Sample Path Large Deviation Principle Research Articles

Queue length asymptotics for the multiple-server queue with heavy-tailed Weibull service times

We study the occurrence of large queue lengths in the GI / GI / d queue with heavy-tailed Weibull-type service times. Our analysis hinges on a recently developed sample path large-deviations principle for Lévy processes and random walks, following a continuous mapping approach. Also, we identify and solve a key variational problem which provides physical insight into the way a large queue length occurs. In contrast to the regularly varying case, we observe several subtle features such as a non-trivial trade-off between the number of big jobs and their sizes and a surprising asymmetric structure in asymptotic job sizes leading to congestion.

Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions

In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models.

Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions

In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncorrelated Gaussian model grows faster than linearly, then, for the asset price process, all the moments of order greater than one are infinite. Similar moment explosion results are obtained for correlated models.

Sample Path Large Deviations for Stochastic Evolutionary Game Dynamics

We study a model of stochastic evolutionary game dynamics in which the probabilities that agents choose suboptimal actions are dependent on payoff consequences. We prove a sample path large deviation principle, characterizing the rate of decay of the probability that the sample path of the evolutionary process lies in a prespecified set as the population size approaches infinity. We use these results to describe excursion rates and stationary distribution asymptotics in settings where the mean dynamic admits a globally attracting state, and we compute these rates explicitly for the case of logit choice in potential games.

Large deviations theory for Markov jump models of chemical reaction networks

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.

Exit time asymptotics for small noise stochastic delay differential equations

Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE). We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for other infinite-dimensional small noise stochastic equations.

Large Deviations for Brownian Motion on Scale Irregular Sierpinski Gaskets

We study sample path large deviation principles for Brownian motion on scale irregular Sierpinski gaskets which are spatially homogeneous but do not have any exact self-similarity. One notable point of our study is that the rate function depends on a large deviation parameter and as such, we can only obtain an example of large deviations in an incomplete form. Instead of showing the large deviations principle we would expect to hold true, we show Varadhan’s integral lemma and exponential tightness by using an incomplete version of such large deviations.

Large deviations for product of sums of random variables

Abstract In this paper, we obtain sample path and scalar large deviation principles for the product of sums of positive random variables. We study the case when the positive random variables are independent and identically distributed and bounded away from zero or the left tail decays to zero sufficiently fast. The explicit formula for the rate function of a scalar large deviation principle is given in the case when random variables are exponentially distributed.

Large deviations for affine diffusion processes on [formula omitted

This paper proves the large deviation principle for affine diffusion processes with initial values in the interior of the state space R+m×Rn. We approach this problem in two different ways. In the first approach, we first prove the large deviation principle for finite dimensional distributions, and then use it to establish the sample path large deviation principle. For this approach, a more careful examination of the affine transform formula is required. The second approach exploits the exponential martingale method of Donati-Martin et al. for the squares of Ornstein–Uhlenbeck processes. We provide an application to importance sampling of affine diffusion models.

Exponential martingale and large deviations for a Cox risk process with Poisson shot noise intensity

We introduce an exponential martingale for a Cox risk process with Poisson shot noise intensity. As applications of the exponential martingale, we obtain the sample path large and moderate deviation principles. Finally, we give an estimate of ruin probability.

Large deviations for a feed-forward network

We consider a feed-forward network with a single-server station serving jobs with multiple levels of priority. The service discipline is preemptive in that the server always serves a job with the current highest level of priority. For this system with discontinuous dynamics, we establish the sample path large deviation principle using a weak convergence argument. In the special case where jobs have two different levels of priority, we also explicitly identify the exponential decay rate of the total population overflow probabilities by examining the geometry of the zero-level sets of the system Hamiltonians.

Large deviations for a feed-forward network

We consider a feed-forward network with a single-server station serving jobs with multiple levels of priority. The service discipline is preemptive in that the server always serves a job with the current highest level of priority. For this system with discontinuous dynamics, we establish the sample path large deviation principle using a weak convergence argument. In the special case where jobs have two different levels of priority, we also explicitly identify the exponential decay rate of the total population overflow probabilities by examining the geometry of the zero-level sets of the system Hamiltonians.

Large deviation principles for sequences of logarithmically weighted means

In this paper we consider several examples of sequences of partial sums of triangular arrays of random variables { X n : n ⩾ 1 } ; in each case X n converges weakly to an infinitely divisible distribution (a Poisson distribution or a centered Normal distribution). For each sequence we prove large deviation results for the logarithmically weighted means { 1 log n ∑ k = 1 n 1 k X k : n ⩾ 1 } with speed function v n = log n . We also prove a sample path large deviation principle for { X n : n ⩾ 1 } defined by X n ( ⋅ ) = ∑ i = 1 n U i ( σ 2 ⋅ ) n , where σ 2 ∈ ( 0 , ∞ ) and { U n : n ⩾ 1 } is a sequence of independent standard Brownian motions.

On the large deviations of a class of modulated additive processes

We prove that the large deviation principle holds for a class of processes inspired by semi-Markov additive processes. For the processes we consider, the sojourn times in the phase process need not be independent and identically distributed. Moreover the state selection process need not be independent of the sojourn times. We assume that the phase process takes values in a finite set and that the order in which elements in the set, called states, are visited is selected stochastically. The sojourn times determine how long the phase process spends in a state once it has been selected. The main tool is a representation formula for the sample paths of the empirical laws of the phase process. Then, based on assumed joint large deviation behavior of the state selection and sojourn processes, we prove that the empirical laws of the phase process satisfy a sample path large deviation principle. From this large deviation principle, the large deviations behavior of a class of modulated additive processes is deduced. As an illustration of the utility of the general results, we provide an alternate proof of results for modulated Levy processes. As a practical application of the results, we calculate the large deviation rate function for a processes that arises as the International Telecommunications Union's standardized stochastic model of two-way conversational speech.

