ORCID Profile
0000-0001-9257-1115
Current Organisations
Uniwersytet Wroclawski
,
Politechnika Wroclawska
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Publisher: Springer Science and Business Media LLC
Date: 07-2005
Publisher: MDPI AG
Date: 07-11-2020
DOI: 10.3390/MATH8111988
Abstract: In this paper, I analyze the distributional properties of the busy period in an on-off fluid queue and the first passage time in a fluid queue driven by a finite state Markov process. In particular, I show that the first passage time has a IFR distribution and the busy period in the Anick-Mitra-Sondhi model has a DFR distribution.
Publisher: Cambridge University Press (CUP)
Date: 12-2011
DOI: 10.1017/S0021900200008573
Abstract: In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ & 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit ex les.
Publisher: MDPI AG
Date: 15-06-2016
DOI: 10.3390/RISKS4020017
Publisher: Elsevier BV
Date: 11-2018
Publisher: Informa UK Limited
Date: 16-02-2018
Publisher: Elsevier BV
Date: 08-2017
Publisher: Cambridge University Press (CUP)
Date: 06-2019
DOI: 10.1017/APR.2019.20
Abstract: We consider a branching random walk on a multitype (with Q types of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index $\\alpha$ , while the other types of particles have lighter tails than the particles of type Q . In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in the n th generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process using the tools developed by Bhattacharya, Hazra, and Roy (2018). As a consequence, we obtain the asymptotic distribution of the position of the rightmost particle in the n th generation.
Publisher: Informa UK Limited
Date: 13-02-2020
Publisher: Politechnika Wroclawska Oficyna Wydawnicza
Date: 2020
Publisher: Elsevier BV
Date: 03-2011
Publisher: Springer Science and Business Media LLC
Date: 27-05-2015
Publisher: MDPI AG
Date: 23-03-2023
Abstract: We find the asymptotics of the value function maximizing the expected utility of discounted idend payments of an insurance company whose reserves are modeled as a classical Cramér risk process, with exponentially distributed claims, when the initial reserves tend to infinity. We focus on the power and logarithmic utility functions. We also perform some numerical analysis.
Publisher: Elsevier BV
Date: 10-2018
Publisher: Springer Science and Business Media LLC
Date: 20-03-2018
Publisher: Springer Science and Business Media LLC
Date: 09-01-2017
Publisher: Cambridge University Press (CUP)
Date: 08-02-2023
DOI: 10.1017/APR.2022.36
Abstract: We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ( $i=1,2,\\dots$ ) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature that is, if the surplus process just before the i th arrival is at level u , then for $a $ the capital jumps up to the level $(1+a)u+C_i$ . The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative idend payments, for the case of a Poisson arrival process of proportional gains. In the idend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.
Publisher: Informa UK Limited
Date: 14-04-2022
Publisher: Institute of Mathematical Statistics
Date: 05-1998
Publisher: Polish Mathematical Society
Date: 09-01-2015
Publisher: Elsevier BV
Date: 02-2008
Publisher: Elsevier BV
Date: 09-2013
Publisher: Springer Berlin Heidelberg
Date: 16-11-2004
Publisher: Springer Science and Business Media LLC
Date: 23-04-2013
Publisher: Springer Science and Business Media LLC
Date: 22-05-2023
DOI: 10.1007/S13385-023-00350-8
Abstract: In this paper we give a solution to the quickest drift change detection problem for a multivariate Lévy process consisting of both continuous (Gaussian) and jump components in the Bayesian approach. We do it for a general 0-modified continuous prior distribution of the change point as well as for a random post-change drift parameter. Classically, our criterion of optimality is based on a probability of false alarm and an expected delay of the detection, which is then reformulated in terms of a posterior probability of the change point. We find a generator of the posterior probability, which in case of general prior distribution is inhomogeneous in time. The main solving technique uses the optimal stopping theory and is based on solving a certain free-boundary problem. We also construct a Generelized Shiryaev-Roberts statistic, which can be used for applications. The paper is supplemented by two ex les, one of which is further used to analyze Polish life tables (after proper calibration) and detect the drift change in the correlated force of mortality of men and women jointly.
