Shot Noise Process Research Articles

On some properties of shock processes in a ‘natural’ scale

We consider shocks modeling in a ‘natural’ scale which is a discrete scale of natural numbers. A system is subject to the shock process and its survival probability and other relevant characteristics are studied in this scale. It turns out that all relations for the probabilities of interest become much easier in the new scale as compared with the conventional chronological time scale. Furthermore, it does not matter what type of the point process of shocks is considered. The shock processes with delays and the analog of a shot-noise process are discussed. Another example of the application of this concept is presented for systems with finite number of components described by signatures.

Modeling the Dependency Structure of Integrated Intensity Processes.

This paper studies an important issue of dependence structure. To model this structure, the intensities within the Cox processes are driven by dependent shot noise processes, where jumps occur simultaneously and their sizes are correlated. The joint survival probability of the integrated intensities is explicitly obtained from the copula with exponential marginal distributions. Subsequently, this result can provide a very useful guide for credit risk management.

Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization

We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure). Our approach is based on an iteration of the classical Poincare inequality, as well as on the use of Malliavin operators, of Stein’s method, and of an (integrated) Mehler’s formula, providing a representation of the Ornstein-Uhlenbeck semigroup in terms of thinned Poisson processes. Our estimates only involve first and second order difference operators, and have consequently a clear geometric interpretation. In particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization, and with non-linear functionals of Poisson shot-noise processes. We discuss two examples of stabilizing functionals in great detail: (i) the edge length of the k-nearest neighbour graph, (ii) intrinsic volumes of k-faces of Voronoi tessellations. In all these examples we obtain rates of convergence (in the Kolmogorov and the Wasserstein distance) that one can reasonably conjecture to be optimal, thus significantly improving previous findings in the literature. As a necessary step in our analysis, we also derive new lower bounds for variances of Poisson functionals.

Modeling forest fires in Mazandaran Province, Iran

We used a spatio-temporal shot-noise Cox process to study the distribution of forest fires reported between 2006 and 2010 in the Mazandaran Province’s forests. The fitted model shows that daily temperature, altitude, and slope-exposure impacted fire occurrence. Forest fire occurred in the region had an aggregated behavior, which increased in radius below 1-km away from fired areas; a periodic pattern of fire occurrence in the region was verified. The risk of forest fire is significantly higher for areas with southern exposure and slope between 30° and 50°, northern exposure and slope between 0° and 50°, and eastern exposure and slope between 0° and 30°. The risk of fire was also significantly higher at altitudes between 1350 and 3000 m asl. Human causes were the main ignition source for forest fires in the region. The fire occurrence rate stayed above average during the drought period from September 2008 to September 2009. Our findings could lead to the development of fire-response and fire-suppression strategies appropriate to specific regions.

Optimal Investment in Markets with Over and Under-Reaction to Information

In this paper we introduce a new jump-diffusion model for stock prices, which takes into account over and under-reaction of the market to incoming news. The jumps' impact on the assets dynamics is twofold: on one hand we use a Poisson process as a driver to obtain discontinuous trajectories and on the other hand the presence of jumps in the drift, via a shot noise process, allows to incorporate fade-away'' effects, meaning that the effects of these abrupt changes fade away as time goes by. Our model is a partial information one: the drift direction after a jump is not accessible to standard investors immediately after the jump.We focus on a maximization of expected utility from terminal wealth problem, providing, in a logarithmic utility setting, the optimal investment strategy in explicit form, both under full (i.e., from the insider point of view) and under partial information (i.e., from the standard investor viewpoint). We test our results on real market data relative to Enron and Ahold.The three main contributions of this paper are: the introduction of a new market model dealing with over and under-reaction to news, the explicit computation of the optimal filter dynamics using an approach based on enlargement of filtrations and the application of the optimal portfolio allocation rule to real market data in both full and partial information setting.

Nonstationary filtered shot-noise processes and applications to neuronal membranes.

