The paper defines and analyzes Poissonian occupation times for negative Lévy processes.
problem Analyzing the time spent below zero for Lévy processes with interruptions.
method Introduces Poissonian occupation times for spectrally negative Lévy processes.
result Extends results on continuous observation to interrupted observation.
New examples of variational bivectors found that are not Poissonian.
problem Identifying variational bivectors that are not Poissonian.
method Constructing examples of variational bivectors.
result Found examples of variational bivectors that are not Poissonian.
Existence of Q-processes for Brownian motion on hyperbolic spaces with Poissonian potentials shown.
problem Existence of path limits (Q-processes) for Brownian motion on hyperbolic spaces with Poissonian potentials.
method Analysis of stationary random potentials with spectral and sup norm bounds, and use of foliated space defined by the point process.
result Existence of Q-processes for Brownian motion on hyperbolic spaces with Poissonian potentials shown.
Proposes a new model to accurately describe random series of events.
problem Accurately and parsimoniously characterize random series of events (RSEs).
method Burstiness Scale (BuSca) model, which views RSEs as a mix of Poissonian and self-exciting processes.
result BuSca accurately describes RSEs in diverse systems, even with only two parameters.
We propose the point process model as the Poissonian-like stochastic sequence with slowly diffusing mean rate and adjust the parameters of the model to the empirical data of trading activity for 26 stocks traded on NYSE. The proposed scaled stochastic differential equation provides the universal description of the trad…
Study of bandit problem with Poisson decision times and Lévy processes.
problem Continuous-time multi-armed bandit problem with Poisson decision times.
method Gittins index policy applied to spectrally one-sided Lévy processes.
result Gittins index converges to classical Lévy bandit index.
Research examines correlations of complex logarithms of lattice points, showing level repulsion and Poissonian behavior.
problem Analyzing correlations of complex logarithms of lattice points.
method Proving existence of pair correlation functions and examining behavior at various scalings.
result Level repulsion observed at linear scaling, Poissonian behavior at sublinear scalings.
Study optimizes dividend strategies for risk processes with Lévy jumps.
problem Optimizing dividend payments in risk processes with Lévy jumps.
method Analyzes spectrally positive and negative Lévy processes, using scale functions.
result Periodic barrier strategy is optimal for spectrally negative Lévy processes with completely monotone Lévy density.
The study examines correlations of logarithms of integers at different scalings.
problem Analyzing pair correlations of logarithms of integers at various scalings.
method Examined correlations of logarithms of positive integers at different scalings, proving the existence of pair correlation functions.
result Level repulsion at linear scaling, total loss of mass at superlinear scalings, and Poissonian behavior at sublinear scalings.
We introduce a deterministic dealer model which implements most of the empirical laws, such as fat tails in the price change distributions, long term memory of volatility and non-Poissonian intervals. We also clarify the causality between microscopic dealers' dynamics and macroscopic market's empirical laws.
Abstract result on correlations of pairs in exponentially growing discrete subsets.
problem Pair correlations in exponentially growing discrete subsets with weight functions.
method Proved abstract result on correlations of pairs of elements in an exponentially growing discrete subset with a weight function.
result Distribution function of unscaled differences is t↦2δe−∣t∣, and pair correlation exhibits Poissonian behavior under certain conditions. We study a linear price impact model including other liquidity takers, whose flow of orders either follows a Poisson or a Hawkes process. The optimal execution problem is solved explicitly in this context, and the closed-formula optimal strategy describes in particular how one should react to the orders of other trader…
Motivated by the desire to bridge the gap between the microscopic description of price formation (agent-based modeling) and the stochastic differential equations approach used classically to describe price evolution at macroscopic time scales, we present a mathematical study of the order book as a multidimensional cont…
Optimizes hybrid dividend strategies in dual models with periodic and continuous payments.
problem Determining the best dividend strategy in a dual model with periodic and continuous payments.
method Generalizes results from a Brownian model to a dual (spectrally positive Lévy) model, using the scale function.
result The optimal strategy is of the hybrid-barrier type and can be expressed using the scale function.
Factor analysis improves PET image interpretation by considering non-standard noise distributions.
problem Improving interpretation of dynamic PET images with non-standard noise distributions.
method Proposes using β-divergence to fit factor models for different noise distributions. result Improves factor analysis results for various noise types in PET images.
