Upper and lower bounds for ruin probability
|
|
- Philippa Cole
- 5 years ago
- Views:
Transcription
1 Upper and lower bounds for ruin probability E. Pancheva,Z.Volkovich and L.Morozensky 3 Institute of Mathematics and Informatics, the Bulgarian Academy of Sciences, 3 Sofia, Bulgaria pancheva@math.bas.bg Software Engineering Department, ORT Braude College of Engineering, Karmiel 9, Israel Affiliate Professor, Department of Mathematics and Statistics, the University of Maryland, Baltimore County, USA vlvolkov@ort.org.il 3 Software Engineering Department, ORT Braude College of Engineering, Karmiel 9, Israel leonatm@bezeqint.net Abstract In this note we discuss upper and lower bound for the ruin probability in an insurance model with very heavy-tailed claims and interarrival times. Key Words and Phrases: compound extremal processes; α-stable approximation; ruin probability Correspondence author
2 Backgrounds The framework of our study is set by a given Bernoulli point process (Bpp) N = {(T k,x k ):k } on the time-state space S =(, ) (, ). By definition (cf. Balkema and Pancheva 99) N is simple in time (T k T j a.s. for k j), its mean measure is finite on compact subsets of S and all restrictions of N to slices over disjoint time intervals are independent. We assume that: a) the sequences {T k } and {X k } are independent and defined on the same probability space; b) the state points {X k } are independent and identically distributed random variables (iid rv s) on (, ) with common distribution function (df) F which is asymptotically continuous at infinity; c) the time points {T k } are increasing to infinity, i.e. <T <T <..., T k a.s. The main problem in the Extreme Value Theory is the asymptotic of the extremal process { X k : T k t} = N(t) X k, associated with N,fort. k k= Here the maximum operation between rv s is denoted by andn(t) := max{k : T k t} is the counting process of N. The method usually used is to choose proper time-space changes ζ n =(τ n (t),u n (x)) of S (i.e. strictly increasing and continuous in both components) such that for n and t> the weak convergence Ỹ n (t) :={ u n (X k):τn (T k) t} = Ỹ (t) () k to a non-degenerate extremal process holds. (For weak convergence of extremal processes consult e.g. Balkema and Pancheva 99.) In fact, the classical Extreme Value Theory deals with Bpp s {(t k,x k ): k } with deterministic time points t k,<t <t <..., t k. One investigates the weak convergence to a non-degenerate extremal process Y n (t) :={ k u n (X k ):t k τ n (t)} = Y (t) () under the assumption that the norming sequence {ζ n } is regular. The later means that for all s > and for n there exist point-wise lim n u n u [ns] (x) = U s (x) lim τ n n τ [ns](t) =σ s (t) and (σ s (t), U s (x)) is a time-space change. As usual means the composition and [s] the integer part of s. The family L = {(σ s (t), U s (x)) : s>} forms a continuous one-parameter group w.r.t. composition. Let us denote the (deterministic) counting function k(t) =max{k : t k t}, and put k n (t) :=k(τ n (t)), k n := k n (). The df of the limit extremal process
3 in() wedenotebyg(t, x) := P(Y (t) < x), and set G(x) := g(,x). Then necessary and sufficient conditions for convergence () are the following. F kn (u n (x)) w G(x), n. kn(t) k n λ(t), n, t >. The regularity of the norming sequence {ζ n } has some important consequences (cf. Pancheva 99). First of all, the limit extremal process Y (t) is self-similar w.r.t. L, i.e. U s Y (t) d = Y σ s (t), s >. Furthermore:. k [ns] k n s a,n,forsomea>andalls>;. the limit df G is max-stable in the sense that G s (x) =G(L s (x)) s >, L s := U a s; (3). the intensity function λ(t) is continuous. Thus, under conditions. and. and the regularity