Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme

We consider the long-time behavior of an explicit tamed exponential Euler scheme applied to a class of parabolic semilinear stochastic partial differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process. We justify the efficiency of using the explicit tamed exponential Euler scheme to approximate the invariant distribution, since the computational cost does not suffer from the at most polynomial growth of the moment bounds. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for SPDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.

Identifying Present Bias from the Timing of Choices

A (partially naïve) quasi-hyperbolic discounter repeatedly chooses whether to complete a task. Her net benefits of task completion are drawn independently between periods from a time-invariant distribution. We show that the probability of completing the task conditional on not having done so earlier increases towards the deadline. Conversely, we establish nonidentifiability by proving that for any time-preference parameters and any dataset with such (weakly increasing) task-completion probabilities, there exists a stationary payoff distribution that rationalizes the agent’s behavior if she is either sophisticated or fully naïve. Additionally, we provide sharp partial identification for the case of observable continuation values. (JEL C14, D11, D15, D90, D91)

Approximation of the invariant distribution for a class of ergodic jump diffusions

In this article, we approximate the invariant distribution ν of an ergodic Jump Diffusion driven by the sum of a Brownian motion and a Compound Poisson process with sub-Gaussian jumps. We first construct an Euler discretization scheme with decreasing time steps. This scheme is similar to those introduced in Lamberton and Pagès Bernoulli 8 (2002) 367-405. for a Brownian diffusion and extended in F. Panloup, Ann. Appl. Probab. 18 (2008) 379-426. to a diffusion with Lévy jumps. We obtain a non-asymptotic quasi Gaussian (asymptotically Gaussian) concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along appropriate test functions f such that f − ν(f) is a coboundary of the infinitesimal generator.

Invariant measures of interacting particle systems: Algebraic aspects

Consider a continuous time particle system ηt = (ηt(k), k ∈ 𝕃), indexed by a lattice 𝕃 which will be either ℤ, ℤ∕nℤ, a segment {1, ⋯ , n}, or ℤd, and taking its values in the set Eκ𝕃 where Eκ = {0, ⋯ , κ − 1} for some fixed κ ∈{∞, 2, 3, ⋯ }. Assume that the Markovian evolution of the particle system (PS) is driven by some translation invariant local dynamics with bounded range, encoded by a jump rate matrix ⊤. These are standard settings, satisfied by the TASEP, the voter models, the contact processes. The aim of this paper is to provide some sufficient and/or necessary conditions on the matrix ⊤ so that this Markov process admits some simple invariant distribution, as a product measure (if 𝕃 is any of the spaces mentioned above), the law of a Markov process indexed by ℤ or [1, n] ∩ ℤ (if 𝕃 = ℤ or {1, …, n}), or a Gibbs measure if 𝕃 = ℤ/nℤ. Multiple applications follow: efficient ways to find invariant Markov laws for a given jump rate matrix or to prove that none exists. The voter models and the contact processes are shown not to possess any Markov laws as invariant distribution (for any memory m). (As usual, a random process X indexed by ℤ or ℕ is said to be a Markov chain with memory m ∈ {0, 1, 2, ⋯ } if ℙ(Xk ∈ A | Xk−i, i ≥ 1) = ℙ(Xk ∈ A | Xk−i, 1 ≤ i ≤ m), for any k.) We also prove that some models close to these models do. We exhibit PS admitting hidden Markov chains as invariant distribution and design many PS on ℤ2, with jump rates indexed by 2 × 2 squares, admitting product invariant measures.

Physics textbooks and its network structures

We can observe self-organised networks all around us. These networks are, in general, scale-invariant networks described by the Barabasi-Albert model. The self-organised networks show certain universalities. These networks, in simplified models, have scale-invariant distribution (power law distribution) and the characteristic parameter α of the distribution has value between 2 and 5. Textbooks are an essential part of the learning process; therefore, we analysed the curriculum in secondary school textbooks of physics from the viewpoint of semantic network structures. We converted the textbook into a tripartite network, where the nodes represented sentences, terms and formulae. We found the same distribution as for self-organised networks. Cluster analysis was applied on the resulting network and we found individual modules—clusters. We obtained nine clusters, three of which were significantly larger. These clusters presented kinematics of point mass, dynamics of point mass and gravitational field with electric field. Keywords: Physics textbook, scale-invariant distribution, semantic network.

Recursive computation of the invariant distributions of Feller processes: Revisited examples and new applications

In this paper, we show that the abstract framework developed in [G. Pagès and C. Rey, Recursive computation of the invariant distribution of Markov and Feller processes, preprint 2017, https://arxiv.org/abs/1703.04557] and inspired by [D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion, Bernoulli 8 2002, 3, 367–405] can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean-reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth.