Showing papers in "Annals of Applied Probability in 2018"

Coexistence and extinction for stochastic Kolmogorov systems

Abstract: In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of $n$ populations that live in a stochastic environment and which can interact nonlinearly (through competition for resources, predator–prey behavior, etc.). Our models are described by $n$-dimensional Kolmogorov systems with white noise (stochastic differential equations—SDE). We give sharp conditions under which the populations converge exponentially fast to their unique stationary distribution as well as conditions under which some populations go extinct exponentially fast. The analysis is done by a careful study of the properties of the invariant measures of the process that are supported on the boundary of the domain. To our knowledge this is one of the first general results describing the asymptotic behavior of stochastic Kolmogorov systems in non-compact domains. We are able to fully describe the properties of many of the SDE that appear in the literature. In particular, we extend results on two dimensional Lotka-Volterra models, two dimensional predator–prey models, $n$ dimensional simple food chains, and two predator and one prey models. We also show how one can use our methods to classify the dynamics of any two-dimensional stochastic Kolmogorov system satisfying some mild assumptions.

Perfect hedging in rough Heston models

Abstract: Rough volatility models are known to reproduce the behavior of historical volatility data while at the same time fitting the volatility surface remarkably well, with very few parameters. However, managing the risks of derivatives under rough volatility can be intricate since the dynamics involve fractional Brownian motion. We show in this paper that surprisingly enough, explicit hedging strategies can be obtained in the case of rough Heston models. The replicating portfolios contain the underlying asset and the forward variance curve, and lead to perfect hedging (at least theoretically). From a probabilistic point of view, our study enables us to disentangle the infinite-dimensional Markovian structure associated to rough volatility models.

A random matrix approach to neural networks

Abstract: This article studies the Gram random matrix model $G=\frac{1}{T}\Sigma^{{\mathsf{T}}}\Sigma$, $\Sigma=\sigma(WX)$, classically found in the analysis of random feature maps and random neural networks, where $X=[x_{1},\ldots,x_{T}]\in\mathbb{R}^{p\times T}$ is a (data) matrix of bounded norm, $W\in\mathbb{R}^{n\times p}$ is a matrix of independent zero-mean unit variance entries and $\sigma:\mathbb{R}\to\mathbb{R}$ is a Lipschitz continuous (activation) function—$\sigma(WX)$ being understood entry-wise. By means of a key concentration of measure lemma arising from nonasymptotic random matrix arguments, we prove that, as $n,p,T$ grow large at the same rate, the resolvent $Q=(G+\gamma I_{T})^{-1}$, for $\gamma>0$, has a similar behavior as that met in sample covariance matrix models, involving notably the moment $\Phi=\frac{T}{n}{\mathrm{E}}[G]$, which provides in passing a deterministic equivalent for the empirical spectral measure of $G$. Application-wise, this result enables the estimation of the asymptotic performance of single-layer random neural networks. This in turn provides practical insights into the underlying mechanisms into play in random neural networks, entailing several unexpected consequences, as well as a fast practical means to tune the network hyperparameters.

The sample size required in importance sampling

Abstract: The goal of importance sampling is to estimate the expected value of a given function with respect to a probability measure $ u$ using a random sample of size $n$ drawn from a different probability measure $\mu$. If the two measures $\mu$ and $ u$ are nearly singular with respect to each other, which is often the case in practice, the sample size required for accurate estimation is large. In this article, it is shown that in a fairly general setting, a sample of size approximately $\exp(D( u\parallel\mu))$ is necessary and sufficient for accurate estimation by importance sampling, where $D( u\parallel\mu)$ is the Kullback–Leibler divergence of $\mu$ from $ u$. In particular, the required sample size exhibits a kind of cut-off in the logarithmic scale. The theory is applied to obtain a general formula for the sample size required in importance sampling for one-parameter exponential families (Gibbs measures).