of the norming sequence, the limit extremal process Y (t) is stochastically continuous with df g(t, x) =G λ(t) (x) and the process Y λ (t) is max-stable in the sense of (3). Let us come back to the point process N with the random time points T k. The Functional Transfer Theorem (FTT) in this framework gives conditions on N for the weak convergence () and determines the explicit form of the limit df f(t, x) :=P(Ỹ (t) <x). In other words, the weak convergence () in the framework with non-random time points can be transfer to the framework of N if some additional condition on the point process N is met. In our case this is condition d) below. Denote by M([, )) the space of all strictly increasing, cadlac functions y :[, ) [, ), y() =, y(t) as t. We assume additionally to a) - c) the following condition d) θ n (s) :=τn (T [skn]) = T (s) where T :[, ) [, ) is a random time change, i.e. stochastically continuous process with sample paths in M([, )). Let us set N n (t) :=N(τ n (t)). In view of condition d) the sequence Λ n (t) := N n(t) = max{k : T k τ n (t)} k n k n = sup{s >:τn (T [skn]) t} = sup{s >:θ n (s) t} is weakly convergent to the inverse process of T (s). Let us denote it by Λ and let Q t (s) =P (Λ(t) <s). 3
4 Now we are ready to state a general FTT for maxima of iid rv s on (, ). Theorem (FTT): Let N = {(T k,x k ):k } be a Bpp described by conditions a) - c). Assume further that there is a regular norming sequence ζ n (t, x) =(τ n (t),u n (x)) of time-space changes of S such that for n and t> conditions.,. and d) hold. Then i) Nn(t) k n ii) P( Nn(t) d Λ(t) k= u n (X k) <x) w E[G(x)] Λ(t) Indeed, we have to show only ii). Observe that for n N n (t) =k n. N n(t) k n.λ(t) k n (λ Λ(t)) k n In the last asymptotic relation we have used condition ). Then by convergence () Ỹ n (t) = Nn(t) (X k )= Y (λ Λ(t)) and k= u n P( Nn(t) k= u n (X k) <x) f(t, x) = G s (x)dq t (s) =E[G(x)] Λ(t) Let us apply these results to a particular insurance risk model. Application to ruin probability The insurance model, we are dealing with here, can be described by a particular Bpp N = {(T k,x k ):k } where a) the claim sizes {X k } are positive iid random variables which df F has a regularly varying tail, i.e. F RV α. We consider the very heavy tail case <α<whenex does not exist, briefly EX = ; b) the claims occur at times {T k } where <T <T <... < T k a.s. We denote the inter-arrival times by J k = T k T k, k, T =and assume the random variables {J k } positive iid with df H. Suppose H RV β, <β<; c) both sequences {X k } and {T k } are independent and defined on the same probability space. The point process N generates the following random processes we are interested in.
5 i) The counting process N(t) =max{k : T k t}. It is a renewal process with N(t) t ast forej =. By the Stable CLT there exists a normalizing sequence {b(n)}, b(n) >, such that [nt] J k k= b(n) converges weakly to a β- stable Levy process S β (t). One can choose b(n) n /β L J (n), where L J denotes a slowly varying function. Let us determine b(n) by the asymptotic relation b( b(n)) n as n. Now the normalized counting process N(nt) is b(n) weakly convergent to the hitting time process E(t) =inf{s : S β (s) >t} of S β, see Meerschaert and Scheffler (). As inverse of S β, E(t) isβ-selfsimilar. ii) The extremal claim process Y (t) ={ X k : T k t} = N(t) X k. In view k= of assumption a) there exist norming constants B(n) n /α L X (n) such that [nt] k= X k B(n) converges weakly to an extremal process Y α(t) with Frechet marginal df, i.e. P (Y α (t) <x)=φ t α(x) =exp tx α.consequently, Y n (t) := N(nt) k= X k B( b(n)) = Y α(e(t)). Below we use the β α - selfsimilarity of the compound extremal process Y α(e(t)) (see