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…
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48 results for “compound Poisson process”
Study shows convergence rates for BSDEs approximated by compound Poisson processes.
problem Analyzing convergence rates of BSDEs driven by Lévy processes.
method Approximating Lévy processes by compound Poisson processes and studying BSDEs.
result Optimal convergence rates derived for BSDEs in -norm and Wasserstein distance.
Characterizes measures preserving compound mixed renewal process properties.
problem Preserving compound mixed renewal process properties under different probability measures.
method Characterization of progressively equivalent probability measures.
result Any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures.
A model for the phenomenological description of tick-by-tick share prices in a stock exchange is introduced. It is based on mixtures of compound Poisson processes. Preliminary results based on Monte Carlo simulation show that this model can reproduce various stylized facts.
We introduce an algorithm for the segmentation of a class of regime switching processes. The segmentation algorithm is a non parametric statistical method able to identify the regimes (patches) of the time series. The process is composed of consecutive patches of variable length, each patch being described by a station…
In this paper we discuss Bayesian nonconvex penalization for sparse learning problems. We explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one-dimensional Lévy processes. We particularly study a family of continuous compound Poisson subordinators and a family of discrete…
We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied bet…
Study parameter sensitivities in bond pricing models with jumps.
problem Analyzing the impact of parameters on bond pricing models with jumps.
method Theoretical analysis and MATLAB simulations of a Brownian motion and compound Poisson process.
result Explicit call price formula and verification of sensitivities.
In this note we study the optimal dividend problem for a company whose surplus process, in the absence of dividend payments, evolves as a generalized compound Poisson model in which the counting process is a generalized Poisson process. This model including the classical risk model and the Polya-Aeppli risk model as sp…
In this paper, we introduce a new model for the risk process based on general compound Hawkes process (GCHP) for the arrival of claims. We call it risk model based on general compound Hawkes process (RMGCHP). The Law of Large Numbers (LLN) and the Functional Central Limit Theorem (FCLT) are proved. We also study the ma…
A method is developed to estimate the parameters of a Levy copula of a discretely observed bivariate compound Poisson process without knowledge of common shocks. The method is tested in a small sample simulation study. Also, the method is applied to a real data set and a goodness of fit test is developed. With the meth…
We present a machine learning model for the analysis of randomly generated discrete signals, modeled as the points of an inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by Mallat, our construction is naturally invariant to translations and reflections, but it decouples th…
Stressing Dynamic Loss Modelsq-fin.RM
Proposes a reverse stress testing framework for dynamic models.
problem Finding plausible models under adverse stresses.
method Compound Poisson process, Kullback-Leibler divergence, optimization problem.
result Intensity and severity of process depend on time and state.
In this paper we assume the insurance wealth process is driven by the compound Poisson process. The discounting factor is modelled as a geometric Brownian motion at first and then as an exponential function of an integrated Ornstein-Uhlenbeck process. The objective is to maximize the cumulated value of expected discoun…
Modeling dependent defaults with multivariate Cox processes.
problem Capturing dependence in default times.
method Multivariate generalized Cox process with càdlàg, increasing processes.
result Closed-form expressions for joint survival probabilities.
Study shows subordinated Cramér-Lundberg model increases ruin probability.
problem Analyzing the impact of subordinated time-changed claims on insurance ruin probability.
method Examined a compound Poisson process modified by a Lévy subordinator.
result Probability of ruin decreases slowly with initial capital, despite unchanged total claim amount.
In this paper we examine the claims reserving problem using Tweedie's compound Poisson model. We develop the maximum likelihood and Bayesian Markov chain Monte Carlo simulation approaches to fit the model and then compare the estimated models under different scenarios. The key point we demonstrate relates to the compar…
Mack's estimator improves chain ladder prediction for large exposure insurance models.
problem Uncertainty quantification in compound Poisson loss models.
method Large exposure asymptotics applied to Mack's estimator.
result Chain ladder prediction uncertainty can be quantified without model assumptions.