Limit theorems for persistence diagrams

Abstract: The persistent homology of a stationary point process on $\mathbf{R}^{N}$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

$N$-player games and mean-field games with absorption

Abstract: We introduce a simple class of mean-field games with absorbing boundary over a finite time horizon. In the corresponding $N$-player games, the evolution of players’ states is described by a system of weakly interacting Ito equations with absorption on first exit from a bounded open set. Once a player exits, her/his contribution is removed from the empirical measure of the system. Players thus interact through a renormalized empirical measure. In the definition of solution to the mean-field game, the renormalization appears in form of a conditional law. We justify our definition of solution in the usual way, that is, by showing that a solution of the mean-field game induces approximate Nash equilibria for the $N$-player games with approximation error tending to zero as $N$ tends to infinity. This convergence is established provided the diffusion coefficient is nondegenerate. The degenerate case is more delicate and gives rise to counter-examples.

On the stability and the uniform propagation of chaos properties of Ensemble Kalman–Bucy filters

Abstract: The ensemble Kalman filter is a sophisticated and powerful data assimilation method for filtering high dimensional problems arising in fluid mechanics and geophysical sciences. This Monte Carlo method can be interpreted as a mean-field McKean–Vlasov-type particle interpretation of the Kalman–Bucy diffusions. In contrast to more conventional particle filters and nonlinear Markov processes, these models are designed in terms of a diffusion process with a diffusion matrix that depends on particle covariance matrices. Besides some recent advances on the stability of nonlinear Langevin-type diffusions with drift interactions, the long-time behaviour of models with interacting diffusion matrices and conditional distribution interaction functions has never been discussed in the literature. One of the main contributions of the article is to initiate the study of this new class of models. The article presents a series of new functional inequalities to quantify the stability of these nonlinear diffusion processes. In the same vein, despite some recent contributions on the convergence of the ensemble Kalman filter when the number of sample tends to infinity very little is known on stability and the long-time behaviour of these mean-field interacting type particle filters. The second contribution of this article is to provide uniform propagation of chaos properties as well as $\mathbb{L}_{n}$-mean error estimates w.r.t. to the time horizon. Our regularity condition is also shown to be sufficient and necessary for the uniform convergence of the ensemble Kalman filter. The stochastic analysis developed in this article is based on an original combination of functional inequalities and Foster–Lyapunov techniques with coupling, martingale techniques, random matrices and spectral analysis theory.

Large deviations theory for Markov jump models of chemical reaction networks

Abstract: We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it, we further establish the corresponding Wentzell–Freidlin (W-F) (infinite time horizon) asymptotic theory. These results apply to jump Markov processes that model the dynamics of chemical reaction networks under mass action kinetics, on a microscopic scale. We provide natural sufficient topological conditions for the applicability of our LDP and W-F results. This then justifies the computation of nonequilibrium potential and exponential transition time estimates between different attractors in the large volume limit, for systems that are beyond the reach of standard chemical reaction network theory.

Random cluster dynamics for the Ising model is rapidly mixing

Abstract: We show that the mixing time of Glauber (single edge update) dynamics for the random cluster model at $q=2$ on an arbitrary $n$-vertex graph is bounded by a polynomial in $n$. As a consequence, the Swendsen–Wang algorithm for the ferromagnetic Ising model at any temperature also has a polynomial mixing time bound.

Local inhomogeneous circular law

Abstract: We consider large random matrices $X$ with centered, independent entries, which have comparable but not necessarily identical variances. Girko’s circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et al. [Probab. Theory Related Fields 159 (2014) 545–595; Probab. Theory Related Fields 159 (2014) 619–660] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of $X$.

