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Florin Avram

Publications and source records attributed to Florin Avram.

At least 37 records · Page 2Linked to original sources
arXiv2021

A Review of First-Passage Theory for the Segerdahl Risk Process and Extensions

The Segerdahl process (Segerdahl (1955)), characterized by exponential claims and affine drift, has drawn a considerable amount of interest -- see, for example, (Tichy (1984); Avram and Usabel (2008), due to its economic interest (it is the simplest risk process which takes into account the effect of interest rates). It is also the simplest non-Levy, non-diffusion example of a spectrally negative Markov risk model. Note that for both spectrally negative Levy and diffusion processes, first passage theories which are based on identifying two basic monotone harmonic functions/martingales have been developped. This means that for these processes many control problems involving dividends, capital injections, etc., may be solved explicitly once the two basic functions have been obtained. Furthermore, extensions to general spectrally negative Markov processes are possible (Landriault et al. (2017), Avram et al. (2018); Avram and Goreac (2019); Avram et al. (2019b). Unfortunately, methods for computing the basic functions are still lacking outside the Levy and diffusion classes, with the notable exception of the Segerdahl process, for which the ruin probability has been computed (Paulsen and Gjessing (1997). However, there is a striking lack of numerical results in both cases. This motivated us to review several approaches, with the purpose of drawing attention to connections between them, and underlying open problems.

math.PR↗
arXiv2021

Optimizing dividends and capital injections limited by bankruptcy, and practical approximations for the Cramér-Lundberg process

The recent papers Gajek-Kucinsky(2017) and Avram-Goreac-Li-Wu(2020) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The first paper works under the spectrally negative Lévy model; the second works under the Cramér-Lundberg model with exponential jumps, where the results are considerably more explicit. The current paper has three purposes. First, it illustrates the fact that quite reasonable approximations of the general problem may be obtained using the particular exponential case studied in Avram-Goreac-Li-Wu(2020). Secondly, it extends the results to the case when a final penalty $P$ is taken into consideration as well besides a proportional cost $k>1$ for capital injections. This requires amending the "scale and Gerber-Shiu functions" already introduced in Gajek-Kucinsky(2017). Thirdly, in the exponential case, the results will be made even more explicit by employing the Lambert-W function. This tool has particular importance in computational aspects and can be employed in theoretical aspects such as asymptotics.

math.OC↗
arXiv2019

The $W,Z$ scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends

First passage problems for spectrally negative Lévy processes with possible absorbtion or/and reflection at boundaries have been widely applied in mathematical finance, risk, queueing, and inventory/storage theory. Historically, such problems were tackled by taking Laplace transform of the associated Kolmogorov integro-differential equations involving the generator operator. In the last years there appeared an alternative approach based on the solution of two fundamental "two-sided exit" problems from an interval (TSE). A spectrally one-sided process will exit smoothly on one side on an interval, and the solution is simply expressed in terms of a "scale function" $W$ (Bertoin 1997). The non-smooth two-sided exit (or ruin) problem suggests introducing a second scale function $Z$ (Avram, Kyprianou and Pistorius 2004). Since many other problems can be reduced to TSE, researchers produced in the last years a kit of formulas expressed in terms of the "$W,Z$ alphabet" for a great variety of first passage problems. We collect here our favorite recipes from this kit, including a recent one (94) which generalizes the classic De Finetti dividend problem. One interesting use of the kit is for recognizing relationships between apparently unrelated problems -- see Lemma 3. Last but not least, it turned out recently that once the classic $W,Z$ are replaced with appropriate generalizations, the classic formulas for (absorbed/ reflected) Lévy processes continue to hold for: a) spectrally negative Markov additive processes (Ivanovs and Palmowski 2012), b) spectrally negative Lévy processes with Poissonian Parisian absorbtion or/and reflection (Avram, Perez and Yamazaki 2017, Avram Zhou 2017), or with Omega killing (Li and Palmowski 2017).

math.PR↗
arXiv2018

A Pontryaghin Maximum Principle Approach For The Optimization Of Dividends/consumption Of Spectrally Negative Markov Processes, Until A Generalized Draw-down Time

