Distributed ledgers – decentralized databases maintained by network consensus – are often modeled as directed acyclic graphs (DAGs) to capture the causal structure of data addition. Although blockchain systems like Bitcoin use linear chains, alternatives such as tangle in IOTA employ random DAGs. In such mechanisms each new transaction approves multiple predecessors selected through a randomized process. Prior work has established a fluid-limit approximation of the tangle’s growth, governed by a delay differential equation. In this paper we go beyond the fluid limit by analyzing the next-order behavior. We show that the fluctuations around the deterministic limit converge to a Gaussian process and derive a stochastic delay differential equation (SDDE) that describes this next-order approximation.

Since stochastic differential equations (SDEs) driven by G-Brownian motion are of great importance in modeling situations that incorporate ambiguity, it is essential to address efficient numerical schemes to approximate the solution of such equations. The stream of research related to the numerical solutions of G-SDEs under standard assumptions is to some extent well understood. In this note, we are interested in designing an implicit $\theta$-Euler–Maruyama scheme to approximate the solution of G-SDEs under locally Lipschitz continuous coefficients. The convergence of the proposed scheme is established using the stopping time technique. In addition, we investigate the exponentially/quasi-surely asymptotic stability property of the scheme.

We study a nonlinear branching diffusion process in the sense of McKean, i.e. where particles are subjected to a mean-field interaction. We consider first a strong formulation of the problem and we provide an existence and uniqueness result by using contraction arguments. Then we consider the notion of weak solution and its equivalent martingale problem formulation. In this setting, we provide a general weak existence result, as well as a propagation of chaos property, i.e. the McKean–Vlasov branching diffusion is the limit of a large-population branching diffusion process with mean-field interaction.

In this paper, the pricing problem of geometric average Asian options under the Vasicek interest rate based on a time-changed mixed fractional Brownian motion is considered. A stochastic process similar to the renewal process is applied to characterize the constant periodicity of the financial asset price in emerging financial markets. The time-changed mixed fractional Brownian motion model $M_{\alpha ,H}(t) = aB(T_\alpha (t)) + bB_H(T_\alpha (t))$ is introduced to describe the underlying asset process of Asian options. When the Hurst exponent satisfies certain conditions, the model is used to price options without arbitrage. By using the hedging and no-arbitrage principle, the partial differential equation satisfied by the price of an Asian option is given. The pricing formula of an Asian call and a put option, and the corresponding parity formula, are obtained, along with their explicit solution.

Biochemical reaction networks (RNs) are widely applied across scientific disciplines to model complex dynamic systems. We investigate the diffusion approximation of RNs with mass-action kinetics, focusing on the identifiability of the stochastic differential equations associated to the reaction network. We derive conditions under which the law of the diffusion approximation is identifiable and provide theorems for verifying identifiability in practice. Notably, our results show that some RNs have non-identifiable reaction rates, even when the law of the corresponding stochastic process is completely known. Moreover, we show that RNs with distinct graphical structures can generate the same diffusion law under specific choices of reaction rates. Finally, we compare our framework with identifiability results in the deterministic ordinary differential equation setting and the discrete continuous-time Markov chain models for RNs.

This work studies time averages of an observable $h(t,X_t)$, where $X_t$ is the solution to a time-inhomogeneous stochastic differential equation (SDE) driven by drift, b(t, x), and diffusion, $\sigma(t{,}{\kern.5pt}x)$, that change sufficiently slowly in time. In this quasistatic regime we derive an approximation to the time average that is computable from properties of the time-homogeneous SDEs driven by $b(t,\cdot)$ and $\sigma(t,\cdot)$ with fixed t; specifically, we utilize $\log$-Sobolev inequalities for the instantaneous invariant distribution and generator for each t. We obtain explicit non-asymptotic error bounds on this quasistatic approximation, both in the form of concentration inequalities and bounds on the expected value. The error bounds demonstrate a competition between the speed of convergence to the instantaneous invariant distributions and their rate of change, matching the intuition that underlies the quasistatic approximation.

This paper focuses mainly on the Euler scheme of stochastic delay differential equations with locally Lipschitz coefficients. The convergence in probability of the Euler scheme and the corresponding weak limit process of the normalized error process are derived. Furthermore, this paper also considers a class of specific degenerate stochastic delay equations and obtains the associated weak limit process for the stronger error process. The error parameter of this stronger error process for such a degenerate system is n instead of $\sqrt{n}$ in the normalized error process. This causes substantial challenges in the analysis and proofs and the weak limit process also becomes more complicated and involves some additional terms. This result is new and interesting even for the non-delay case.