e.g. Pancheva et al. 3). iii) The accumulated claim process S(t) = N(t) k= X k. Using the same norming sequence as above we observe that S n (t) := N(nt) k= X k B( b(n)) = Z α(e(t)). Here Z α is an α-stable Levy process and the composition Z α (E(t)) is β α - selfsimilar. iv) The risk process R(t) =c(t) S(t). Here u := c() is the initial capital and c(t) denotes the premium income up to time t, hence it is an increasing curve. We assume c(t) right-continuous. Note, the extremal claim process Y (t) and the accumulated claim process S(t) need the same time-space changes ζ n (t, x) =(nt, x B( b(n)) )toachieveweak convergence to a proper limiting process. In fact, {ζ n } makes the claim sizes smaller and compensates this by increasing their number in the interval [,t]. Both processes Y n (t) and S n (t) are generated by the point process N n = {( T k n, X k ) : k }. B( b(n)) risk processes R n (t) = the condition d) c(nt) B( b(n)) With the latter we also associate the sequence of c(nt) B( b(n)) S n(t). Let us assume additionally to a) - c) w c (t), c increasing curve with c () >. 5
6 Under conditions a) - d) the sequence R n converges weakly to the risk process (cf Furrer et al. 997) R α,β (t) =c (t) Z α (E(t)) with initial capital u = c (). Using the R α,β - approximation of the initial risk process R(t), when time and initial capital increase with n, we next obtain upper ( ψ) and lower(ψ) bound for the ruin probability Ψ(c, t) :=P (inf s t R(s) < ). Let Z α () and E() have df s G α and Q, resp. Then we have : ψ(c,t) := P ( inf α,β(s) < ) s t P (sup s t Z α (E(s)) >u ) P (Z α (E(t)) >u ) ) α = Q(( u )dg xt β α (x) =: ψ(c,t) α Here Q = Q. On the other hand ψ(c,t) P (Y α (E(t)) >c (t)) ( ) α c (t) = Q( )dφ α (x) =:ψ(c,t) xt β α Here we have used the self-similarity of the processes Z α, Y α and E. Thus, finally we get ψ(c,t) ψ(c,t) ψ(c,t) Remember, our initial insurance model was described by the point process N with the associated risk process R(t). We have denoted the corresponding ruin probability by Ψ(c, t) withu = c(). Then N(s) Ψ(c, t) =P ( inf {c(s) X k } < ) s t k= = P ( inf { c(ns) N(ns) s t n B( b(n)) X k B( b(n)) } < ) k= Now let initial capital u and time t increase with n in such a way that u = u t B( b(n)), n = t. We observe that under conditions a) - d) we may approximate Ψ(c, t) ψ(c,t ) and consequently for u and t large enough ψ(c,t ) Ψ(c, t) ψ(c,t ) ()
7 3 Examples Assume that our model is characterized by α =, i.e. the df of Z α () is the Levy df G α (x) =( Φ( x )). Here Φ is the standard normal df. We suppose also that the random variable E() is Exp()-distributed, namely Q(s) = e s, s. Further, let us take the income curve c to be of the special form c (t) =u + t β α c, c positive constant, that agrees with the selfsimilarity of the process Z α (E(t)). Now the upper bound depends on (u,t,β) and the lower bound depends on (u,t,β,c). We calculate the bounds ψ and ψ in two cases α > β = and α < β =.75 by using MATLAB7. The results of the calculations show clearly that in case β>α, when large claims arrive often, the bounds of the ruin probability are larger than in the case β<α, even in small time interval. Note,ifwechoosetheincomecurveintheabovespecialform,wemay calculate the ruin probability ψ(c,t ) in the approximating model exactly, namely ψ(c,t ) = P ( inf {u + s β α c Zα (E(s))} < ) s t = P ( inf {s β α (c Zα (E()))} < u ) s t = P ( inf {s β α (c Zα (E()))} < u, s t c Z α (E()) < ) = P (t β α (c Z α (E())) < u, c Z α (E()) < ) = P (Z α (E()) >c+ u ) t β α ( ) α c (t ) = Q( )dg xt β α (x) α Below we give graphical results related to the computation of ψ(c,t ), ψ(c,t )and ψ(c,t ) in the cases: c=., c=, c= when α = and β = β =.75. 7