The first motivation of our paper is to explore further the idea that, in risk control problems, it may be profitable to base decisions both on the position of the underlying process Xt and on its supremum Xt := sup 0$\le$s$\le$t Xs. Strongly connected to Azema-Yor/generalized draw-down/trailing stop time (see [AY79]), this framework provides a natural unification of draw-down and classic first passage times. We illustrate here the potential of this unified framework by solving a variation of the De Finetti problem of maximizing expected discounted cumulative dividends/consumption gained under a barrier policy, until an optimally chosen Azema-Yor time, with a general spectrally negative Markov model. While previously studied cases of this problem [APP07, SLG84, AS98, AVZ17, AH18, WZ18] assumed either L{é}vy or diffusion models, and the draw-down function to be fixed, we describe, for a general spectrally negative Markov model, not only the optimal barrier but also the optimal draw-down function. This is achieved by solving a variational problem tackled by Pontryaghins maximum principle. As a by-product we show that in the L{é}vy case the classic first passage solution is indeed optimal; in the diffusion case, we obtain the optimality equations, but the existence of solutions improving the classic ones is left for future work.

math.OC↗
arXiv2018

General drawdown of general tax model in a time-homogeneous Markov framework

Drawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure) and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes -- see Avram, Vu, Zhou(2017), Li, Vu, Zhou(2017). In this paper, we further examine general drawdown related quantities for taxed time-homogeneous Markov processes, using the pathwise connection between general drawdown and tax.

math.PR↗
arXiv2018

First passage problems for upwards skip-free random walks via the $Φ,W,Z$ paradigm

We develop the theory of the $W$ and $Z$ scale functions for right-continuous (upwards skip-free) discrete-time discrete-space random walks, along the lines of the analogue theory for spectrally negative Lévy processes. Notably, we introduce for the first time in this context the one and two-parameter scale functions $Z$, which appear for example in the joint problem of deficit at ruin and time of ruin, and in problems concerning the walk reflected at an upper barrier. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous. The theory is shown to be fruitful by providing a convenient unified framework for studying dividends-capital injection problems under various objectives, for the so-called compound binomial risk model of actuarial science.

math.PR↗
arXiv2017

Spectrally negative Lévy processes with Parisian reflection below and classical reflection above

We consider a company that receives capital injections so as to avoid ruin. Differently from the classical bail-out settings where the underlying process is restricted to stay at or above zero, we study the case bail-out can only be made at independent Poisson times. Namely, we study a version of the reflected process that is pushed up to zero only on Poisson observation times at which the process is below zero. We also study the case with additional classical reflection above so as to model a company that pays dividends according to a barrier strategy. Focusing on the spectrally negative Lévy case, we compute, using the scale function, various fluctuation identities including capital injections and dividends.

math.PR↗
arXiv2017

Steady-state analysis of single exponential vacation in a $PH/MSP/1/\infty$ queue using roots

We consider an infinite-buffer single-server queue where inter-arrival times are phase-type ($PH$), the service is provided according to Markovian service process $(MSP)$, and the server may take single, exponentially distributed vacations when the queue is empty. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function (VGF) of system-length distribution at a pre-arrival epoch. Also, we obtain the steady-state system-length distribution at an arbitrary epoch along with some important performance measures such as the mean number of customers in the system and the mean system sojourn time of a customer. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. At the end, we present numerical results in the form of tables to show the effect of model parameters on the performance measures.

math.PR↗
arXiv2016

On fluctuation theory for spectrally negative Levy processes with Parisian reflection below, and applications