Publisher: Informa UK Limited
Date: 04-2011
Publisher: Cambridge University Press (CUP)
Date: 09-2006
Abstract: In this paper we study the busy period in GI/G/1 work-conserving queues. We give the exact asymptotics of the tail distribution of the busy period under the light tail assumptions. We also study the workload process in the M/G/1 system conditioned to stay positive.
Publisher: Association for Computing Machinery (ACM)
Date: 09-03-2012
Abstract: We consider a two-dimensional stochastic fluid model with N ONOFF inputs and temporary assistance, which is an extension of the same model with N = 1 in Mahabhashyam et al. (2008). The rates of change of both buffers are piecewise constant and dependent on the underlying Markovian phase of the model, and the rates of change for Buffer 2 are also dependent on the specific level of Buffer 1. This is because both buffers share a fixed output capacity, the precise proportion of which depends on Buffer 1. The generalization of the number of ON-OFF inputs necessitates modifications in the original rules of output-capacity sharing from Mahabhashyam et al. (2008) and considerably complicates both the theoretical analysis and the numerical computation of various performance measures. We derive the marginal probability distribution of Buffer 1, and bounds for that of Buffer 2. Furthermore, restricting Buffer 1 to a finite size, we determine its marginal probability distribution in the specific case of N = 1, thus providing numerical comparisons to the corresponding results in Mahabhashyam et al. (2008) where Buffer 1 is assumed to be infinite.
Publisher: Elsevier BV
Date: 06-2022
Publisher: Cambridge University Press (CUP)
Date: 03-2010
Abstract: We investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.
Publisher: Wiley
Date: 11-07-2019
DOI: 10.1111/MAFI.12218
Publisher: Springer Science and Business Media LLC
Date: 17-07-2020
DOI: 10.1007/S00780-020-00431-6
Abstract: This paper revisits the optimal capital structure model with endogenous bankruptcy, first studied by Leland (J. Finance 49:1213–1252, 1994) and Leland and Toft (J. Finance 51:987–1019, 1996). Unlike in the standard case where shareholders continuously observe the asset value and bankruptcy is executed instantaneously without delay, the information of the asset value is assumed to be updated periodically at the jump times of an independent Poisson process. Under a spectrally negative Lévy model, we obtain the optimal bankruptcy strategy and the corresponding capital structure. A series of numerical studies provide an analysis of the sensitivity, with respect to the observation frequency, of the optimal strategies, optimal leverage and credit spreads.
Publisher: Informa UK Limited
Date: 07-12-2016
Publisher: Springer New York
Date: 28-10-2012
Publisher: MDPI AG
Date: 20-03-2021
DOI: 10.3390/JRFM14030130
Abstract: The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process St is a geometric Lévy process with negative exponential jumps, i.e., St=seζt+σBt−∑i=1NtYi. The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function VAPutω(s) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that VAPutω(s) takes a simplified form.
Publisher: Cambridge University Press (CUP)
Date: 06-2020
DOI: 10.1017/APR.2020.2
Abstract: In this paper, we solve exit problems for a one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\\omega(\\cdot,\\cdot)$ dependent on the present level of the process and the current state of the environment. Moreover, we analyze the respective resolvents. All identities are expressed in terms of new generalizations of classical scale matrices for MAPs. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the Omega model, where bankruptcy takes place at rate $\\omega(\\cdot,\\cdot)$ when the surplus process becomes negative. Finally, we consider Markov-modulated Brownian motion (MMBM) as a special case and present analytical and numerical results for a particular choice of piecewise intensity function $\\omega(\\cdot,\\cdot)$ .
