In this paper we employ a Gaussian-type heat kernel estimate to establish Krylov’s estimate and Khasminskii’s estimate for the Euler–Maruyama (EM) algorithm. For applications, by taking Zvonkin’s transformation into account, we investigate the convergence rate of the EM algorithm for a class of multidimensional stochastic differential equations (SDEs) with low regular drifts, which need not be piecewise Lipschitz.

We obtain a generalisation of the Stroock–Varadhan support theorem for a large class of systems of subcritical singular stochastic partial differential equations driven by a noise that is either white or approximately self-similar. The main problem that we face is the presence of renormalisation. In particular, it may happen in general that different renormalisation procedures yield solutions with different supports. One of the main steps in our construction is the identification of a subgroup $\mathcal {H}$ of the renormalisation group such that any renormalisation procedure determines a unique coset $g\circ \mathcal {H}$ . The support of the solution then depends only on this coset and is obtained by taking the closure of all solutions obtained by replacing the driving noises by smooth functions in the equation that is renormalised by some element of $g\circ \mathcal {H}$ .

One immediate corollary of our results is that the $\Phi ^4_3$ measure in finite volume has full support, and the associated Langevin dynamic is exponentially ergodic.

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.

Taking the consideration of two-dimensional stochastic Navier–Stokes equations with multiplicative Lévy noises, where the noises intensities are related to the viscosity, a large deviation principle is established by using the weak convergence method skillfully, when the viscosity converges to 0. Due to the appearance of the jumps, it is difficult to close the energy estimates and obtain the desired convergence. Hence, one cannot simply use the weak convergence approach. To overcome the difficulty, one introduces special norms for new arguments and more careful analysis.

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.

Our aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

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.

We define a new family of multivariate stochastic processes over a finite time horizon that we call generalised Liouville processes (GLPs). GLPs are Markov processes constructed by splitting Lévy random bridges into non-overlapping subprocesses via time changes. We show that the terminal values and the increments of GLPs have generalised multivariate Liouville distributions, justifying their name. We provide various other properties of GLPs and some examples.

A set of data with positive values follows a Pareto distribution if the log–log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf’s law if the log–log plot has a slope of $-1$. Since many types of ranked data follow Zipf’s law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of strictly positive continuous semimartingales with parameters that depend only on rank. We show that the stationary distribution of an Atlas model will follow Zipf’s law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf’s law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf’s law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.

It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).

In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.

Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

Rumours have become part of our daily lives, and their spread has a negative impact on a variety of human affairs. Therefore, how to control the spread of rumours is an important topic. In this paper, we extend the classic Maki–Thompson model from a deterministic framework to a stochastic framework with a forgetting mechanism, because real-world person-to-person communications are inevitably affected by random factors. By constructing suitable stochastic Lyapunov functions, we show that the asymptotic behaviour of the stochastic rumour model is governed by the basic reproductive number. If this number is less than one, then the solution of the stochastic rumour model oscillates around the rumour-free equilibrium under extra mild conditions, indicating the extinction of the rumour with a probability of one. Otherwise, the solution always fluctuates around the endemic equilibrium under certain parametric restrictions, implying that the rumour will continually persist. In addition, we discuss a possible intervention strategy that stops the spread of rumours by strengthening the intensity of white noise, which is very different from the deterministic rumour model without white noise. Also, numerical simulations are conducted to support our analytical results.

In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter β2 > 0, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range 0 < β2 < 4π via the variational approach due to Barashkov-Gubinelli (2018). We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range 0 < β2 < 2π. Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range 0 < β2 < 4π.

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.

We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

This work proposes and analyzes a family of spatially inhomogeneous epidemic models. This is our first effort to use stochastic partial differential equations (SPDEs) to model epidemic dynamics with spatial variations and environmental noise. After setting up the problem, the existence and uniqueness of solutions of the underlying SPDEs are examined. Then, definitions of permanence and extinction are given, and certain sufficient conditions are provided for permanence and extinction. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.

For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits. We also discuss our program on rare transitions in noisy heteroclinic networks.