Tail measure and spectral tail process of regularly varying time series

Abstract: The goal of this paper is an exhaustive investigation of the link between the tail measure of a regularly varying time series and its spectral tail process, independently introduced in [Owada and Samorodnitsky (2012)] and [Stochastic Process. Appl. 119 (2009) 1055–1080]. Our main result is to prove in an abstract framework that there is a one-to-one correspondence between these two objects, and given one of them to show that it is always possible to build a time series of which it will be the tail measure or the spectral tail process. For nonnegative time series, we recover results explicitly or implicitly known in the theory of max-stable processes.

Mutation frequencies in a birth–death branching process

Abstract: First, we revisit the stochastic Luria–Delbruck model: a classic two-type branching process which describes cell proliferation and mutation. We prove limit theorems and exact results for the mutation times, clone sizes and number of mutants. Second, we extend the framework to consider mutations at multiple sites along the genome. The number of mutants in the two-type model characterises the mean site frequency spectrum in the multiple-site model. Our predictions are consistent with previously published cancer genomic data.

Zooming in on a lévy process at its supremum

Abstract: Let $M$ and $\tau$ be the supremum and its time of a Levy process $X$ on some finite time interval. It is shown that zooming in on $X$ at its supremum, that is, considering $((X_{\tau+t\varepsilon}-M)/a_{\varepsilon})_{t\in\mathbb{R}}$ as $\varepsilon\downarrow0$, results in $(\xi_{t})_{t\in\mathbb{R}}$ constructed from two independent processes having the laws of some self-similar Levy process $\widehat{X}$ conditioned to stay positive and negative. This holds when $X$ is in the domain of attraction of $\widehat{X}$ under the zooming-in procedure as opposed to the classical zooming out [Trans. Amer. Math. Soc. 104 (1962) 62–78]. As an application of this result, we establish a limit theorem for the discretization errors in simulation of supremum and its time, which extends the result in [Ann. Appl. Probab. 5 (1995) 875–896] for a linear Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a Levy process is provided.

Limit distributions for KPZ growth models with spatially homogeneous random initial conditions.

Abstract: For stationary KPZ growth in 1+1 dimensions, the height fluctuations are governed by the Baik–Rains distribution. Using the totally asymmetric single step growth model, alias TASEP, we investigate height fluctuations for a general class of spatially homogeneous random initial conditions. We prove that for TASEP there is a one-parameter family of limit distributions, labeled by the diffusion coefficient of the initial conditions. The distributions are defined through a variational formula. We use Monte Carlo simulations to obtain their numerical plots. Also discussed is the connection to the six-vertex model at its conical point.

A Liouville theorem for elliptic systems with degenerate ergodic coefficients

Abstract: We study the behavior of second-order degenerate elliptic systems in divergence form with random coefficients which are stationary and ergodic. Assuming moment bounds like Chiarini and Deuschel (2014) on the coefficient field $a$ and its inverse, we prove an intrinsic large-scale $C^{1,\alpha}$-regularity estimate for $a$-harmonic functions and obtain a first-order Liouville theorem for $a$-harmonic functions.

Stochastic approximation of quasi-stationary distributions on compact spaces and applications

Abstract: In the continuity of a recent paper ([6]), dealing with finite Markov chains, this paper proposes and analyzes a recursive algorithm for the approximation of the quasi-stationary distribution of a general Markov chain living on a compact metric space killed in finite time. The idea is to run the process until extinction and then to bring it back to life at a position randomly chosen according to the (possibly weighted) empirical occupation measure of its past positions. General conditions are given ensuring the convergence of this measure to the quasi-stationary distribution of the chain. We then apply this method to the numerical approximation of the quasi-stationary distribution of a diffusion process killed on the boundary of a compact set and to the estimation of the spectral gap of irreducible Markov processes. Finally, the sharpness of the assumptions is illustrated through the study of the algorithm in a non-irreducible setting.