Order cancellation process plays a crucial role in the dynamics of price formation in order-driven stock markets and is important in the construction and validation of computational finance models. Based on the order flow data of 18 liquid stocks traded on the Shenzhen Stock Exchange in 2003, we investigate the empiric…
The usual development of the continuous-time random walk (CTRW) proceeds by assuming that the present is one of the jumping times. Under this restrictive assumption integral equations for the propagator and mean escape times have been derived. We generalize these results to the case when the present is an arbitrary tim…
We study the activity, i.e., the number of transactions per unit time, of financial markets. Using the diffusion entropy technique we show that the autocorrelation of the activity is caused by the presence of peaks whose time distances are distributed following an asymptotic power law which ultimately recovers the Pois…
We present a simple microstructure model of financial returns that combines (i) the well-known ARFIMA process applied to tick-by-tick returns, (ii) the bid-ask bounce effect, (iii) the fat tail structure of the distribution of returns and (iv) the non-Poissonian statistics of inter-trade intervals. This model allows us…
This paper consists of two parts. The first part is devoted to empirical analysis of consolidated order book (COB) for the index RTS futures. In the second part we consider Poissonian multi--agent model of the COB. By varying parameters of different groups of agents submitting orders to the book we are able to model va…
Develops a method for solving FBSDEs with jumps.
problem Solving FBSDEs with jumps and random Poisson measures.
method Asymptotic expansion method for FBSDEs driven by Poisson and Brownian motions.
result Provides a semi-analytic solution technique with error estimate.
We present an empirical study of the subordination hypothesis for a stochastic time series of a stock price. The fluctuating rate of trading is identified with the stochastic variance of the stock price, as in the continuous-time random walk (CTRW) framework. The probability distribution of the stock price changes (log…
We explore the possibilities of importance sampling in the Monte Carlo pricing of a structured credit derivative referred to as Collateralized Debt Obligation (CDO). Modeling a CDO contract is challenging, since it depends on a pool of (typically about 100) assets, Monte Carlo simulations are often the only feasible ap…
In this paper we build a link between the Teichmuller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincare' uniformization. In the case of a one-sheeted hyperboloid with n orb…
Study on signal recovery from low-rank matrix with sparse noise.
problem Inference of a rank-one signal in the presence of sparse noise.
method Replica method from statistical physics, recursive distributional equations, population dynamics algorithm.
result Critical signal strength for recovery via top eigenvector identified.
Social, technological and economic time series are divided by events which are usually assumed to be random albeit with some hierarchical structure. It is well known that the interevent statistics observed in these contexts differs from the Poissonian profile by being long-tailed distributed with resting and active per…
New method improves sampling for weakly log-concave posteriors.
problem Sampling from weakly log-concave posterior distributions.
method Stochastic Langevin Monte Carlo with over-damped diffusion.
result Simulation horizon is (dlog(n)2)(1+r)2 with Poisson subsampling. Introduces a new class of hybrid processes combining Markov chains and Hawkes processes.
problem Characterize and ensure existence and uniqueness of complex hybrid marked point processes.
method Defines hybrid marked point processes implicitly via intensity and state process interactions, proving existence and uniqueness under general assumptions.
result Proves existence and uniqueness of hybrid marked point processes, extending existing results.
This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known Lévy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson proc…
A deep Neyman-Scott process uses Poisson processes for efficient inference in complex point processes.
problem Efficient inference in complex hierarchical point processes.
method Developed an efficient posterior sampling via Markov chain Monte Carlo for likelihood-based inference.
result More hidden Poisson processes improve likelihood fitting and event prediction.
The study examines Hawkes processes and their long-term behavior.
problem Understanding the long-term behavior of Hawkes processes.
method Proving functional limit theorems under various conditions on the dispersion of child events.
result Functional limit theorems hold for Hawkes processes with different levels of child event dispersion.
Paper discovers process models from online event streams.
problem Discovering process models from continuous event streams.
method Generic architecture for process discovery in event streams.
result The proposed architecture enables process discovery from event streams.
