An infinite depth dam with Poisson input and Poisson release

Abstract

In this paper the content Z(t) of a dam where Z(t) ε( - ∞, h], h an arbitrary constant, is studied. Inputs to the dam form a sequence of i.i.d. r.v.'s and occur according to a Poisson process with parameter λ > 0 while releases from the dam form a sequence of i.i.d. r.v.'s and occur according to a Poisson process with parameter μ > 0. The Laplace transform of the process Z(t) is derived and expressions for the moments are given. It is shown that Z(t) has a nondegenerate limiting distribution, as t -- ∞, if average inputs per unit time exceed average releases per unit time.     

Le XIImecongrès international d'actuaires

Abstract

We study the following inverse thinning problem for renewal processes: for which completely monotone functions f is f/(p+qf), 0?p?1, q=1-p, completely monotone? A characterisation of such f's is given. We also study the case when f comes from a gamma distribution, and present some ideas for more general results.

The intention of this note is to add some information to a paper by Yannaros (1985), in which thinned renewal processes are considered. Let X_n , n?1, be i.i.d. non-negative random variables, distributed according to a probability measure µ, and let S_n = X ₁+...+X_n (with S ₀=0) be the corresponding renewal process. Replacing µ by the probability measure ν=∑_n?1 pq^n-1 µ^n* (µ^n* =µ* ... µ*, n times) we get a new renewal process, obtained from the original one by independently at each stage preserving the process with probability p. Here and below q= 1-p, and to avoid trivialities we assume that 0 Let µ^(s) = ∫[0,∞) exp (-sx)µ(dx) , s?0, denote the Laplace transform of µ. Then ν^=pµ/(1-µ^). We will study the inverse problem: given a completely monotone function ψ, when does ψ(p+qψ) define a completely monotone function. A complete characterisation, and some of its consequences, is given in §§ 1–3 below. In §§ 4–5 we study the gamma distribution. It is proved that the inverse problem has a negative solution when the parameter a > 1, i.e. 1/(p + q(1 + s) ^a ) is not completely monotone then. In Yannaros (1985) this was proved for a=2, 3, ... with entirely different methods. (That 1/(p+q(1+s)^a is completely monotone for 0?a?1 is easily seen; cf. Yannaros (1985). Finally, in § 6 we give some suggestions to more general results related to thinning. Perhaps the most interesting problem is to find sufficiently general conditions for an absolutely monotone function to have a Bernstein function as its inverse.     

A note on different models of stochastic processes dealt with in the collective theory of risk

Abstract

1. For the definition of general processes with special regard to those concerned in Collective Risk Theory reference is made to Cramér (Collective Risk Theory, Skandia Jubilee Volume, Stockholm, 1955). Let the independent parameter of such a process be denoted by τ, with the origin at the point of departure of the process and on a scale independent of the number of expected changes of the random function. Denote with p(τ, n)dt the asymptotic expression for the conditional probability of one change in the random function while the parameter passes from τ to τ + dτ: relative to the hypothesis that n changes have occurred, while the parameter passes from 0 to τ. Assume further—unless the contrary is stated—that the probability of more than one change, while the parameter passes from τ to τ + dτ, is of smaller order than dτ.     

Risk theory in a Markovian environment

Abstract

We consider risk processes _{t t?0} with the property that the rate β of the Poisson arrival process and the distribution of B of the claim sizes are not fixed in time but depend on the state of an underlying Markov jump process {Z_t } _t?0 such that β=β _i and B=B_i when Z_t=i . A variety of methods, including approximations, simulation and numerical methods, for assessing the values of the ruin probabilities are studied and in particular we look at the Cramér-Lundberg approximation and diffusion approximations with correction terms. The mathematical framework is Markov-modulated random walks in discrete and continuous time, and in particular Wiener-Hopf factorisation problems and conjugate distributions (Esscher transforms) are involved.     

On the asymptotic normality of some unbiased estimates

Abstract

Let X _m(n) =(X _j , n, ..., X _{j m,n} ) be a subset of observations of a sample X_n = (X_1n X _2n ... , X _nn ). Here the X_jn 'S in Xn are not necessarily independent or identically distributed, and m(n) mayor may not tend to infinity as n tends to infinity. Suppose the joint density function h_n =h_n (x _m (n); θ) of the X _jn 's in Xm(n) is completely specified except the values of the parameters in the parameter vector θ = (θ1 θ2, ... , θ _k ), where θ belongs to a non-degenerate open subset H of the k-dimensional Euclidean space Rk and k?m(n).     