8 Graphics of ψ(c,t ),ψ(c,t ) and ψ(c,t ) Figure : α =,β =,c= Figure : α =,β =,c=
9 Figure 3: α =,β =,c= Figure : α =,β =.75,c=. 9
10 Figure 5: α =,β =.75,c= Figure : α =,β =.75,c=.
11 References [] Balkema A., Pancheva E. (99) Decomposition for Multivariate Extremal Processes. Commun. Statist.-Theory Meth., Vol. 5, Nr [] Furrer H., Michna Zb. (997) Stable Levy Motion Approximation in Collective Risk Theory. Insurance : Mathem. and Econom., Vol [3] Meerschaert M.M., Scheffler H-P. () Limit Theorems for Continuous Time Random Walks. Submitted. [] Pancheva E., Kolkovska E., Yordanova P. (3) Random Time-Changed Extremal Processes. Comun. Tecnica No I-3-/7--3 PE/CIMAT. [5] Pancheva E. (99) Self-similar Extremal Processes. J. Math. Sci., Vol. 9, Nr
Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications
Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications Remigijus Leipus (with Yang Yang, Yuebao Wang, Jonas Šiaulys) CIRM, Luminy, April 26-30, 2010 1. Preliminaries
More informationStability of the Defect Renewal Volterra Integral Equations
19th International Congress on Modelling and Simulation, Perth, Australia, 12 16 December 211 http://mssanz.org.au/modsim211 Stability of the Defect Renewal Volterra Integral Equations R. S. Anderssen,
More informationFinite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims
Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. Finite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims Tao Jiang Abstract
More informationModèles de dépendance entre temps inter-sinistres et montants de sinistre en théorie de la ruine
Séminaire de Statistiques de l'irma Modèles de dépendance entre temps inter-sinistres et montants de sinistre en théorie de la ruine Romain Biard LMB, Université de Franche-Comté en collaboration avec
More informationPoisson Processes. Stochastic Processes. Feb UC3M
Poisson Processes Stochastic Processes UC3M Feb. 2012 Exponential random variables A random variable T has exponential distribution with rate λ > 0 if its probability density function can been written
More informationA Note On The Erlang(λ, n) Risk Process
A Note On The Erlangλ, n) Risk Process Michael Bamberger and Mario V. Wüthrich Version from February 4, 2009 Abstract We consider the Erlangλ, n) risk process with i.i.d. exponentially distributed claims
More informationFunctional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals
Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico
More informationRENEWAL PROCESSES AND POISSON PROCESSES
1 RENEWAL PROCESSES AND POISSON PROCESSES Andrea Bobbio Anno Accademico 1997-1998 Renewal and Poisson Processes 2 Renewal Processes A renewal process is a point process characterized by the fact that the
More informationRuin Probabilities of a Discrete-time Multi-risk Model
Ruin Probabilities of a Discrete-time Multi-risk Model Andrius Grigutis, Agneška Korvel, Jonas Šiaulys Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 4, Vilnius LT-035, Lithuania
More informationMath 576: Quantitative Risk Management
Math 576: Quantitative Risk Management Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 11 Haijun Li Math 576: Quantitative Risk Management Week 11 1 / 21 Outline 1
More informationModelling the risk process
Modelling the risk process Krzysztof Burnecki Hugo Steinhaus Center Wroc law University of Technology www.im.pwr.wroc.pl/ hugo Modelling the risk process 1 Risk process If (Ω, F, P) is a probability space
More informationMultivariate Risk Processes with Interacting Intensities
Multivariate Risk Processes with Interacting Intensities Nicole Bäuerle (joint work with Rudolf Grübel) Luminy, April 2010 Outline Multivariate pure birth processes Multivariate Risk Processes Fluid Limits
More informationExtremes and ruin of Gaussian processes
International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo. June 28-30, 2006 Extremes and ruin of Gaussian processes Jürg Hüsler Department of Math. Statistics, University
More informationRare event simulation for the ruin problem with investments via importance sampling and duality
Rare event simulation for the ruin problem with investments via importance sampling and duality Jerey Collamore University of Copenhagen Joint work with Anand Vidyashankar (GMU) and Guoqing Diao (GMU).