Elliptical processes generalize Gaussian and Student-t models with fat tails and computational efficiency.
problem Need for models with fat tails and computational tractability.
method Represent elliptical distributions as continuous mixtures of Gaussian distributions, derive closed-form expressions for marginal and conditional distributions.
result Elliptical processes offer advantages in robust regression compared to Gaussian processes.
Directly proves CRP from stick-breaking process without measure theory.
problem Indirect proof of CRP from stick-breaking process is complex.
method Direct proof using stick-breaking process to CRP, avoiding measure theory.
result Direct proof connects stick-breaking process to CRP.
We show that the stick-breaking construction of the beta process due to Paisley, et al. (2010) can be obtained from the characterization of the beta process as a Poisson process. Specifically, we show that the mean measure of the underlying Poisson process is equal to that of the beta process. We use this underlying re…
SNP extends Neural Processes to handle temporal dependencies in sequences.
problem Handling temporal dependencies in sequences of stochastic processes.
method Integrates a temporal state-transition model into Neural Processes.
result First 4D model capable of dynamic 3D scene modeling.
We investigate the Student-t process as an alternative to the Gaussian process as a nonparametric prior over functions. We derive closed form expressions for the marginal likelihood and predictive distribution of a Student-t process, by integrating away an inverse Wishart process prior over the covariance kernel of a G…
Efficient methods for Lévy models using SINH-regular processes.
problem Efficient numerical methods for evaluating Lévy models.
method Defining SL-processes and sSL-processes, deriving properties of characteristic exponent, and showing all popular Lévy processes can be subordinated to Brownian motion.
result All crucial properties of characteristic exponent are consequences of a specific representation, and all popular Lévy processes are SL- or sSL-subordinated Brownian motion.
GRM uses graph neural networks to score process activity relevance.
problem Improving business processes with performance measures.
method Graph Relevance Miner (GRM) based on graph neural networks.
result Quantitatively evaluated relevance scores with four datasets.
Researchers study the geometric properties of a specific type of stable processes.
problem Understanding the information geometry of tempered stable processes.
method Derivation of α-divergence, Fisher information matrices, and α-connections.
result Obtained Fisher information matrices and α-connections for statistical manifolds.
This study bridges discrete and continuous state spaces using the Ehrenfest process and diffusion models.
problem Understanding the relationship between discrete and continuous state spaces in stochastic processes.
method Investigates time-continuous Markov jump processes on discrete state spaces and their correspondence to state-continuous diffusion processes.
result The time-reversal of the Ehrenfest process converges to the time-reversed Ornstein-Uhlenbeck process, bridging discrete and continuous state spaces.
Paper proposes a new method for online process discovery.
problem Online process discovery requires limited memory.
method Mapped online process discovery to cache memory management and applied cache replacement policies.
result Implemented and evaluated a new approach for online process discovery.
Student's-T processes improve on Gaussian processes by handling outliers and variance more flexibly.
problem Outliers and variance limitations in Gaussian processes.
method Generalization of Gaussian processes using Student's-T distribution, with new kernel function and update rule.
result Student's-T processes provide better performance in Bayesian optimization, especially with outliers.
The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution. This process is very useful in several fields of applied and theoretical physics including models for anomalous diffusion. Contrary to the well-known Poiss…
New analysis shows differences in complexity between ergodic and nonergodic processes.
problem Understanding structural complexity in infinite-memory processes.
method Simple analysis of structural complexity using random samples of stationary, ergodic finite-memory component processes.
result Alternative view of predictability, complexity, and learning in nonergodic processes.
Proposes a new BSP-Tree process for flexible space partition modeling.
problem Limited modelling flexibility of axis-aligned partitions in Mondrian process.
method Introduces a self-consistent Binary Space Partitioning (BSP)-Tree process with oblique cuts.
result Clear inferential improvements over standard Mondrian process and related methods.
Elliptical processes extend Gaussian models with heavier tails.
problem Regression and classification with non-Gaussian likelihoods or heavy tails.
method Spline normalizing flow for variational inference of elliptical distributions.
result Elliptical processes outperform Gaussian processes in non-Gaussian settings.
Introduces Dirac processes for financial derivative pricing.
problem High implied volatility for CDS swaptions in hazard rate setups.
method Uses Dirac delta functions to add spikes to short-rate models.
result Dirac processes enable high implied volatility for CDS swaptions.