First crossing time, overshoot and Appell-Hessenberg type functions

We consider a general insurance risk model with extended flexibility under which claims arrive according to a point process with independent increments, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous, real valued function. Point processes with independent increments are in general non-stationary, allowing for an arbitrary (possibly discontinuous) claim arrival cumulative intensity function which is appealing for insurance applications. Under these general assumptions, we derive a closed form expression for the joint distribution of the time to ruin and the deficit at ruin, which is remarkable, since as we show, it involves a new interesting class of what we call Appell-Hessenberg type functions. The latter are shown to coincide with the classical Appell polynomials in the Poisson case and to yield a new class of the so called Appell-Hessenberg factorial polynomials in the case of negative binomial claim arrivals. Corollaries of our main result generalize previous ruin formulas e.g., those obtained for the case of stationary Poisson claim arrivals

Finite Time Non-Ruin Probability Formulae for Erlang Claim Interarrivals and Continuous Interdependent Claim Severities

A closed form expression, in terms of some functions which we call exponential Appell polynomials, for the probability of non-ruin of an insurance company, in a finite-time interval is derived, assuming independent, non-identically Erlang distributed claim inter-arrival times, τi ∼ Erlang (gi, λi) , i = 1, 2, . . ., any continuous joint distribution of the claim amounts and any non-negative, non-decreasing real function, representing its premium income. In the special case when τi ∼ Erlang (gi, λ) , i = 1, 2, . . . it is shown that our main result yields a formula for the probability of non-ruin expressed in terms of the classical Appell polynomials. We give another special case of our non-ruin probability formula for τi ∼ Erlang (1, λi) , i = 1, 2, . . ., i.e., when the inter-arrival times are non-identically exponentially distributed and also show that it coincides with the formula for Poisson claim arrivals, given in [18], when τi ∼ Erlang(1, λ), i = 1, 2, . . .. The main result is extended further to a risk model in which inter-arrival times are dependent random variables, obtained by randomizing the Erlang shape or/and rate parameters. We give also some useful auxiliary results which characterize and express explicitly (and recurrently) the exponential Appell polynomials which appear in our finite time non-ruin probability formulae

Ruin and Deficit Under Claim Arrivals with the Order Statistics Property

We consider an insurance risk model with extended flexibility, under which claims arrive according to a point process with an order statistics (OS) property, their amounts may have any joint distri- bution and the premium income is accumulated following any non- decreasing, possibly discontinuous real valued function. We generalize the definition of an OS point process, assuming it is generated by an arbitrary cdf allowing jump discontinuities, which corresponds to an arbitrary (possibly discontinuous) claim arrival cumulative intensity function. The latter feature is appealing for insurance applications since it allows to consider clusters of claims arriving instantaneously. Under these general assumptions, a closed form expression for the joint distribution of the time to ruin and the deficit at ruin is derived, which remarkably involves classical Appell polynomials. Corollaries of our main result generalize previous non-ruin formulas e.g., those obtained by Ignatov and Kaishev (2000, 2004, 2006) and Lef`evre and Loisel (2009) for the case of stationary Poisson claim arrivals and by Lef`evre and Picard (2011, 2014), for OS claim arrivals

Operational risk and insurance: a ruin probabilistic reserving approach

A new methodology for financial and insurance operational risk capital estimation is proposed. It is based on using the finite time probability of (non-)ruin as an operational risk measure, within a general risk model. It allows for inhomogeneous operational loss frequency (dependent inter-arrival times) and dependent loss severities which may have any joint discrete or continuous distribution. Under the proposed methodology, operational risk capital assessment is viewed not as a one off exercise, performed at some moment of time, but as dynamic reserving, following a certain risk capital accumulation function. The latter describes the accumulation of risk capital with time and may be any nondecreasing, mpositive real function hHtL. Under these reasonably general assumptions, the probability of mnon-ruin is explicitly expressed using closed form expressions, derived by Ignatov and Kaishev (2000, 2004, 2007) and Ignatov, Kaishev and Krachunov (2001) and by setting it to a high enough preassigned mvalue, say 0.99, it is possible to obtain not just a value for the capital charge but a (dynamic) risk capital accumulation strategy, hHtL. In view of its generality, the proposed methodology is capable of accommodating any (heavy tailed) mdistributions, such as the Generalized Pareto Distribution, the Lognormal distribution the g-and-h mdistribution and the GB2 distribution. Applying this methodology on numerical examples, we demonstrate that dependence in the loss severities may have a dramatic effect on the estimated risk capital. In addition, we show also that one and the same high enough survival probability may be achieved by mdifferent risk capital accumulation strategies one of which may possibly be preferable to accumulating capital just linearly, as has been assumed by Embrechts et al. (2004). The proposed methodology takes into account also the effect of insurance on operational losses, in which case it is proposed to take the probability of joint survival of the financial institution and the insurance provider as a joint operational risk measure. The risk capital allocation strategy is then obtained in such a way that the probability of joint survival is equal to a preassigned high enough value, say 99.9