Publisher: Cambridge University Press (CUP)
Date: 09-2004
Abstract: For a K -stage cyclic queueing network with N customers and general service times, we provide bounds on the n th departure time from each stage. Furthermore, we analyze the asymptotic tail behavior of cycle times and waiting times given that at least one service-time distribution is subexponential.
Publisher: Cambridge University Press (CUP)
Date: 08-2011
DOI: 10.1017/S0021900200099150
Abstract: In this paper we consider a general Lévy process X reflected at a downward periodic barrier A t and a constant upper barrier K , giving a process V K t = X t + L A t − L K t . We find the expression for a loss rate defined by l K =E L K 1 and identify its asymptotics as K →∞ when X has light-tailed jumps and E X 1 & .
Publisher: Cambridge University Press (CUP)
Date: 09-2021
DOI: 10.1017/APR.2020.71
Abstract: In this paper we analyse the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of models in the theory of matrix-analytic methods. So far, only transient and stationary analyses of SFMs have been considered in the literature. The limiting conditional distribution gives useful insights into what happens when the process has been evolving for a long time, given that its busy period has not ended yet. We derive expressions for the Yaglom limit in terms of the singularity˜ $s^*$ such that the key matrix of the SFM, ${\\boldsymbol{\\Psi}}(s)$ , is finite (exists) for all $s\\geq s^*$ and infinite for $s s^*$ . We show the uniqueness of the Yaglom limit and illustrate the application of the theory with simple ex les.
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2017
Publisher: Informa UK Limited
Date: 02-10-2013
Publisher: Springer Science and Business Media LLC
Date: 20-09-2022
DOI: 10.1007/S10687-021-00427-1
Abstract: Motivated by a seminal paper of Kesten et al. ( Ann. Probab. , 3(1) , 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters A n , n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of $\\xi _{n}:=\\log ((1-A_{n})/A_{n})$ ξ n : = log ( ( 1 − A n ) / A n ) is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n th generation which becomes even heavier with increase of n . More precisely, we prove that, for all n , the distribution tail $\\mathbb {P}(Z_{n} \\ge m)$ ℙ ( Z n ≥ m ) of the n th population size Z n is asymptotically equivalent to $n\\overline F(\\log m)$ n F ¯ ( log m ) as m grows. In this way we generalise Bhattacharya and Palmowski ( Stat. Probab. Lett. , 154 , 108550, 2019) who proved this result in the case n = 1 for regularly varying environment F with parameter α 1. Further, for a subcritical branching process with subexponentially distributed ξ n , we provide the asymptotics for the distribution tail $\\mathbb {P}(Z_{n} m)$ ℙ ( Z n m ) which are valid uniformly for all n , and also for the stationary tail distribution. Then we establish the “principle of a single atypical environment” which says that the main cause for the number of particles to be large is the presence of a single very small environmental parameter A k .
Publisher: Elsevier BV
Date: 03-2017
Publisher: Informa UK Limited
Date: 09-11-2020
Publisher: Cambridge University Press (CUP)
Date: 13-06-2022
DOI: 10.1017/APR.2021.49
Abstract: Let $X_t^\\sharp$ be a multivariate process of the form $X_t =Y_t - Z_t$ , $X_0=x$ , killed at some terminal time T , where $Y_t$ is a Markov process having only jumps of length smaller than $\\delta$ , and $Z_t$ is a compound Poisson process with jumps of length bigger than $\\delta$ , for some fixed $\\delta $ . Under the assumptions that the summands in $Z_t$ are subexponential, we investigate the asymptotic behaviour of the potential function $u(x)= \\mathbb{E}^x \\int_0^\\infty \\ell\\big(X_s^\\sharp\\big)ds$ . The case of heavy-tailed entries in $Z_t$ corresponds to the case of ‘big claims’ in insurance models and is of practical interest. The main approach is based on the fact that u ( x ) satisfies a certain renewal equation.
