open research questions.

The deferred correction (DC) method is a classical method for solving ordinary differential equations; one of its key features is to iteratively use lower order numerical methods so that high-order numerical scheme can be obtained. The main advantage of the DC approach is its simplicity and robustness. In this paper, the DC idea will be adopted to solve forward backward stochastic differential equations (FBSDEs) which have practical importance in many applications. Noted that it is difficult to design high-order and relatively “clean” numerical schemes for FBSDEs due to the involvement of randomness and the coupling of the FSDEs and BSDEs. This paper will describe how to use the simplest Euler method in each DC step–leading to simple computational complexity–to achieve high order rate of convergence.