Nash equilibria of threshold type for two-player nonzero-sum games of stopping

Abstract: This paper analyses two-player nonzero-sum games of optimal stopping on a class of linear regular diffusions with not nonsingular boundary behaviour [in the sense of Ito and McKean (Diffusion Processes and Their Sample Paths (1974) Springer, page 108)]. We provide sufficient conditions under which Nash equilibria are realised by each player stopping the diffusion at one of the two boundary points of an interval. The boundaries of this interval solve a system of algebraic equations. We also provide conditions sufficient for the uniqueness of the equilibrium in this class.

Duality and fixation in $\Xi$-Wright–Fisher processes with frequency-dependent selection

Abstract: A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of \emph{potential} parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using sampling- and moment-duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the population's ancestral process. The scaling limits are, respectively, a two-types $\Xi$-Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branching-coalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process' ergodic properties.

Stochastic Cucker–Smale models: Old and new

Abstract: In this paper we revisit and generalize various stochastic models extending the deterministic Cucker-Smale model for self organization. We study flocking and swarming properties. We show how these properties strongly depend on the structure and on the variance of the noise.

BSDEs with mean reflection

Abstract: In this paper, we study a new type of BSDE, where the distribution of the Y-component of the solution is required to satisfy an additional constraint, written in terms of the expectation of a loss function. This constraint is imposed at any deterministic time t and is typically weaker than the classical pointwise one associated to reflected BSDEs. Focusing on solutions (Y, Z, K) with deterministic K, we obtain the well-posedness of such equation, in the presence of a natural Skorokhod type condition. Such condition indeed ensures the minimality of the enhanced solution, under an additional structural condition on the driver. Our results extend to the more general framework where the constraint is written in terms of a static risk measure on Y. In particular, we provide an application to the super hedging of claims under running risk management constraint.

Spectral gap of random hyperbolic graphs and related parameters

Abstract: Millennium Nucleus Information and Coordination in Networks ICM/FIC P10-024F CONICYT via Basal in Applied Mathematics

Tracy–Widom fluctuations for perturbations of the log-gamma polymer in intermediate disorder

Abstract: The free-energy fluctuations of the discrete directed polymer in 1+1 dimensions is conjecturally in the Tracy-Widom universality class at all finite temperatures and in the intermediate disorder regime. Seppalainen's log-gamma polymer was proven to have GUE Tracy-Widom fluctuations in a restricted temperature range by Borodin et. al. (2013). We remove this restriction, and extend this result into the intermediate disorder regime. This result also identifies the scale of fluctuations of the log-gamma polymer in the intermediate disorder regime, and thus verifies a conjecture of Alberts et. al. (2010). Using a perturbation argument, we show that any polymer that matches a certain number of moments with the log-gamma polymer also has Tracy-Widom fluctuations in intermediate disorder.

A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

Abstract: In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form $\mathcal{Q}=TX(TX)^{*}$, where $X$ is an $M_{2}\times N$ random matrix with $X_{ij}=N^{-1/2}q_{ij}$ such that $q_{ij}$ are i.i.d. random variables with zero mean and unit variance, and $T$ is an $M_{1}\times M_{2}$ deterministic matrix such that $T^{*}T$ is diagonal. We study the asymptotic behavior of the largest eigenvalues of $\mathcal{Q}$ when $M:=\min\{M_{1},M_{2}\}$ and $N$ tend to infinity with $\lim_{N\to\infty}{N}/{M}=d\in(0,\infty)$. We prove that the Tracy–Widom law holds for the largest eigenvalue of $\mathcal{Q}$ if and only if $\lim_{s\rightarrow\infty}s^{4}\mathbb{P}(\vert q_{ij}\vert\geq s)=0$ under mild assumptions of $T$. The necessity and sufficiency of this condition for the edge universality was first proved for Wigner matrices by Lee and Yin [Duke Math. J. 163 (2014) 117–173].