Publisher: Elsevier BV
Date: 08-2009
Publisher: Springer Science and Business Media LLC
Date: 1999
Publisher: Elsevier BV
Date: 12-1996
Publisher: IOP Publishing
Date: 08-2023
Abstract: Stochastic resetting is a rapidly developing topic in the field of stochastic processes and their applications. It denotes the occasional reset of a diffusing particle to its starting point and effects, inter alia, optimal first-passage times to a target. Recently the concept of partial resetting, in which the particle is reset to a given fraction of the current value of the process, has been established and the associated search behaviour analysed. Here we go one step further and we develop a general technique to determine the time-dependent probability density function (PDF) for Markov processes with partial resetting. We obtain an exact representation of the PDF in the case of general symmetric Lévy flights with stable index 0 α ⩽ 2 . For Cauchy and Brownian motions (i.e. α = 1 , 2 ), this PDF can be expressed in terms of elementary functions in position space. We also determine the stationary PDF. Our numerical analysis of the PDF demonstrates intricate crossover behaviours as function of time.
Publisher: Springer International Publishing
Date: 2019
Publisher: Springer Science and Business Media LLC
Date: 25-04-2018
Publisher: Elsevier BV
Date: 10-2002
Publisher: Springer Science and Business Media LLC
Date: 13-03-2021
DOI: 10.1007/S11134-021-09698-8
Abstract: This paper presents an analysis of the stochastic recursion $$W_{i+1} = [V_iW_i+Y_i]^+$$ W i + 1 = [ V i W i + Y i ] + that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model’s stability condition. Writing $$Y_i=B_i-A_i$$ Y i = B i - A i , for independent sequences of nonnegative i.i.d. random variables $$\\{A_i\\}_{i\\in {\\mathbb N}_0}$$ { A i } i ∈ N 0 and $$\\{B_i\\}_{i\\in {\\mathbb N}_0}$$ { B i } i ∈ N 0 , and assuming $$\\{V_i\\}_{i\\in {\\mathbb N}_0}$$ { V i } i ∈ N 0 is an i.i.d. sequence as well (independent of $$\\{A_i\\}_{i\\in {\\mathbb N}_0}$$ { A i } i ∈ N 0 and $$\\{B_i\\}_{i\\in {\\mathbb N}_0}$$ { B i } i ∈ N 0 ), we then consider three special cases (i) $$V_i$$ V i equals a positive value a with certain probability $$p\\in (0,1)$$ p ∈ ( 0 , 1 ) and is negative otherwise, and both $$A_i$$ A i and $$B_i$$ B i have a rational LST, (ii) $$V_i$$ V i attains negative values only and $$B_i$$ B i has a rational LST, (iii) $$V_i$$ V i is uniformly distributed on [0, 1], and $$A_i$$ A i is exponentially distributed. In all three cases, we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.
Publisher: Elsevier BV
Date: 08-2017
Publisher: Institute of Mathematical Statistics
Date: 2015
DOI: 10.1214/ECP.V20-2999
Publisher: Cambridge University Press (CUP)
Date: 10-1999
DOI: 10.1017/S026996489913403X
Abstract: In this paper we consider an infinite buffer fluid model whose input is driven by independent semi-Markov processes. The output capacity of the buffer is a constant. We derive upper and lower bounds for the limiting distribution of the stationary buffer content process. We discuss ex les and applications where the results can be used to determine bounds on the loss probability in telecommunication networks.
Publisher: Springer Science and Business Media LLC
Date: 2003
Publisher: Cambridge University Press (CUP)
Date: 12-2010
Publisher: Cambridge University Press (CUP)
Date: 30-08-2023
DOI: 10.1017/JPR.2022.27
Abstract: We analyse an additive-increase and multiplicative-decrease (also known as growth–collapse) process that grows linearly in time and that, at Poisson epochs, experiences downward jumps that are (deterministically) proportional to its present position. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All eight Laplace transforms can be described in terms of two so-called scale functions associated with the upward one-sided exit time and with the upward two-sided exit time. All other Laplace transforms can be obtained from the above scale functions by taking limits, derivatives, integrals, and combinations of these.