On definition of compound poisson processes

Abstract

Some authors define the (elementary) compound Poisson process in wide sense {χ _t , 0 ? t < ∞} with help of probability distributions where τ is a so-called operational time, a continuous non-decreasing function of t vanishing for t = 0, and V(q, t) is a non-negative distribution function for every t.     

Ruin probabilities and aggregrate claims distributions for shot noise Cox processes

We consider a risk process R _t where the claim arrival process is a superposition of a homogeneous Poisson process and a Cox process with a Poisson shot noise intensity process, capturing the effect of sudden increases of the claim intensity due to external events. The distribution of the aggregate claim size is investigated under these assumptions. For both light-tailed and heavy-tailed claim size distributions, asymptotic estimates for infinite-time and finite-time ruin probabilities are derived. Moreover, we discuss an extension of the model to an adaptive premium rule that is dynamically adjusted according to past claims experience.     

Ruin probabilities expressed in terms of ladder height distributions

Abstract

During the latest few years much attention has been given to the study of the ruin problem of a risk business when the epochs of claims form a renewal process. The study of this problem was initiated by E. S. Andersen (1957). Thorin has then in a series of papers (Thorin, 1970, 1971a, 1971b) shown that the Wiener-Hopf technique, originally developed by Cramer (1955) in the case of a Poisson process, can be used in this more general case, and Takacs (1970) has derived results similar to those of Thorin by an entirely new technique.     

On stationary point processes and Markov chains

Abstract

Introductory. In the theory of random processes we may distinguish between ordinary processes and point processes. The former are concerned with a quantity, say x (t), which varies with time t, the latter with events, incidences, which may be represented as points along the time axis. For both categories, the stationary process is of great importance, i. e., the special case in which the probability structure is independent of absolute time. Several examples of stationary processes of the ordinary type have been examined in detail (see e. g. H. Wold 1). The literature on stationary point processes, on the other hand, has exclusively been concerned with the two simplest cases, viz. the Poisson process and the slightly more general process arising in renewal theory (see e. g. J. Doob 3).     

The representation of American options prices under stochastic volatility and jump-diffusion dynamics

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square-root process as used by Heston [Rev. Financial Stud., 1993, 6, 327–343], and by a Poisson jump process as introduced by Merton [J. Financial Econ., 1976, 3, 125–144]. Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalizes in an intuitive way the structure of the solution to the corresponding European option pricing problem obtained by Scott [Math. Finance, 1997, 7(4), 413–426], but here in the case of a call option and constant interest rates.     

Estimation of waiting-time parameters in the GI/G/1 queuing system

Abstract

In a previous paper [2] the author has studied the distribution of the average value of n consecutive observations of the waiting-time in an M/G/1 queuing system, i.e. a system where the arrival epochs form a Poisson process with constant intensity. The observations were assumed to be made during the equilibrium state of the process.     

Some limit theorems in urn models with indistinguishable balls

Abstract

When n indistinguishable balls are distributed into m cells, there are two possible models (known as Bose-Einstein Statistics) depending on whether the cells are allowed to be empty or not. In these models, the possible limit laws of the number of cells containing k balls each are identified which are Normal, Poisson and degenerate. It is interesting to see that these are the only three limit laws of the occupancy variables studied here.     

A ruin model with a resampled environment

ABSTRACT

This paper considers a Cramér–Lundberg risk setting, where the components of the underlying model change over time. We allow the more general setting of the cumulative claim process being modeled as a spectrally positive Lévy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs, we resample the model components from a finite number of d settings. It results in a setup that is particularly suited to describe situations in which the risk reserve dynamics are affected by external processes. We extend the classical Cramér–Lundberg approximation (asymptotically characterizing the all-time ruin probability in a light-tailed setting) to this more general setup. In addition, for the situation that the driving Lévy processes are sums of Brownian motions and compound Poisson processes, we find an explicit uniform bound on the ruin probability. In passing we propose an importance-sampling algorithm facilitating efficient estimation, and prove it has bounded relative error. In a series of numerical experiments we assess the accuracy of the asymptotics and bounds, and illustrate that neglecting the resampling can lead to substantial underestimation of the risk.     