Well-posedness is established for multi-dimensional mean-field stochastic Volterra equations with Lipschitz-continuous coefficients, allowing for singular kernels as well as for one-dimensional mean-field stochastic Volterra equations with Hölder-continuous diffusion coefficients and sufficiently regular kernels. In these different settings, quantitative, pointwise propagation of chaos results are derived for the associated Volterra-type interacting particle systems.

Given a sequence of graphs $G_n$ and a fixed graph H, denote by $T(H, G_n)$ the number of monochromatic copies of the graph H in a uniformly random c-coloring of the vertices of $G_n$. In this paper we study the joint distribution of a finite collection of monochromatic graph counts in networks with multiple layers (multiplex networks). Specifically, given a finite collection of graphs $H_1, H_2, \ldots, H_d$ we derive the joint distribution of $(T(H_1, G_n^{(1)}), T(H_2, G_n^{(2)}), \ldots, T(H_d, G_n^{(d)}))$, where $\mathbf{G}_n = (G_n^{(1)}, G_n^{(2)}, \ldots, G_n^{(d)})$ is a collection of dense graphs on the same vertex set converging in the multiplex cut-metric. The limiting distribution is the sum of two independent components: a multivariate Gaussian and a sum of independent bivariate stochastic integrals. This extends previous results on the marginal convergence of monochromatic subgraphs in a sequence of graphs to the joint convergence of a finite collection of monochromatic subgraphs in a sequence of multiplex networks. Several applications and examples are discussed.

Past research has indicated that the covariance of the stochastic gradient descent (SGD) error done via minibatching plays a critical role in determining its regularization and escape from low potential points. Motivated by some new research in this area, we prove universality results by showing that noise classes that have the same mean and covariance structure of SGD via minibatching have similar properties. We mainly consider the SGD algorithm, with multiplicative noise, introduced in previous work (Wu et al (2016) Int. Conf. on Machine Learning, PMLR, pp. 10367–10376), which has a much more general noise class than the SGD algorithm done via minibatching. We establish non-asymptotic bounds for the multiplicative SGD algorithm in the Wasserstein distance. We also show that the error term for the algorithm is approximately a scaled Gaussian distribution with mean 0 at any fixed point.

In this paper, we are mainly devoted to the limiting behaviour of smooth inertial manifolds for a class of random retarded differential equations with a singular parameter $\delta$ and their Galerkin approximations, which have not been considered before. Under appropriate conditions, we show not only that the inertial manifolds for this class of random retarded equations converge pointwise to those of the corresponding stochastic equations driven by white noise as $\delta\rightarrow 0$, but also that the inertial manifolds of their Galerkin approximations converge pointwise to those of stochastic equations driven by white noise described above under the simultaneous limits $\delta\rightarrow 0$ and $M\rightarrow +\infty$, where $M$ denotes the dimension index of the orthogonal projection operator $P_M$ in the Galerkin scheme.

We consider d-dimensional stochastic differential equations (SDEs) of the form $\textrm{d}U_t = b(U_t)\,\textrm{d}t + \sigma\,\textrm{d}Z_t$. Let $X_t$ denote the solution if the driving noise $Z_t$ is a d-dimensional rotationally symmetric $\alpha$-stable process ($1\lt \alpha\lt 2$), and let $Y_t$ be the solution if the driving noise is a d-dimensional Brownian motion. Continuing the work started in Deng et al. (2025), we derive an estimate of the total variation distance $\|\textrm{law}(X_{t})-\textrm{law}(Y_{t})\|_\textrm{TV}$ for all $t \gt 0$, and we show that the ergodic measures $\mu_\alpha$ and $\mu_2$ of $X_t$ and $Y_t$, respectively, satisfy $\|\mu_\alpha-\mu_2\|_\textrm{TV} \leq {Cd\log(1+d)}(2-\alpha)/({\alpha-1})$. We show that this bound is optimal with respect to $\alpha$ by an Ornstein–Uhlenbeck SDE. Combining this bound with a recent interpolation result from Huang et al. (2023), we can derive a bound in the Wasserstein-p distance ($0 \lt p \lt 1$): $\|\mu_\alpha-\mu_2\|_{W_p} \leq {Cd^{(p+3)/2}\log(1+d)}(2-\alpha)/{\alpha-1}$.