Publisher: Informa UK Limited
Date: 31-08-2017
Publisher: Elsevier BV
Date: 2012
Publisher: Informa UK Limited
Date: 22-11-2024
Publisher: Informa UK Limited
Date: 07-2012
Publisher: Springer Science and Business Media LLC
Date: 21-02-2013
Publisher: Cambridge University Press (CUP)
Date: 09-2006
DOI: 10.1017/S0001867800001270
Abstract: In this paper we study the busy period in GI/G/1 work-conserving queues. We give the exact asymptotics of the tail distribution of the busy period under the light tail assumptions. We also study the workload process in the M/G/1 system conditioned to stay positive.
Publisher: Springer Berlin Heidelberg
Date: 2008
Publisher: Cambridge University Press (CUP)
Date: 03-2003
Abstract: This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W ( t ) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜 . In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜 . In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .
Publisher: Institute of Mathematical Statistics
Date: 2013
DOI: 10.1214/EJP.V18-1958
Publisher: Cambridge University Press (CUP)
Date: 12-2011
Abstract: In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit ex les.
Publisher: Institute of Mathematical Statistics
Date: 12-2008
DOI: 10.1214/08-AAP529
Publisher: Springer Science and Business Media LLC
Date: 22-08-2022
DOI: 10.1007/S11134-022-09858-4
Abstract: We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy-period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a Balancing Reduces Asymptotic Variance of Outputs (BRAVO) phenomenon for the work-output process (namely the busy time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy period. In the current paper, we strengthen the previous results by reducing to assumptions to existence of $$2+\\epsilon $$ 2 + ϵ moments.
Publisher: Springer International Publishing
Date: 2019
Publisher: Springer Science and Business Media LLC
Date: 09-07-2011
Publisher: Bernoulli Society for Mathematical Statistics and Probability
Date: 05-2013
DOI: 10.3150/11-BEJ404
Publisher: Institute of Mathematical Statistics
Date: 02-2007
Publisher: Institute of Mathematical Statistics
Date: 08-2015
DOI: 10.1214/14-AAP1038
Publisher: Elsevier BV
Date: 12-2018
Publisher: Cambridge University Press (CUP)
Date: 06-2007
DOI: 10.1017/S0021900200117875
Abstract: We give precise asymptotic estimates of the tail behavior of the distribution of the supremum of a process with regenerative increments. Our results cover four qualitatively different regimes involving both light tails and heavy tails, and are illustrated with ex les arising in queueing theory and insurance risk.
Publisher: Springer Science and Business Media LLC
Date: 10-08-2020
DOI: 10.1007/S11134-020-09664-W
Abstract: In this note, we prove that the speed of convergence of the workload of a Lévy-driven queue to the quasi-stationary distribution is of order 1/ t . We identify also the Laplace transform of the measure giving this speed and provide some ex les.
Publisher: Springer Science and Business Media LLC
Date: 2001
Publisher: Wiley
Date: 15-03-2021
DOI: 10.1111/MAFI.12301
Publisher: Cambridge University Press (CUP)
Date: 08-2011
Abstract: In this paper we consider a general Lévy process X reflected at a downward periodic barrier A t and a constant upper barrier K , giving a process V K t = X t + L A t − L K t . We find the expression for a loss rate defined by l K =E L K 1 and identify its asymptotics as K →∞ when X has light-tailed jumps and E X 1 .
Publisher: Institute of Mathematical Statistics
Date: 08-2005
Publisher: Elsevier BV
Date: 07-2020
Publisher: MDPI AG
Date: 29-01-2023
DOI: 10.3390/JRFM16020082
Abstract: We derive the explicit price of the perpetual American put option canceled at the last-passage time of the underlying above some fixed level. We assume that the asset process is governed by a geometric spectrally negative Lévy process. We show that the optimal exercise time is the first moment when the asset price process drops below an optimal threshold. We perform numerical analysis considering classical Black–Scholes models and the model where the logarithm of the asset price has additional exponential downward shocks. The proof is based on some martingale arguments and the fluctuation theory of Lévy processes.