We study optimal trade execution strategies in financial markets with discrete order flow. The agent has a finite liquidation horizon and must minimize price impact given a random number of incoming trade counterparties. Assuming that the order flow is given by a Poisson process, we give a full analysis of the prop…
The paper shows robustness of Hilbert space-valued stochastic volatility models to perturbations.
problem Robustness of Hilbert space-valued stochastic volatility models to measurement or approximation errors.
method Quantifying the error induced by volatility perturbations and studying robustness of volatility process with finite dimensional approximations.
result Explicit bounds for the induced error in terms of approximation of the underlying parameter.
A new sequential method estimates Poisson means in streaming data, achieving optimality and efficiency.
problem Estimating Poisson means in a streaming, or online, framework.
method A quasi-Bayesian approach based on Newton's algorithm for a sequential estimate.
result Established frequentist guarantees including consistency and asymptotic optimality.
This paper considers magnitude, asymptotics and duration of drawdowns for some Lévy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative Lévy processes using an approximation approach. For any spectrally negative Lévy process whose scale functions are well-behaved at …
Investigates RI strategies for life insurers with LRD mortality rates.
problem Effect of long-range dependent mortality rates on RI strategies.
method Volterra mortality model, compound Poisson process, open-loop equilibrium mean-variance criterion.
result Explicit equilibrium RI controls derived and uniqueness studied.
Study optimal reinsurance pricing under model uncertainty for multiple insurers.
problem Optimal reinsurance pricing in the presence of multiple sources of model uncertainty.
method Solves a continuous-time Stackelberg game for general reinsurance contracts, considering entropy penalties and ambiguity in insurers' models.
result Reinsurer prices under a distortion of the barycentre of insurers' models, maximizing expected wealth with an entropy penalty.
Researchers calculated EVaR for various distributions using Lambert function.
problem Difficulty in finding analytical representation of EVaR measure.
method Used Lambert function to calculate EVaR for multiple distributions.
result Successfully calculated EVaR for 7 specific distributions.
Paper establishes sufficient condition for comparing linear combinations of infinite-mean risks.
problem Comparing linear combinations of infinite-mean risks under stochastic dominance.
method Introduced a new class of distributions and used majorization order to compare weights.
result Linear combinations of random variables are stochastically larger when their weight vectors are smaller in majorization order.
Count data are often used in recommender systems: they are widespread (song play counts, product purchases, clicks on web pages) and can reveal user preference without any explicit rating from the user. Such data are known to be sparse, over-dispersed and bursty, which makes their direct use in recommender systems chal…
Under the Basel II standards, the Operational Risk (OpRisk) advanced measurement approach is not prescriptive regarding the class of statistical model utilised to undertake capital estimation. It has however become well accepted to utlise a Loss Distributional Approach (LDA) paradigm to model the individual OpRisk loss…
New algorithm identifies best arm in rare event scenarios.
problem Identifying the best arm with tiny reward probability.
method Approximated Compound Poisson process for faster algorithms.
result Improved computational efficiency with minor sample complexity increase.
We consider the problem of pricing derivatives written on some industrial loss index via utility indifference pricing. The industrial loss index is modelled by a compound Poisson process and the insurer can adjust her portfolio by choosing the risk loading, which in turn determines the demand. We compute the price of a…
Non-negative matrix factorization models based on a hierarchical Gamma-Poisson structure capture user and item behavior effectively in extremely sparse data sets, making them the ideal choice for collaborative filtering applications. Hierarchical Poisson factorization (HPF) in particular has proved successful for scala…
We present a general framework, the coupled compound Poisson factorization (CCPF), to capture the missing-data mechanism in extremely sparse data sets by coupling a hierarchical Poisson factorization with an arbitrary data-generating model. We derive a stochastic variational inference algorithm for the resulting model …
We consider a two-dimensional optimal dividend problem in the context of two branches of an insurance company with compound Poisson surplus processes dividing claims and premia in some specified proportions. We solve the stochastic control problem of maximizing expected cumulative discounted dividend payments (among al…
We study tick-by-tick financial returns belonging to the FTSE MIB index of the Italian Stock Exchange (Borsa Italiana). We can confirm previously detected non-stationarities. However, scaling properties reported in the previous literature for other high-frequency financial data are only approximately valid. As a conseq…
Study short-maturity VIX and European option prices with jumps.