Filtered shot noise processes have proven to be very effective in modeling the evolution of systems exposed to shot noise sources and have been applied to a wide variety of fields ranging from electronics through biology. In particular, they can model the membrane potential V(m) of neurons driven by stochastic input, where these filtered processes are able to capture the nonstationary characteristics of V(m) fluctuations in response to presynaptic input with variable rate. In this paper we apply the general framework of Poisson point processes transformations to analyze these systems in the general case of nonstationary input rates. We obtain exact analytic expressions, as well as different approximations, for the joint cumulants of filtered shot noise processes with multiplicative noise. These general results are then applied to a model of neuronal membranes subject to conductance shot noise with a continuously variable rate of presynaptic spikes. We propose very effective approximations for the time evolution of the V(m) distribution and a simple method to estimate the presynaptic rate from a small number of V(m) traces. This work opens the perspective of obtaining analytic access to important statistical properties of conductance-based neuronal models such as the first passage time.

Large deviations for multidimensional state-dependent shot-noise processes

Large deviations for multidimensional state-dependent shot-noise processes

Nonparametric estimation of mark’s distribution of an exponential shot-noise process

In this paper, we consider a nonlinear inverse problem occurring in nuclear science. Gamma rays randomly hit a semiconductor detector which produces an impulse response of electric current. Because the sampling period of the measured current is larger than the mean inter arrival time of photons, the impulse responses associated to different gamma rays can overlap: this phenomenon is known as pileup. In this work, it is assumed that the impulse response is an exponentially decaying function. We propose a novel method to infer the distribution of gamma photon energies from the indirect measurements obtained from the detector. This technique is based on a formula linking the characteristic function of the photon density to a function involving the characteristic function and its derivative of the observations. We establish that our estimator converges to the mark density in uniform norm at a logarithmic rate. A limited Monte-Carlo experiment is provided to support our findings.

A Reduced-Form Model for Correlated Defaults with Regime-Switching Shot Noise Intensities

In this paper, we consider a two-dimensional reduced form contagion model with regime-switching interacting default intensities. The model assumes that the intensities of the default times are driven by macro-economy described by a homogenous Markov chain and that the default of one firm may trigger a positive jump, associated with the state of Markov chain, in the default intensity of the other firm. The intensities before the default of the other firm are modeled by a two-dimensional regime-switching shot noise process with common shocks. By using the idea of “change of measure” and some closed-form formulas for the joint conditional Laplace transforms of the regime-switching shot noise processes and the integrated regime-switching shot noise processes, we derive the two-dimensional conditional and unconditional joint distributions of the default times. Based on these results, we can express the single-name credit default swap (CDS) spread, the first and second-to-default CDS spreads on two underlyings in terms of fundamental matrix solutions of linear, matrix-valued, ordinary differential equations.

Regime-switching shot-noise processes and longevity bond pricing

In this paper, we consider the valuation of longevity bonds under a regime-switching interest rate and a regimeswitching force of mortality model. The model assumes that the interest rate is driven by economic and environmental conditions described by a homogenous Markov chain and that the stochastic force of mortality is modeled by the sum of a regime-switching Gompertz–Makeham model and a regime-switching shot-noise process. Using the conditional Laplace transform of the regime-switching shot-noise process, we give a formula for the longevity bond price in terms of a couple of system partial differential equations. The pricing formula is also derived by using the concept of stochastic flows and the idea of change of measure.

A Multivariate Regime-Switching Mean Reverting Process and Its Application to the Valuation of Credit Risk

In this article, we study the counterparty risk on a credit default swap (CDS) and the valuation of a first-to-default basket swap on three underlyings under a common shock model with regime-switching intensities. We assume that the defaults of all the names are driven by some shock events, whose arrivals are governed by a multivariate regime-switching shot noise process. Based on some expressions for the joint Laplace transform of the regime-switching shot noise processes, we give explicit formulas for the spread of the CDS contract with and without counterparty risk and the spread of the first-to-default basket swap on the three underlyings.