First passage time for compound Poisson processes with diffusion: ruin theoretical and financial applications

In this paper, we propose to revisit Kendall’s identity (see, e.g. Kendall (1957)) related to the distribution of the first passage time for spectrally negative Lévy processes. We provide an alternative proof to Kendall’s identity for a given class of spectrally negative Lévy processes, namely compound Poisson processes with diffusion, through the application of Lagrange’s expansion theorem. This alternative proof naturally leads to an extension of this well-known identity by further examining the distribution of the number of jumps before the first passage time. In the process, we generalize some results of Gerber (1990 Gerber, H. U. (1990). When does the surplus reach a given target? Insurance: Mathematics and Economics 9, 115–119.  [Google Scholar]) to the class of compound Poisson processes perturbed by diffusion. We show that this main result is particularly relevant to further our understanding of some problems of interest in actuarial science. Among others, we propose to examine the finite-time ruin probability of a dual Poisson risk model with diffusion or equally the distribution of a busy period in a specific fluid flow model. In a second example, we make use of this result to price barrier options issued on an insurer’s stock price.     

The skewed multifractal random walk with applications to option smiles

Abstract

We generalize the construction of the multifractal random walk (MRW) due to Bacry et al (Bacry E, Delour J and Muzy J-F 2001 Modelling financial time series using multifractal random walks Physica A 299 84) to take into account the asymmetric character of financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. We compute the leading behaviour of q-moments of the process, which behave as power laws of the time lag with an exponent ζ _q =p?2p(p?1)λ² for even q=2p, as in the symmetric MRW, and as ζ _q =p + 1?2p ²λ²?α (q=2p + 1), where λ and α are parameters. We show that this extended model reproduces the ‘HARCH’ effect or ‘causal cascade’ reported by some authors. We illustrate the usefulness of this ‘skewed’ MRW by computing the resulting shape of the volatility smiles generated by such a process, which we compare with approximate cumulant expansion formulae for the implied volatility. A large variety of smile surfaces can be reproduced.     

The power of patience: a behavioural regularity in limit-order placement

Abstract

In this paper we demonstrate a striking regularity in the way people place limit orders in financial markets, using a data set consisting of roughly two million orders from the London Stock Exchange. We define the relative limit price as the difference between the limit price and the best price available. Merging the data from 50 stocks, we demonstrate that for both buy and sell orders, the unconditional cumulative distribution of relative limit prices decays roughly as a power law with exponent approximately –1.5. This behaviour spans more than two decades, ranging from a few ticks to about 2000 ticks. Time series of relative limit prices show interesting temporal structure, characterized by an autocorrelation function that asymptotically decays as C(τ)～τ^?0.4. Furthermore, relative limit price levels are positively correlated with and are led by price volatility. This feedback may potentially contribute to clustered volatility.     

A diffusion approximation for the ruin function of a risk process with compounding assets

Abstract

The traditional theory of collective risk is concerned with fluctuations in the capital reserve {Y(t): t ?O} of an insurance company. The classical model represents {Y(t)} as a positive constant x (initial capital) plus a deterministic linear function (cumulative income) minus a compound Poisson process (cumulative claims). The central problem is to determine the ruin probability ψ(x) that capital ever falls to zero. It is known that, under reasonable assumptions, one can approximate {Y(t)} by an appropriate Wiener process and hence ψ(.) by the corresponding exponential function of (Brownian) first passage probabilities. This paper considers the classical model modified by the assumption that interest is earned continuously on current capital at rate β > O. It is argued that Y(t) can in this case be approximated by a diffusion process Y*(t) which is closely related to the classical Ornstein-Uhlenbeck process. The diffusion {Y*(t)}, which we call compounding Brownian motion, reduces to the ordinary Wiener process when β = O. The first passage probabilities for Y*(t) are found to form a truncated normal distribution, which approximates the ruin function ψ(.) for the model with compounding assets. The approximate expression for ψ(.) is compared against the exact expression for a special case in which the latter is known. Assuming parameter values for which one would anticipate a good approximation, the two expressions are found to agree extremely well over a wide range of initial asset levels.     

Efficient estimation of drift parameters in stochastic volatility models

We study the parametric problem of estimating the drift coefficient in a stochastic volatility model , where Y is a log price process and V the volatility process. Assuming that one can recover the volatility, precisely enough, from the observation of the price process, we construct an efficient estimator for the drift parameter of the diffusion V. As an application we present the efficient estimation based on the discrete sampling with δ _n →0 and n δ _n →∞. We show that our setup is general enough to cover the case of ‘microstructure noise’ for the price process as well.      

On the “Output” process of a state dependent queue

Abstract

It is well known that if in a queueing situation the arrivals occur in a Poisson stream with intensity λ, and probability differential of a service completion is μd t+o(dt), then in the steady state the successive epochs of service completion also occur in a Poisson stream with intensity λ, that of arrivals. Here, we shall show that this criteria could be proved in the more general case in which the probability differential of a service completion is directly proportional to the number of customers momentarily present in the system.