More informationLimit Laws for Maxima of Functions of Independent Non-identically Distributed Random Variables
ProbStat Forum, Volume 07, April 2014, Pages 26 38 ISSN 0974-3235 ProbStat Forum is an e-journal. For details please visit www.probstat.org.in Limit Laws for Maxima of Functions of Independent Non-identically
More informationRegular Variation and Extreme Events for Stochastic Processes
1 Regular Variation and Extreme Events for Stochastic Processes FILIP LINDSKOG Royal Institute of Technology, Stockholm 2005 based on joint work with Henrik Hult www.math.kth.se/ lindskog 2 Extremes for
More informationLecture Notes on Risk Theory
Lecture Notes on Risk Theory February 2, 21 Contents 1 Introduction and basic definitions 1 2 Accumulated claims in a fixed time interval 3 3 Reinsurance 7 4 Risk processes in discrete time 1 5 The Adjustment
More informationReinsurance and ruin problem: asymptotics in the case of heavy-tailed claims
Reinsurance and ruin problem: asymptotics in the case of heavy-tailed claims Serguei Foss Heriot-Watt University, Edinburgh Karlovasi, Samos 3 June, 2010 (Joint work with Tomasz Rolski and Stan Zachary
More informationUniversity Of Calgary Department of Mathematics and Statistics
University Of Calgary Department of Mathematics and Statistics Hawkes Seminar May 23, 2018 1 / 46 Some Problems in Insurance and Reinsurance Mohamed Badaoui Department of Electrical Engineering National
More informationUpper Bounds for the Ruin Probability in a Risk Model With Interest WhosePremiumsDependonthe Backward Recurrence Time Process
Λ4flΛ4» ν ff ff χ Vol.4, No.4 211 8fl ADVANCES IN MATHEMATICS Aug., 211 Upper Bounds for the Ruin Probability in a Risk Model With Interest WhosePremiumsDependonthe Backward Recurrence Time Process HE
More informationIntroduction to Rare Event Simulation
Introduction to Rare Event Simulation Brown University: Summer School on Rare Event Simulation Jose Blanchet Columbia University. Department of Statistics, Department of IEOR. Blanchet (Columbia) 1 / 31
More informationStochastic Renewal Processes in Structural Reliability Analysis:
Stochastic Renewal Processes in Structural Reliability Analysis: An Overview of Models and Applications Professor and Industrial Research Chair Department of Civil and Environmental Engineering University
More informationMS&E 321 Spring Stochastic Systems June 1, 2013 Prof. Peter W. Glynn Page 1 of 7
MS&E 321 Spring 12-13 Stochastic Systems June 1, 213 Prof. Peter W. Glynn Page 1 of 7 Section 9: Renewal Theory Contents 9.1 Renewal Equations..................................... 1 9.2 Solving the Renewal
More informationAsymptotics of the Finite-time Ruin Probability for the Sparre Andersen Risk. Model Perturbed by an Inflated Stationary Chi-process
Asymptotics of the Finite-time Ruin Probability for the Sparre Andersen Risk Model Perturbed by an Inflated Stationary Chi-process Enkelejd Hashorva and Lanpeng Ji Abstract: In this paper we consider the
More informationClass 11 Non-Parametric Models of a Service System; GI/GI/1, GI/GI/n: Exact & Approximate Analysis.