Law of large numbers for the largest component in a hyperbolic model of complex networks

Abstract: We consider the component structure of a recent model of random graphs on the hyperbolic plane that was introduced by Krioukov et al. The model exhibits a power law degree sequence, small distances and clustering, features that are associated with so-called complex networks. The model is controlled by two parameters $\alpha$ and $ u$ where, roughly speaking, $\alpha$ controls the exponent of the power law and $ u$ controls the average degree. Refining earlier results, we are able to show a law of large numbers for the largest component. That is, we show that the fraction of points in the largest component tends in probability to a constant $c$ that depends only on $\alpha, u$, while all other components are sublinear. We also study how $c$ depends on $\alpha, u$. To deduce our results, we introduce a local approximation of the random graph by a continuum percolation model on $\mathbb{R}^{2}$ that may be of independent interest.

Hypoelliptic stochastic FitzHugh–Nagumo neuronal model: Mixing, up-crossing and estimation of the spike rate

Abstract: The FitzHugh–Nagumo is a well-known neuronal model that describes the generation of spikes at the intracellular level We study a stochastic version of the model from a probabilistic point of view The hypoellipticity is proved, as well as the existence and uniqueness of the stationary distribution The bi-dimensional stochastic process is $\beta$-mixing The stationary density can be estimated with an adaptive nonparametric estimator Then we focus on the distribution of the length between successive spikes Spikes are difficult to define directly from the continuous stochastic process We study the distribution of the number of up-crossings We link it to the stationary distribution and propose an estimator of its expectation We finally prove mathematically that the mean length of inter-up-crossings interval is equal to its up-crossings rate We illustrate the proposed estimators on a simulation study Different regimes are explored, with no, few or high generation of spikes

Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

Abstract: We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic topology It then turns out that the growth rate of the Betti numbers and the properties of the limiting processes all depend on the distance of the region of interest from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes We also derive the limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology

Backward SDEs for optimal control of partially observed path-dependent stochastic systems: A control randomization approach

Abstract: We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in mathematical finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality beyond the Markovian case, in particular our model may include path-dependence in the coefficients (both with respect to the state and the control), and does not require any nondegeneracy condition on the controlled equation. We use a randomization method, previously adopted only for cases of full observation, and consisting in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary (“randomized”) control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. In a second step, we prove that the latter can be associated by duality to a BSDE, which then characterizes the value of the original problem as well.

Volatility and arbitrage

Abstract: The capitalization-weighted cumulative variation d i=1 0 µi(t)d(log µi)(t) in an equity market consisting of a fixed number d of assets with capitalization weights µi(·) ; is an observable and a nondecreasing function of time. If this observable of the market is not just nondecreasing but actually grows at a rate bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.

An impossibility result for reconstruction in the degree-corrected stochastic block model

Abstract: We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on $n$ nodes, having i.i.d. weights $(\phi_{u})_{u=1}^{n}$ (possibly heavy-tailed), partitioned into $q\geq2$ asymptotically equal-sized clusters. The model parameters are two constants $a,b>0$ and the finite second moment of the weights $\Phi^{(2)}$. Vertices $u$ and $v$ are connected by an edge with probability $\frac{\phi_{u}\phi_{v}}{n}a$ when they are in the same class and with probability $\frac{\phi_{u}\phi_{v}}{n}b$ otherwise. We prove that it is information-theoretically impossible to estimate the clusters in a way positively correlated with the true community structure when $(a-b)^{2}\Phi^{(2)}\leq q(a+b)$. As by-products of our proof we obtain $(1)$ a precise coupling result for local neighbourhoods in DC-SBMs, that we use in Gulikers, Lelarge and Massoulie (2016) to establish a law of large numbers for local-functionals and $(2)$ that long-range interactions are weak in (power-law) DC-SBMs.

Multiple-priors optimal investment in discrete time for unbounded utility function

Abstract: This paper investigates the problem of maximizing expected terminal utility in a discrete-time financial market model with a finite horizon under nondominated model uncertainty. We use a dynamic programming framework together with measurable selection arguments to prove that under mild integrability conditions, an optimal portfolio exists for an unbounded utility function defined on the half-real line.