Serial Amplify-and-Forward Relay Transmission Systems in Nakagami- <inline-formula> <tex-math notation="TeX">$m$</tex-math></inline-formula> Fading Channels With a Poisson Interference Field

In this paper, the end-to-end performance of a wireless relay transmission system that employs amplify-and-forward (AF) relays and operates in an interference-limited Nakagami- m fading environment is studied. The wireless links from one relay node to another experience Nakagami- m fading, and the number of interferers per hop is Poisson distributed. The aggregate interference at each relay node is modeled as a shot-noise process whose distribution follows an α-stable process. For the considered system, analytical expressions for the moments of the end-to-end signal-to-interference ratio (SIR), the end-to-end outage probability (OP), the average bit-error probability (ABEP), and the average channel capacity are obtained. General asymptotic expressions for the end-to-end ABEP are also derived. The results provide useful insights regarding the factors affecting the performance of the considered system. Monte Carlo simulation results are further provided to demonstrate the validity of the proposed mathematical analysis.

Dual-Branch MRC Receivers Under Spatial Interference Correlation and Nakagami Fading

Despite being ubiquitous in practice, the performance of maximal-ratio combining (MRC) in the presence of interference is not well understood. Because the interference received at each antenna originates from the same set of interferers, but partially de-correlates over the fading channel, it possesses a complex correlation structure. This work develops a realistic analytic model that accurately accounts for the interference correlation using stochastic geometry. Modeling interference by a Poisson shot noise process with independent Nakagami fading, we derive the link success probability for dual-branch interference-aware MRC. Using this result, we show that the common assumption that all receive antennas experience equal interference power underestimates the true performance, although this gap rapidly decays with increasing the Nakagami parameter $m_{\text{I}}$ of the interfering links. In contrast, ignoring interference correlation leads to a highly optimistic performance estimate for MRC, especially for large $m_{\text{I}}$. In the low outage probability regime, our success probability expression can be considerably simplified. Observations following from the analysis include: (i) for small path loss exponents, MRC and minimum mean square error combining exhibit similar performance, and (ii) the gains of MRC over selection combining are smaller in the interference-limited case than in the well-studied noise-limited case.

Catastrophe Insurance Modeled by Shot-Noise Processes

Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. We give a general account of shot-noise processes with time-inhomogeneous drivers inspired by recent results in credit risk. Moreover, we derive a number of useful results for modeling and pricing with shot-noise processes. Besides this, we obtain some highly tractable examples and constitute a useful modeling tool for dynamic claims processes. The results can in particular be used for pricing Catastrophe Bonds (CAT bonds), a traded risk-linked security. Additionally, current results regarding the estimation of shot-noise processes are reviewed.

Limit theorems for renewal shot noise processes with eventually decreasing response functions

We consider shot noise processes (X(t))t≥0 with deterministic response function h and the shots occurring at the renewal epochs 0=S0<S1<S2⋯ of a zero-delayed renewal process. We prove convergence of the finite-dimensional distributions of (X(ut))u≥0 as t→∞ in different regimes. If the response function h is directly Riemann integrable, then the finite-dimensional distributions of (X(ut))u≥0 converge weakly as t→∞. Neither scaling nor centering are needed in this case. If the response function is eventually decreasing, non-integrable with an integrable power, then, after suitable shifting, the finite-dimensional distributions of the process converge. Again, no scaling is needed. In both cases, the limit is identified. If the distribution of S1 is in the domain of attraction of an α-stable law and the response function is regularly varying at ∞ with index −β (with 0≤β<1/α or 0≤β≤α, depending on whether ES1<∞ or ES1=∞), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of (X(ut))u≥0. The limits are fractionally integrated stable Lévy motions if ES1<∞ and fractionally integrated inverse stable subordinators if ES1=∞.

Asymptotic tail behavior of Poisson shot-noise processes with interdependence between shock and arrival time

Abstract This paper studies the tail behavior of the Poisson shot-noise processes with interdependent and heavy-tailed random shocks. In the presence of statistical dependence between the shock and its arrival time we establish the asymptotic behavior of the tail probability. Two examples are presented as illustrations of the main results as well.