Service Engineering Class 11 Non-Parametric Models of a Service System; GI/GI/1, GI/GI/n: Exact & Approximate Analysis. G/G/1 Queue: Virtual Waiting Time (Unfinished Work). GI/GI/1: Lindley s Equations
More informationRare Events in Random Walks and Queueing Networks in the Presence of Heavy-Tailed Distributions
Rare Events in Random Walks and Queueing Networks in the Presence of Heavy-Tailed Distributions Sergey Foss Heriot-Watt University, Edinburgh and Institute of Mathematics, Novosibirsk The University of
More informationSolutions For Stochastic Process Final Exam
Solutions For Stochastic Process Final Exam (a) λ BMW = 20 0% = 2 X BMW Poisson(2) Let N t be the number of BMWs which have passes during [0, t] Then the probability in question is P (N ) = P (N = 0) =
More informationStatistical test for some multistable processes
Statistical test for some multistable processes Ronan Le Guével Joint work in progress with A. Philippe Journées MAS 2014 1 Multistable processes First definition : Ferguson-Klass-LePage series Properties
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationRuin Probability for Non-standard Poisson Risk Model with Stochastic Returns
Ruin Probability for Non-standard Poisson Risk Model with Stochastic Returns Tao Jiang Abstract This paper investigates the finite time ruin probability in non-homogeneous Poisson risk model, conditional
More informationRMSC 2001 Introduction to Risk Management
RMSC 2001 Introduction to Risk Management Tutorial 4 (2011/12) 1 February 20, 2012 Outline: 1. Failure Time 2. Loss Frequency 3. Loss Severity 4. Aggregate Claim ====================================================
More informationQuantile-quantile plots and the method of peaksover-threshold
Problems in SF2980 2009-11-09 12 6 4 2 0 2 4 6 0.15 0.10 0.05 0.00 0.05 0.10 0.15 Figure 2: qqplot of log-returns (x-axis) against quantiles of a standard t-distribution with 4 degrees of freedom (y-axis).
More informationRuin probabilities in multivariate risk models with periodic common shock
Ruin probabilities in multivariate risk models with periodic common shock January 31, 2014 3 rd Workshop on Insurance Mathematics Universite Laval, Quebec, January 31, 2014 Ruin probabilities in multivariate
More informationChapter 2. Poisson Processes. Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan
Chapter 2. Poisson Processes Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan Outline Introduction to Poisson Processes Definition of arrival process Definition
More informationRMSC 2001 Introduction to Risk Management
RMSC 2001 Introduction to Risk Management Tutorial 4 (2011/12) 1 February 20, 2012 Outline: 1. Failure Time 2. Loss Frequency 3. Loss Severity 4. Aggregate Claim ====================================================
More informationarxiv: v1 [q-fin.rm] 27 Jun 2017
Risk Model Based on General Compound Hawkes Process Anatoliy Swishchuk 1 2 arxiv:1706.09038v1 [q-fin.rm] 27 Jun 2017 Abstract: In this paper, we introduce a new model for the risk process based on general
More informationABC methods for phase-type distributions with applications in insurance risk problems
ABC methods for phase-type with applications problems Concepcion Ausin, Department of Statistics, Universidad Carlos III de Madrid Joint work with: Pedro Galeano, Universidad Carlos III de Madrid Simon
More informationVaR vs. Expected Shortfall
VaR vs. Expected Shortfall Risk Measures under Solvency II Dietmar Pfeifer (2004) Risk measures and premium principles a comparison VaR vs. Expected Shortfall Dependence and its implications for risk measures
More informationMULTIVARIATE COMPOUND POINT PROCESSES WITH DRIFTS
MULTIVARIATE COMPOUND POINT PROCESSES WITH DRIFTS By HUAJUN ZHOU A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department
More informationNon-Life Insurance: Mathematics and Statistics
ETH Zürich, D-MATH HS 08 Prof. Dr. Mario V. Wüthrich Coordinator Andrea Gabrielli Non-Life Insurance: Mathematics and Statistics Solution sheet 9 Solution 9. Utility Indifference Price (a) Suppose that
More informationRandomly Weighted Sums of Conditionnally Dependent Random Variables
Gen. Math. Notes, Vol. 25, No. 1, November 2014, pp.43-49 ISSN 2219-7184; Copyright c ICSRS Publication, 2014 www.i-csrs.org Available free online at http://www.geman.in Randomly Weighted Sums of Conditionnally
More informationAsymptotic Statistics-III. Changliang Zou
Asymptotic Statistics-III Changliang Zou The multivariate central limit theorem Theorem (Multivariate CLT for iid case) Let X i be iid random p-vectors with mean µ and and covariance matrix Σ. Then n (
More informationProbability Models. 4. What is the definition of the expectation of a discrete random variable?