Publisher: MDPI AG
Date: 25-03-2019
DOI: 10.3390/RISKS7010034
Abstract: We study a portfolio selection problem in a continuous-time Itô–Markov additive market with prices of financial assets described by Markov additive processes that combine Lévy processes and regime switching models. Thus, the model takes into account two sources of risk: the jump diffusion risk and the regime switching risk. For this reason, the market is incomplete. We complete the market by enlarging it with the use of a set of Markovian jump securities, Markovian power-jump securities and impulse regime switching securities. Moreover, we give conditions under which the market is asymptotic-arbitrage-free. We solve the portfolio selection problem in the Itô–Markov additive market for the power utility and the logarithmic utility.
Publisher: Cambridge University Press (CUP)
Date: 06-2007
Abstract: We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of idends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the n th moment of the net present value of idends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the n th moment of the net present value of idends, makes a new link with the distribution of integrated exponential subordinators.
Publisher: Springer Science and Business Media LLC
Date: 24-10-2018
Publisher: Elsevier BV
Date: 11-2020
Publisher: MDPI AG
Date: 28-08-2020
DOI: 10.3390/RISKS8030090
Abstract: In this work, we adapt a Monte Carlo algorithm introduced by Broadie and Glasserman in 1997 to price a π-option. This method is based on the simulated price tree that comes from discretization and replication of possible trajectories of the underlying asset’s price. As a result, this algorithm produces the lower and the upper bounds that converge to the true price with the increasing depth of the tree. Under specific parametrization, this π-option is related to relative maximum drawdown and can be used in the real market environment to protect a portfolio against volatile and unexpected price drops. We also provide some numerical analysis.
Publisher: Springer Science and Business Media LLC
Date: 31-05-2012
Publisher: Elsevier BV
Date: 09-2019
Publisher: ACM
Date: 2012
Publisher: MDPI AG
Date: 26-08-2021
DOI: 10.3390/RISKS9090157
Abstract: In this paper, we generate boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and a hypoexponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations, we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are linearly dependent on reserves, representing, for instance, returns on risk-free investments of the insurance capital, we firstly derive explicit solutions of the ordinary differential equations under considerations, in terms of special mathematical functions and integrals, from which we can further determine their asymptotics. This allows us to recover the ruin probabilities obtained for general premiums dependent on reserves. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve owned by the insurance company.
Publisher: Elsevier BV
Date: 07-2011
Publisher: Elsevier BV
Date: 03-2018
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2013
DOI: 10.1137/110852000
Publisher: Bernoulli Society for Mathematical Statistics and Probability
Date: 08-2006
Publisher: Elsevier BV
Date: 11-2019
Publisher: Informa UK Limited
Date: 26-06-2017
Publisher: Cambridge University Press (CUP)
Date: 06-2007
DOI: 10.1017/S0021900200003016
Abstract: We give precise asymptotic estimates of the tail behavior of the distribution of the supremum of a process with regenerative increments. Our results cover four qualitatively different regimes involving both light tails and heavy tails, and are illustrated with ex les arising in queueing theory and insurance risk.
Publisher: Cambridge University Press (CUP)
Date: 06-2007
Abstract: We give precise asymptotic estimates of the tail behavior of the distribution of the supremum of a process with regenerative increments. Our results cover four qualitatively different regimes involving both light tails and heavy tails, and are illustrated with ex les arising in queueing theory and insurance risk.
Publisher: Cambridge University Press (CUP)
Date: 03-2010
DOI: 10.1017/S0021900200006409
Abstract: We investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.
Publisher: Institute of Mathematical Statistics
Date: 2018
DOI: 10.1214/18-EJP133
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