problem Analyzing VIX and European options with jumps in short-maturity models.
method Local-stochastic volatility models with compound Poisson jumps, leading-order asymptotics in closed-form.
result Closed-form solutions for VIX and European option prices in short-maturity models.
We investigate the existence of affine realizations for Lévy driven interest rate term structure models under the real-world probability measure, which so far has only been studied under an assumed risk-neutral probability measure. For models driven by Wiener processes, all results obtained under the risk-neutral appro…
Most energy and commodity markets exhibit mean-reversion and occasional distinctive price spikes, which results in demand for derivative products which protect the holder against high prices. To this end, in this paper we present exact and fast methodologies for the simulation of the spot price dynamics modeled as the …
Model-based collaborative filtering analyzes user-item interactions to infer latent factors that represent user preferences and item characteristics in order to predict future interactions. Most collaborative filtering algorithms assume that these latent factors are static, although it has been shown that user preferen…
We investigate the pricing of cliquet options in a jump-diffusion model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a drifted Lévy process entailing a Brownian diffusion component as well as compound Poisson jumps. We also derive representations for the density…
In this note we investigate the consistency under inversion of jump diffusion processes in the Foreign Exchange (FX) market. In other terms, if the EUR/USD FX rate follows a given type of dynamics, under which conditions will USD/EUR follow the same type of dynamics? In order to give a numerical description of this pro…
The paper predicts cryptocurrency prices using a path-dependent Monte Carlo simulation.
problem Forecasting cryptocurrency prices with volatility and jumps.
method Merton's jump diffusion model with machine learning, traditional, and statistical methods.
result Introduced a path-dependent Monte Carlo simulation for cryptocurrency price prediction.
New MGCPP model for order flow in financial markets.
problem Modeling order flow dynamics in financial markets.
method Developed MGCPP, proved LLN and FCLTs, applied to real data.
result Validated MGCPP model with real trading data.
In a dual risk model, the premiums are considered as the costs and the claims are regarded as the profits. The surplus can be interpreted as the wealth of a venture capital, whose profits depend on research and development. In most of the existing literature of dual risk models, the profits follow the compound Poisson …
Local signature-based expansionsq-fin.MF
Study local expansions of continuous-time processes using Ito signature properties.
problem Analyzing local expansions of continuous-time processes and their moments.
method Using the Ito signature, a basis of iterated integrals, to conduct expansions of the process' characteristic function.
result Explicit coefficients and stochastic representations for asymptotics as time shrinks or diverges.
We model the price of a stock via a Langévin equation with multi-dimensional fluctuations coupled in the price and in time. We generalize previous models in that we assume that the fluctuations conditioned on the time step are compound Poisson processes with operator stable jump intensities. We derive exact relations f…
In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regards to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional Central Limit Theorems (FCLT) for seve…
New insights into empirical Bayes and compound decision problems with improved regret bounds.
problem Estimating means of normally or Poisson distributed vectors under squared loss.
method Combines Bayesian and frequentist approaches using data-driven estimators.
result Optimal regret bounds for Poisson and normal mean models, resolving conjectures.
In this paper we introduce two new Hawkes processes, namely, compound and regime-switching compound Hawkes processes, to model the price processes in limit order books. We prove Law of Large Numbers and Functional Central Limit Theorems (FCLT) for both processes. The two FCLTs are applied to limit order books where we …