The Distribution of Multiple Shot Noise Process and Its Integral

In this paper, we study multiple shot noise process and its integral. We analyse these two processes systematically for their theoretical distributions, based on the piecewise deterministic Markov process theory developed by Davis [1] and the martingale methodology used by Dassios and Jang [2]. The analytic expressions of the Laplace transforms of these two processes are presented. We also obtain the multivariate probability generating function for the number of jumps, for which we use a multivariate Cox process. To derive these, we assume that the Cox processes jumps, intensity jumps and primary event jumps are independent of each other. Using the Laplace transform of the integral of multiple shot noise process, we obtain the tail of multivariate distributions of the first jump times of the Cox processes, i.e. the multivariate survival functions. Their numerical calculations and other relevant joint distributions’ numerical values are also presented.

On the performance of asynchronous ad‐hoc networks using interference alignment with renewal process

The performance of ad-hoc networks is greatly affected by interference, particularly interference from nearby nodes. Based on stochastic geometry, this work studies a form of multiple-input-multiple-output (interference alignment (IA) that eliminates interference from transmitters within a range and treats the remaining interference as a shot noise process. Adapting to the bursty nature of ad-hoc network traffic, the authors introduce a novel distributed ad-hoc network with renewal process, whereby cooperation between interferer and receiver is implemented in one-way process, which results from the difficulties for the node to change the beamforming or receiving matrix whereas communicating with a proposed transmitter. In addition, this scheme takes advantage of IA, leaving the desired signal with higher degree of freedom (DoF) compared with the partially zero-forcing method. Outage probability and transmission capacity are derived with and without Rayleigh fading. Simulation results show the effect of IA in ad-hoc network, including the incremental in the DoF of the desired signal as time goes by. Monte Carlo simulations are implemented and show that IA outperforms successive interference cancellation by at least 10 dB in terms of the signal-to-interference ratio with large path-loss exponent.

Shot Noise analysis of subthreshold membrane potential activity in neurons

Shot noise is one of the most extensively harnessed phenomena used to model different aspects of neural activity in neuroscience. Shot noise processes [1] display very characteristic fluctuations that correspond to the sum of causal system responses to unitary event arrivals that follow a Poisson distribution. In general, applications of shot noise [2-4] have been confined to the so-called Campbell Limit corresponding to steady-state regime, or to specific post-synaptic responses that make the process Markovian or with the assumption of homogeneous Poisson event arrivals. In this work we go beyond this context to show that the shot-noise formalism can be used to analyze the synaptic activity of neurons beyond the Campbell limit, for any type of bounded post-synaptic response function and non-homogeneous Poisson event arrivals. This enhancement of technique allows for two main applications. 1) Computation of the full density for current synapses and statistical moments for conductance synapses. This yields a statistical description of the subthreshold membrane potential. This includes considering distributions on the afferent populations over biological parameters such as peak intensity, number of independent channels, or dynamical parameters such as distribution of incoming firing rate or variable incoming firing rate. The picture obtained is of particular interest for various applications such as calculating the transfer function. 2) Under stationarity assumptions, this technique offers the potential to reverse the moments of membrane potential integration. This offers a candidate method to infer characteristics of the synaptic activity (and therefore of the network activity) from the analysis of the subthreshold variations of the membrane potential. This application is of particular interest for the analysis of intracellular recordings in vivo. We illustrate these techniques by comparing analytic expressions with numerical estimates, as well as to neurons subject to a large range of synaptic activity (including low release rate far from the diffusion limit). In conclusion, due to their flexibility as a modeling tool, shot noise analysis appears as a very useful tool in computational neuroscience to model post-synaptic and sub-threshold membrane potential responses to incoming network activity, and also as a tool for electrophysiological analysis of network activity from intracellular recordings.

Functional limit theorems for renewal shot noise processes with increasing response functions

We consider renewal shot noise processes with response functions which are eventually nondecreasing and regularly varying at infinity. We prove weak convergence of renewal shot noise processes, properly normalized and centered, in the space D[0,∞) under the J1 or M1 topology. The limiting processes are either spectrally nonpositive stable Lévy processes, including the Brownian motion, or inverse stable subordinators (when the response function is slowly varying), or fractionally integrated stable processes or fractionally integrated inverse stable subordinators (when the index of regular variation is positive). The proof exploits fine properties of renewal processes, distributional properties of stable Lévy processes and the continuous mapping theorem.