Ruin and deficit at ruin under an extended order statistics risk process

We consider an insurance risk model with extended flexibility, under which claims arrive according to a point process with an order statistics (OS) property, their amounts may have any joint distribution and the premium income is accumulated following any nondecreasing, possibly discontinuous real valued function. We generalize the definition of an OS point process, assuming it is generated by an arbitrary cdf, allowing jump discontinuities which corresponds to an arbitrary (possibly discontinuous) claim arrival cumulative intensity function. The latter feature is appealing for insurance applications since it allows to consider clusters of claims arriving instantaneously. Under these general assumptions, a closed form expression for the joint distribution of the time to ruin and the deficit at ruin is derived, which remarkably involves classical Appell polynomials. Corollaries of our main result generalize previous non-ruin formulas e.g., those obtained by Ignatov and Kaishev (2000, 2004, 2006) and Lef`evre and Loisel (2009) for the case of stationary Poisson claim arrivals and by Lef`evre and Picard (2011, 2014), for OS claim arrivals

Dissipative Chaos in Semiconductor Superlattices

We consider the motion of ballistic electrons in a miniband of a semiconductor superlattice (SSL) under the influence of an external, time-periodic electric field. We use the semi-classical balance-equation approach which incorporates elastic and inelastic scattering (as dissipation) and the self-consistent field generated by the electron motion. The coupling of electrons in the miniband to the self-consistent field produces a cooperative nonlinear oscillatory mode which, when interacting with the oscillatory external field and the intrinsic Bloch-type oscillatory mode, can lead to complicated dynamics, including dissipative chaos. For a range of values of the dissipation parameters we determine the regions in the amplitude-frequency plane of the external field in which chaos can occur. Our results suggest that for terahertz external fields of the amplitudes achieved by present-day free electron lasers, chaos may be observable in SSLs. We clarify the nature of this novel nonlinear dynamics in the superlattice-external field system by exploring analogies to the Dicke model of an ensemble of two-level atoms coupled with a resonant cavity field and to Josephson junctions.Comment: 33 pages, 8 figure

Effect of temperature on resonant electron transport through stochastic conduction channels in superlattices

We show that resonant electron transport in semiconductor superlattices with an applied electric and tilted magnetic field can, surprisingly, become more pronounced as the lattice and conduction electron temperature increases from 4.2 K to room temperature and beyond. It has previously been demonstrated that at certain critical field parameters, the semiclassical trajectories of electrons in the lowest miniband of the superlattice change abruptly from fully localised to completely unbounded. The unbounded electron orbits propagate through intricate web patterns, known as stochastic webs, in phase space, which act as conduction channels for the electrons and produce a series of resonant peaks in the electron drift velocity versus electric field curves. Here, we show that increasing the lattice temperature strengthens these resonant peaks due to a subtle interplay between thermal population of the conduction channels and transport along them. This enhances both the electron drift velocity and the influence of the stochastic webs on the current-voltage characteristics, which we calculate by making self-consistent solutions of the coupled electron transport and Poisson equations throughout the superlattice. These solutions reveal that increasing the temperature also transforms the collective electron dynamics by changing both the threshold voltage required for the onset of self-sustained current oscillations, produced by propagating charge domains, and the oscillation frequency.Comment: 8 figures, 12 page

Measurement of the radiative decay of polarized muons in the MEG experiment

We studied the radiative muon decay $\mu^+ \to e^+\nu\bar{\nu}\gamma$ by using for the first time an almost fully polarized muon source. We identified a large sample (~13000) of these decays in a total sample of 1.8x10^14 positive muon decays collected in the MEG experiment in the years 2009--2010 and measured the branching ratio B($\mu^+ \to e^+\nu\bar{\nu}\gamma$) = (6.03+-0.14(stat.)+-0.53(sys.))x10^-8 for E_e > 45 MeV and E_{\gamma} > 40 MeV, consistent with the Standard Model prediction. The precise measurement of this decay mode provides a basic tool for the timing calibration, a normalization channel, and a strong quality check of the complete MEG experiment in the search for $\mu^+ \to e^+\gamma$ process.Comment: 8 pages, 7 figures. Added an introduction to NLO calculation which was recently calculated. Published versio