1 Probability Models The list of questions below is provided in order to help you to prepare for the test and exam. It reflects only the theoretical part of the course. You should expect the questions
More informationMartingale Theory for Finance
Martingale Theory for Finance Tusheng Zhang October 27, 2015 1 Introduction 2 Probability spaces and σ-fields 3 Integration with respect to a probability measure. 4 Conditional expectation. 5 Martingales.
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 Discrete Stochastic Processes Midterm Quiz April 6, 2010 There are 5 questions, each with several parts.
More informationThe finite-time Gerber-Shiu penalty function for two classes of risk processes
The finite-time Gerber-Shiu penalty function for two classes of risk processes July 10, 2014 49 th Actuarial Research Conference University of California, Santa Barbara, July 13 July 16, 2014 The finite
More informationAsymptotic statistics using the Functional Delta Method
Quantiles, Order Statistics and L-Statsitics TU Kaiserslautern 15. Februar 2015 Motivation Functional The delta method introduced in chapter 3 is an useful technique to turn the weak convergence of random
More informationQualifying Exam in Probability and Statistics. https://www.soa.org/files/edu/edu-exam-p-sample-quest.pdf
Part 1: Sample Problems for the Elementary Section of Qualifying Exam in Probability and Statistics https://www.soa.org/files/edu/edu-exam-p-sample-quest.pdf Part 2: Sample Problems for the Advanced Section
More informationSUMMARY OF RESULTS ON PATH SPACES AND CONVERGENCE IN DISTRIBUTION FOR STOCHASTIC PROCESSES
SUMMARY OF RESULTS ON PATH SPACES AND CONVERGENCE IN DISTRIBUTION FOR STOCHASTIC PROCESSES RUTH J. WILLIAMS October 2, 2017 Department of Mathematics, University of California, San Diego, 9500 Gilman Drive,
More informationA stretched-exponential Pinning Model with heavy-tailed Disorder
A stretched-exponential Pinning Model with heavy-tailed Disorder Niccolò Torri Université Claude-Bernard - Lyon 1 & Università degli Studi di Milano-Bicocca Roma, October 8, 2014 References A. Auffinger
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 383 2011 215 225 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Uniform estimates for the finite-time ruin
More informationThe exact asymptotics for hitting probability of a remote orthant by a multivariate Lévy process: the Cramér case
The exact asymptotics for hitting probability of a remote orthant by a multivariate Lévy process: the Cramér case Konstantin Borovkov and Zbigniew Palmowski Abstract For a multivariate Lévy process satisfying
More informationStability of optimization problems with stochastic dominance constraints
Stability of optimization problems with stochastic dominance constraints D. Dentcheva and W. Römisch Stevens Institute of Technology, Hoboken Humboldt-University Berlin www.math.hu-berlin.de/~romisch SIAM
More informationState-dependent Importance Sampling for Rare-event Simulation: An Overview and Recent Advances
State-dependent Importance Sampling for Rare-event Simulation: An Overview and Recent Advances By Jose Blanchet and Henry Lam Columbia University and Boston University February 7, 2011 Abstract This paper
More information18.175: Lecture 14 Infinite divisibility and so forth
18.175 Lecture 14 18.175: Lecture 14 Infinite divisibility and so forth Scott Sheffield MIT 18.175 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin
More informationHawkes Processes and their Applications in Finance and Insurance
Hawkes Processes and their Applications in Finance and Insurance Anatoliy Swishchuk University of Calgary Calgary, Alberta, Canada Hawks Seminar Talk Dept. of Math. & Stat. Calgary, Canada May 9th, 2018
More informationAsymptotic Analysis of Exceedance Probability with Stationary Stable Steps Generated by Dissipative Flows