Most likely paths to error when estimating the mean of a reflected random walk

It is known that simulation of the mean position of a Reflected Random Walk (RRW) { W n } exhibits non-standard behavior, even for light-tailed increment distributions with negative drift. The Large Deviation Principle (LDP) holds for deviations below the mean, but for deviations at the usual speed above the mean the rate function is null. This paper takes a deeper look at this phenomenon. Conditional on a large sample mean, a complete sample path LDP analysis is obtained. Let I denote the rate function for the one dimensional increment process. If I is coercive, then given a large simulated mean position, under general conditions our results imply that the most likely asymptotic behavior, ψ ∗ , of the paths n − 1 W ⌊ t n ⌋ is to be zero apart from on an interval [ T 0 , T 1 ] ⊂ [ 0 , 1 ] and to satisfy the functional equation ∇ I ( d d t ψ ∗ ( t ) ) = λ ∗ ( T 1 − t ) whenever ψ ( t ) ≠ 0 . If I is non-coercive, a similar, but slightly more involved, result holds. These results prove, in broad generality, that Monte Carlo estimates of the steady-state mean position of a RRW have a high likelihood of over-estimation. This has serious implications for the performance evaluation of queueing systems by simulation techniques where steady state expected queue-length and waiting time are key performance metrics. The results show that naïve estimates of these quantities from simulation are highly likely to be conservative.

Sample path Large Deviations and optimal importance sampling for stochastic volatility models

Sample path Large Deviation Principles (LDP) of the Freidlin–Wentzell type are derived for a class of diffusions, which govern the price dynamics in common stochastic volatility models from Mathematical Finance. LDP are obtained by relaxing the non-degeneracy requirement on the diffusion matrix in the standard theory of Freidlin and Wentzell. As an application, a sample path LDP is proved for the price process in the Heston stochastic volatility model. Using the sample path LDP for the Heston model, the problem is considered of selecting an importance sampling change of drift, for both the price and volatility, which minimize the variance of Monte Carlo estimators for path-dependent option prices. An asymptotically optimal change of drift is identified as a solution to a two-dimensional variational problem. The case of the arithmetic average Asian put option is solved in detail.

Sample path large and moderate deviations for risk model with delayed claims

Sample path large and moderate deviation principles for Markov modulated risk models with delayed claims are proved by the exponential martingale method. As applications, asymptotic estimates and exponential bounds of the ruin probability are also studied.

Functional continuity and large deviations for the behavior of single-class queueing networks

We consider a single-class queueing network in which the functional network primitives describe the cumulative exogenous arrivals, service times and routing decisions of the queues. The behavior of the network consisting of the cumulative total arrival, cumulative idle time, and queue length developments for each node is specified by conditions which relate the network primitives to the network behavior. For a broad class of network primitives, including discrete customer and fluid models, a network behavior exists, but need not be unique. Nevertheless, the mapping from network primitives to the set of associated network behavior is upper semicontinuous at network primitives with continuous routing. As an application we consider a sequence of random network primitives satisfying a sample path large deviation principle. We take advantage of the partial functional set-valued upper semicontinuity in order to derive a large deviation principle for the sequence of associated random queue length processes and to identify the rate function. This extends the results of Puhalskii (Markov Process. Relat. Fields 13(1), 99---136, 2007) about large deviations for the tail probabilities of generalized Jackson networks. Since the analysis is carried out on the doubly-infinite time axis ?, we can directly treat stationary situations.

Large Deviations of Queues Sharing a Randomly Time-Varying Server

We consider a discrete-time model where multiple queues, each with its own exogenous arrival process, are served by a server whose capacity varies randomly and asynchronously with respect to different queues. This model is primarily motivated by the problem of efficient scheduling of transmissions of multiple data flows sharing a wireless channel. We address the following problem of controlling large deviations of the queues: find a scheduling rule, which is optimal in the sense of maximizing 0.1 $$\min_{i}\biggl[\lim_{n\to\infty}\frac{-1}{n}\log P(a_{i}Q_{i}>n)\biggr],$$ where Q i is the length of the i-th queue in a stationary regime, and a i >0 are parameters. Thus, we seek to maximize the minimum of the exponential decay rates of the tails of distributions of weighted queue lengths a i Q i . We give a characterization of the upper bound on (0.1) under any scheduling rule, and of the lower bound on (0.1) under the exponential (EXP) rule. We prove that the two bounds match, thus proving optimality of the EXP rule. The EXP rule is very parsimonious in that it does not require any "pre-computation" of its parameters, and uses only current state of the queues and of the server. The EXP rule is not invariant with respect to scaling of the queues, which complicates its analysis in the large deviations regime. To overcome this, we introduce and prove a refined sample path large deviations principle, or refined Mogulskii theorem, which is of independent interest.

Large Deviations of Poisson Cluster Processes

In this paper we prove scalar and sample path large deviation principles for a large class of Poisson cluster processes. As a consequence, we provide a large deviation principle for ergodic Hawkes point processes.