The truncated EM method for stochastic differential equations with Poisson jumps

In this paper, we use the truncated Euler–Maruyama (EM) method to study the finite time strong convergence for SDEs with Poisson jumps under the Khasminskii-type condition. We establish the finite time L r (r≥2)-convergence order when the drift and diffusion coefficients satisfy the super-linear growth condition and the jump coefficient satisfies the linear growth condition. The result shows that the optimal L r - convergence order is close to 1. This is significantly different from the result on SDEs without jumps. When all the three coefficients of SDEs are allowing to grow super-linearly, the L r (0<r<2)-convergence results are also investigated and the optimal L r - convergence order is shown to be not greater than 1∕4. Moreover, we prove that the truncated EM method preserves nicely the mean square exponential stability and asymptotic boundedness of the underlying SDEs with Poisson jumps. Several examples are given to illustrate our results

Convergence of the split-step θ-method for stochastic age-dependent population equations with Markovian switching and variable delay

We present a stochastic age-dependent population model that accounts for Markovian switching and variable delay. By using the approximate value at the nearest grid-point on the left of the delayed argument to estimate the delay function, we propose a class of split-step θ -method for solving stochastic delay age-dependent population equations (SDAPEs) with Markovian switch- ing. We show that the numerical method is convergent under the given conditions. Numerical examples are provided to illustrate our results

Mean-square Stability and Convergence of Compensated Split-Step thetatheta-method for Nonlinear Jump Diffusion Systems

In this paper, we analyze the strong convergence and stability of the Compensated Splite-step $theta$ (CSS$theta$) and Forward-Backward Euler-Maruyama (FBEM) methods for Numerical solutions of Stochastic Differential Equations with jumps (SDEwJs),where ‎$sqrt{2}-1leqthetaleq 1‎$. The drift term $f$ has a one-sided Lipschitz condition, the diffusion term $g$ and jump term $h$ satisfy global Lipschitz condition. Furthermore, we discuss about the stability of SDEwJs with constant coefficients and present new useful relations between their coefficients. Finally we examine the correctness and efficiency of theorems with some examples.In this paper, we analyze the strong convergence and stability of the Compensated Splite-step $theta$ (CSS$theta$) and Forward-Backward Euler-Maruyama (FBEM) methods for Numerical solutions of Stochastic Differential Equations with jumps (SDEwJs),where ‎$sqrt{2}-1leqthetaleq 1‎$. The drift term $f$ has a one-sided Lipschitz condition, the diffusion term $g$ and jump term $h$ satisfy global Lipschitz condition. Furthermore, we discuss about the stability of SDEwJs with constant coefficients and present new useful relations between their coefficients. Finally we examine the correctness and efficiency of theorems with some examples

Environmental management and restoration under unified risk and uncertainty using robustified dynamic Orlicz risk

Environmental management and restoration should be designed such that the risk and uncertainty owing to nonlinear stochastic systems can be successfully addressed. We apply the robustified dynamic Orlicz risk to the modeling and analysis of environmental management and restoration to consider both the risk and uncertainty within a unified theory. We focus on the control of a jump-driven hybrid stochastic system that represents macrophyte dynamics. The dynamic programming equation based on the Orlicz risk is first obtained heuristically, from which the associated Hamilton-Jacobi-Bellman (HJB) equation is derived. In the proposed Orlicz risk, the risk aversion of the decision-maker is represented by a power coefficient that resembles a certainty equivalence, whereas the uncertainty aversion is represented by the Kullback-Leibler divergence, in which the risk and uncertainty are handled consistently and separately. The HJB equation includes a new state-dependent discount factor that arises from the uncertainty aversion, which leads to a unique, nonlinear, and nonlocal term. The link between the proposed and classical stochastic control problems is discussed with a focus on control-dependent discount rates. We propose a finite difference method for computing the HJB equation. Finally, the proposed model is applied to an optimal harvesting problem for macrophytes in a brackish lake that contains both growing and drifting populations

Taming neuronal noise with large networks

How does reliable computation emerge from networks of noisy neurons? While individual neurons are intrinsically noisy, the collective dynamics of populations of neurons taken as a whole can be almost deterministic, supporting the hypothesis that, in the brain, computation takes place at the level of neuronal populations. Mathematical models of networks of noisy spiking neurons allow us to study the effects of neuronal noise on the dynamics of large networks. Classical mean-field models, i.e., models where all neurons are identical and where each neuron receives the average spike activity of the other neurons, offer toy examples where neuronal noise is absorbed in large networks, that is, large networks behave like deterministic systems. In particular, the dynamics of these large networks can be described by deterministic neuronal population equations. In this thesis, I first generalize classical mean-field limit proofs to a broad class of spiking neuron models that can exhibit spike-frequency adaptation and short-term synaptic plasticity, in addition to refractoriness. The mean-field limit can be exactly described by a multidimensional partial differential equation; the long time behavior of which can be rigorously studied using deterministic methods. Then, we show that there is a conceptual link between mean-field models for networks of spiking neurons and latent variable models used for the analysis of multi-neuronal recordings. More specifically, we use a recently proposed finite-size neuronal population equation, which we first mathematically clarify, to design a tractable Expectation-Maximization-type algorithm capable of inferring the latent population activities of multi-population spiking neural networks from the spike activity of a few visible neurons only, illustrating the idea that latent variable models can be seen as partially observed mean-field models. In classical mean-field models, neurons in large networks behave like independent, identically distributed processes driven by the average population activity -- a deterministic quantity, by the law of large numbers. The fact the neurons are identically distributed processes implies a form of redundancy that has not been observed in the cortex and which seems biologically implausible. To show, numerically, that the redundancy present in classical mean-field models is unnecessary for neuronal noise absorption in large networks, I construct a disordered network model where networks of spiking neurons behave like deterministic rate networks, despite the absence of redundancy. This last result suggests that the concentration of measure phenomenon, which generalizes the ``law of large numbers'' of classical mean-field models, might be an instrumental principle for understanding the emergence of noise-robust population dynamics in large networks of noisy neurons

Large-scale games in large-scale systems

Many real-world problems modeled by stochastic games have huge state and/or action spaces, leading to the well-known curse of dimensionality. The complexity of the analysis of large-scale systems is dramatically reduced by exploiting mean field limit and dynamical system viewpoints. Under regularity assumptions and specific time-scaling techniques, the evolution of the mean field limit can be expressed in terms of deterministic or stochastic equation or inclusion (difference or differential). In this paper, we overview recent advances of large-scale games in large-scale systems. We focus in particular on population games, stochastic population games and mean field stochastic games. Considering long-term payoffs, we characterize the mean field systems using Bellman and Kolmogorov forward equations.Comment: 30 pages. Notes for the tutorial course on mean field stochastic games, March 201