Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

<abstract><p>We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the deterministic one, such as conditional, partial and asymptotic symmetries. A number of explicit examples are presented.</p></abstract>

Regularization of the backward stochastic heat conduction problem

In this paper, we study the backward problem for the stochastic parabolic heat equation driven by a Wiener process. We show that the problem is ill-posed by violating the continuous dependence on the input data. In order to restore stability, we apply a filter regularization method which is completely new in the stochastic setting. Convergence rates are established under different a priori assumptions on the sought solution.

Controlling a random population

Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial setting, and posed as an open problem the stochastic setting, where the agent is represented by a Markov decision process. In this paper, we show that the stochastic control problem is decidable. Our solution makes significant uses of well quasi orders, of the max-flow min-cut theorem, and of the theory of regular cost functions. We introduce an intermediate problem of independence interest called the sequential flow problem and study its complexity.

Computational characterization of recombinase circuits for periodic behaviors

In nature, recombinases are site-specific proteins capable of rearranging DNA, and they are expanding the repertoire of gene editing tools used in synthetic biology. The on/off response of recombinases, achieved by inverting the direction of a promoter, makes them suitable for Boolean logic computation; however, recombinase-based logic gate circuits are single-use due to the irreversibility of the DNA rearrangement, and it is still unclear how a dynamical circuit, such as an oscillator, could be engineered using recombinases. Preliminary work has demonstrated that recombinase- based circuits can yield periodic behaviors in a deterministic setting. However, since a few molecules of recombinase are enough to perform the inverting function, it is crucial to assess how the inherent stochasticity at low copy number affects the periodic behavior. Here, we propose six different circuit designs for recombinase-based oscillators. We model them in a stochastic setting, leveraging the Gillespie algorithm for extensive simulations, and we show that they can yield periodic behaviors. To evaluate the incoherence of oscillations, we use a metric based on the statistical properties of auto-correlation functions. The main core of our design consists of two self-inhibitory, recombinase-based modules coupled by a common promoter. Since each recombinase inverts its own promoter, the overall circuit can give rise to switching behavior characterized by a regular period. We introduce different molecular mechanisms (transcriptional regulation, degradation, sequestration) to tighten the control of recombinase levels, which slows down the response timescale of the system and thus improves the coherence of oscillations. Our results support the experimental realization of recombinase-based oscillators and, more generally, the use of recombinases to generate dynamic behaviors in synthetic biology.

The impact of noise and topology on opinion dynamics in social networks

We investigate the impact of noise and topology on opinion diversity in social networks. We do so by extending well-established models of opinion dynamics to a stochastic setting where agents are subject both to assimilative forces by their local social interactions, as well as to idiosyncratic factors preventing their population from reaching consensus. We model the latter to account for both scenarios where noise is entirely exogenous to peer influence and cases where it is instead endogenous, arising from the agents’ desire to maintain some uniqueness in their opinions. We derive a general analytical expression for opinion diversity, which holds for any network and depends on the network’s topology through its spectral properties alone. Using this expression, we find that opinion diversity decreases as communities and clusters are broken down. We test our predictions against data describing empirical influence networks between major news outlets and find that incorporating our measure in linear models for the sentiment expressed by such sources on a variety of topics yields a notable improvement in terms of explanatory power.

STOCHASTICITY AND COOPERATIVE HUNTING IN PREDATOR–PREY INTERACTIONS

We derive models of stochastic differential equations describing predator–prey interactions with cooperative hunting in predators based on a deterministic system proposed by Alves and Hilker. The deterministic model is analyzed first by providing a critical degree of cooperation below which the predators go extinct globally. Above the critical threshold, the deterministic model has two coexisting steady states and predators may persist depending on initial conditions. One of the stochastic models is derived from a continuous-time Markov chain while the other is based on a mean reverting process. Using Euler–Maruyama approximations, we investigate the stochastic systems numerically by providing estimated probabilities of predator extinction in the parameter regimes for which the predators cooperate intensively. It is found that predators may go extinct in the stochastic setting when they can otherwise survive indefinitely in the deterministic setting. The estimated probabilities of extinction are overall larger if populations are oscillating in the ODE system.

Almost conservation laws for stochastic nonlinear Schrödinger equations

In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the I-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on $${\mathbb {R}}^3$$ R 3 with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the I-method with Ito’s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.

Performance Analysis of Hybrid MTS/MTO Systems with Stochastic Demand and Production

We present a comprehensive numerical approach with reasonably light complexity in terms of implementation and computation for assessing the performance of hybrid make-to-stock (MTS)/make-to-order (MTO) systems. In such hybrid systems, semi-finished products are produced up front and stored in a decoupling inventory. When an order arrives, the products are completed and possibly customised. We study this system in a stochastic setting: demand and production are modelled by random processes. In particular, our model includes two coupled Markovian queues: one queue represents the decoupling inventory and the other the order backlog. These queues are coupled as order processing can only occur when both queues are non-empty. We rely on matrix analytic techniques to study the performance of the MTO/MTS system under non-restrictive stochastic assumptions. In particular, we allow for arrival correlation and non-exponential setup and MTS and MTO processing times, while the hybrid MTS/MTO system is managed by an (s,S)-type threshold policy that governs switching from MTO to MTS and back. By some numerical examples, we assess the impact of inventory control, irregular order arrivals, setup and order processing times on inventory levels and lead times.

Transboundary water allocation in critical scarcity conditions: a stochastic bankruptcy approach

A common problem in water resource allocation is to design a stable and feasible mechanism of water sharing in critical scarcity conditions. The task becomes very challenging when the water demand exceeds the available water resources reserves. To address this pervasive allocation problem related to transboundary rivers, the bankruptcy method is used. The bankruptcy method distributes water among riparian states when their total demand exceeds the total available water. This paper describes a new methodology for the allocation of scarce water resources in a complex system using a stochastic game theory which is an extension of bankruptcy theory. The authors have also proposed ‘weighted bankruptcy’ approach that can be used under a stochastic setting. The weighted bankruptcy approach favors agents with ‘high agricultural productivity’. The bankruptcy rules have been applied in the water resource system in four critical scarcity scenarios. The available water is allocated under the simple and weighted bankruptcy rules. The results showed that under all four scenarios, the weighted bankruptcy rules favor the agents which have a high agricultural productivity. The stochastic bankruptcy approach under the simple and the weighted bankruptcy rules can provide important strategic information for better management and sustainable sharing of water resources.

On the Max-Min Fair Stochastic Allocation of Indivisible Goods

We study the problem of fairly allocating a set of indivisible goods to risk-neutral agents in a stochastic setting. We propose an (approximation) algorithm to find a stochastic allocation that maximizes the minimum utility among the agents. The algorithm runs by repeatedly finding an (approximate) allocation to maximize the total virtual utility of the agents. This implies that the problem is solvable in polynomial time when the utilities are gross-substitutes (which is a subclass of submodular). When the utilities are submodular, we can find a (1 − 1/e)-approximate solution for the problem and this is best possible unless P=NP. We also extend the problem where a stochastic allocation must satisfy the (ex ante) envy-freeness. Under this condition, we demonstrate that the problem is NP-hard even when every agent has an additive utility with a matroid constraint (which is a subclass of gross-substitutes). Furthermore, we propose a polynomial-time algorithm for the setting with a restriction that the matroid constraint is common to all agents.