As well known, all functionals of a Markov process may be expressed in terms of the generator operator, modulo some analytic work. In the case of spectrally negative Markov processes however, it is conjectured that everything can be expressed in a more direct way using the $W$ scale function which intervenes in the two-sided first passage problem, modulo performing various integrals. This conjecture arises from work on Levy processes \cite{AKP,Pispot,APP,Iva,IP, ivanovs2013potential,AIZ,APY}, where the $W$ scale function has explicit Laplace transform, and is therefore easily computable; furthermore it was found in the papers above that a second scale function $Z$ introduced in \cite{AKP} greatly simplifies first passage laws, especially for reflected processes. This paper gathers a collection of first passage formulas for spectrally negative Parisian Lévy processes, expressed in terms of $W,Z$ which may serve as an "instruction kit" for computing quantities of interest in applications, for example in risk theory and mathematical finance. To illustrate the usefulness of our list, we construct a new index for the valuation of financial companies modeled by spectrally negative Lévy processes, based on a Dickson-Waters modifications of the de Finetti optimal expected discounted dividends objective. We offer as well an index for the valuation of conglomerates of financial companies. An implicit question arising is to investigate analog results for other classes of spectrally negative Markovian processes.

math.PR↗
arXiv2015

On the central management of risk networks

We introduce a family of risk networks composed from a) several subsidiary branches $U_i(t), i=1,...,I$ necessary for coping with different types of risks, which must all be kept above $0$, and b) a central branch (CB) which bails out the subsidiaries whenever necessary. Ruin occurs when the central branch is ruined. We find out that with one subsidiary ($I=1$), the finite time ruin probability of the central branch may be explicitly written out in terms of the finite time ruin probability of the subsidiary, provided that the CB in the absence of subsidiary bailouts is a deterministic drift. To study other problems, like for example the optimization of dividends to the CB with one subsidiary over a barrier, it is convenient to restrict to the case of phase-type claims to the subsidiary, and study the Markovian phase process at the moments when the CB process reaches new minima. The resulting structure is quite close to that of the phase of a PH/G/1 queue at the moments when it reaches new minima, and this yields in principle numeric approaches to several problems, based on the iterative calculation of the Laplace transforms of the upwards and downwards ladder times (also called excursions, and busy periods). In this work we propose a different approach of further providing matrix exponential approximations for the distributions of the ladder times. The advantage of this approach is that once a SNMAP approximation is obtained, many similar problems may be solved just by applying the recently developed scale matrix methodology. A numeric experiment for the CB barrier optimization problem is provided.

math.PR↗
arXiv2014

Semi-infinite optimization with sums of exponentials via polynomial approximation

We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by low-degree polynomials. Thus, the constraints can be approximated with polynomial inequalities that can be implemented with linear matrix inequalities. Convexity is preserved, but the problem has now a finite number of constraints. We show how to take advantage of the properties of the exponential function in order to build quickly accurate approximations. The problem used for illustration is the least-squares fitting of a positive sum of exponentials to an empirical density. When the exponents are given, the problem is convex, but we also give a procedure for optimizing the exponents. Several examples show that the method is flexible, accurate and gives better results than other methods for the investigated problems.

math.OC↗
arXiv2012

On matrix exponential approximations of the infimum of a spectrally negative Levy process

We recall four open problems concerning constructing high-order matrix-exponential approximations for the infimum of a spectrally negative Levy process (with applications to first-passage/ruin probabilities, the waiting time distribution in the M/G/1 queue, pricing of barrier options, etc). On the way, we provide a new approximation, for the perturbed Cramer-Lundberg model, and recall a remarkable family of (not minimal order) approximations of Johnson and Taaffe, which fit an arbitrarily high number of moments, greatly generalizing the currently used approximations of Renyi, De Vylder and Whitt-Ramsay. Obtaining such approximations which fit the Laplace transform at infinity as well would be quite useful.

math.PR↗
arXiv2009

Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cramér light-tail assumption on the claim size distribution.

math.PR↗
arXiv2008

On a Szego Type Limit Theorem, the Holder-Young-Brascamp-Lieb Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of Stationary Fields

Many statistical applications require establishing central limit theorems for sums, integrals, or for quadratic forms of functions of a stationary process. A particularly important case is that of Appell polynomials, since the Appell expansion rank" determines typically the type of central limit theorem satisfied by these functionals. We review and extend here to multidimensional indices a functional analysis approach to this problem proposed by Avram and Brown (1989), based on the method of cumulants and on integrability assumptions in the spectral domain; several applications are presented as well.

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