We show that, under certain conditions, a strongly continuous semigroup admits an almost surely frequently hypercyclic random vector defined as a stochastic integral in Fréchet spaces with respect to the Brownian motion. Two criteria are given. We will apply the second criterion to three examples: translation semigroups on spaces of integrable functions, the exponential of weighted shifts, and the translation operators on the space of entire functions. This last example, with a stochastic approach, seems to be new in the literature. Some other examples are given.

Hybrid stochastic differential equations (SDEs) are a useful tool for modeling continuously varying stochastic systems modulated by a random environment, which may depend on the system state itself. In this paper we establish the pathwise convergence of solutions to hybrid SDEs using space-grid discretizations. Though time-grid discretizations are a classical approach for simulation purposes, our space-grid discretization provides a link with multi-regime Markov-modulated Brownian motions. This connection allows us to explore aspects that have been largely unexplored in the hybrid SDE literature. Specifically, we exploit our convergence result to obtain efficient and computationally tractable approximations for first-passage probabilities and expected occupation times of the solutions to hybrid SDEs. Lastly, we illustrate the effectiveness of the resulting approximations through numerical examples.

A general way to represent stochastic differential equations (SDEs) on smooth manifolds is based on the Schwartz morphism. In this manuscript, we are interested in SDEs on a smooth manifold $M$ that are driven by p-dimensional Wiener process $W_t \in \mathbb{R}^p$ and time $t$. In terms of the Schwartz morphism, such an SDE is represented by a Schwartz morphism that morphs the semimartingale $(t,W_t)\in\mathbb{R}^{p+1}$ into a semimartingale on the manifold $M$. We show that it is possible to construct such Schwartz morphisms using special maps that we call diffusion generators. We show that one of the ways to construct a diffusion generator is by considering the flow of differential equations. One particular case is the construction of diffusion generators using Lagrangian vector fields. Using the diffusion generator approach, we also give the extended Itô formula (also known as generalized Itô formula or Itô–Wentzell formula) for SDEs on manifolds.

We give a new proof of the singular continuity of Minkowski’s $?$-function. Our proof follows by showing that the maximal Lyapunov exponent of a specific pair of $3\times 3$ nonnegative integer matrices related to Stern’s diatomic sequence is strictly greater than $\log 2$.

We study a queueing system with a fixed number of parallel service stations of infinite servers, each having a dedicated arrival process, and one flexible arrival stream that is routed to one of the service stations according to a ‘weighted’ shortest queue policy. We consider the model with general arrival processes and general service time distributions. Assuming that the dedicated arrival rates are of order n and the flexible arrival rate is of order $\sqrt{n}$, we show that the diffusion-scaled queueing processes converge to a stochastic Volterra integral equation with ‘ranks’ driven by a continuous Gaussian process. It reduces to the limiting diffusion with a discontinuous drift in the Markovian setting.

We study the long time dynamic properties of the nonlocal Kuramoto–Sivashinsky (KS) equation with multiplicative white noise. First, we consider the dynamic properties of the stochastic nonlocal KS equation via a transformation into the associated conjugated random differential equation. Next, we prove the existence and uniqueness of solution for the conjugated random differential equation in the theory of random dynamical systems. We also establish the existence and uniqueness of a random attractor for the stochastic nonlocal equation.

We prove a scaling limit theorem for two-type Galton–Watson branching processes with interaction. The limit theorem gives rise to a class of mixed-state branching processes with interaction used to simulate evolution for cell division affected by parasites. Such processes can also be obtained by the pathwise-unique solution to a stochastic equation system. Moreover, we present sufficient conditions for extinction with probability 1 and the exponential ergodicity in the $L^1$-Wasserstein distance of such processes in some cases.

In this paper, we introduce a unified framework based on the pathwise expansion method to derive explicit recursive formulas for cumulative distribution functions, option prices, and transition densities in multivariate diffusion models. A key innovation of our approach is the introduction of the quasi-Lamperti transform, which normalizes the diffusion matrix at the initial time. This transformation facilitates expansions using uncorrelated Brownian motions, effectively reducing multivariate problems to one-dimensional computations. Consequently, both the analysis and the computation are significantly simplified. We also present two novel applications of the pathwise expansion method. Specifically, we employ the proposed framework to compute the value-at-risk for stock portfolios and to evaluate complex derivatives, such as forward-starting options. Our method has the flexibility to accommodate models with diverse features, including stochastic risk premiums, stochastic volatility, and nonaffine structures. Numerical experiments demonstrate the accuracy and computational efficiency of our approach. In addition, as a theoretical contribution, we establish an equivalence between the pathwise expansion method and the Hermite polynomial-based expansion method in the literature.