Asymptotic Analysis of Exceedance Probability with Stationary Stable Steps Generated by Dissipative Flows Uğur Tuncay Alparslan a, and Gennady Samorodnitsky b a Department of Mathematics and Statistics,
More informationPoint Process Control
Point Process Control The following note is based on Chapters I, II and VII in Brémaud s book Point Processes and Queues (1981). 1 Basic Definitions Consider some probability space (Ω, F, P). A real-valued
More informationB. Appendix B. Topological vector spaces
B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function
More informationPrecise Large Deviations for Sums of Negatively Dependent Random Variables with Common Long-Tailed Distributions
This article was downloaded by: [University of Aegean] On: 19 May 2013, At: 11:54 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer
More informationASYMPTOTIC BEHAVIOR OF THE FINITE-TIME RUIN PROBABILITY WITH PAIRWISE QUASI-ASYMPTOTICALLY INDEPENDENT CLAIMS AND CONSTANT INTEREST FORCE
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 44, Number 5, 214 ASYMPTOTIC BEHAVIOR OF THE FINITE-TIME RUIN PROBABILITY WITH PAIRWISE QUASI-ASYMPTOTICALLY INDEPENDENT CLAIMS AND CONSTANT INTEREST FORCE
More informationAsymptotics of minimax stochastic programs
Asymptotics of minimax stochastic programs Alexander Shapiro Abstract. We discuss in this paper asymptotics of the sample average approximation (SAA) of the optimal value of a minimax stochastic programming
More informationInfinite Divisibility and Max-Infinite Divisibility with Random Sample Size
Statistical Methods 5 (2), 2003, 126-139 Infinite Divisibility and Max-Infinite Divisibility with Random Sample Size S. Satheesh Department of Statistics Cochin University of Science and Technology Cochin
More informationNotes 9 : Infinitely divisible and stable laws
Notes 9 : Infinitely divisible and stable laws Math 733 - Fall 203 Lecturer: Sebastien Roch References: [Dur0, Section 3.7, 3.8], [Shi96, Section III.6]. Infinitely divisible distributions Recall: EX 9.
More informationPractical conditions on Markov chains for weak convergence of tail empirical processes
Practical conditions on Markov chains for weak convergence of tail empirical processes Olivier Wintenberger University of Copenhagen and Paris VI Joint work with Rafa l Kulik and Philippe Soulier Toronto,
More informationUses of Asymptotic Distributions: In order to get distribution theory, we need to norm the random variable; we usually look at n 1=2 ( X n ).
1 Economics 620, Lecture 8a: Asymptotics II Uses of Asymptotic Distributions: Suppose X n! 0 in probability. (What can be said about the distribution of X n?) In order to get distribution theory, we need
More informationESTIMATING BIVARIATE TAIL
Elena DI BERNARDINO b joint work with Clémentine PRIEUR a and Véronique MAUME-DESCHAMPS b a LJK, Université Joseph Fourier, Grenoble 1 b Laboratoire SAF, ISFA, Université Lyon 1 Framework Goal: estimating
More informationThe Convergence Rate for the Normal Approximation of Extreme Sums
The Convergence Rate for the Normal Approximation of Extreme Sums Yongcheng Qi University of Minnesota Duluth WCNA 2008, Orlando, July 2-9, 2008 This talk is based on a joint work with Professor Shihong
More informationOperational Risk and Pareto Lévy Copulas
Operational Risk and Pareto Lévy Copulas Claudia Klüppelberg Technische Universität München email: cklu@ma.tum.de http://www-m4.ma.tum.de References: - Böcker, K. and Klüppelberg, C. (25) Operational VaR
More informationQuantitative Modeling of Operational Risk: Between g-and-h and EVT
: Between g-and-h and EVT Paul Embrechts Matthias Degen Dominik Lambrigger ETH Zurich (www.math.ethz.ch/ embrechts) Outline Basel II LDA g-and-h Aggregation Conclusion and References What is Basel II?
More informationOn heavy tailed time series and functional limit theorems Bojan Basrak, University of Zagreb
On heavy tailed time series and functional limit theorems Bojan Basrak, University of Zagreb Recent Advances and Trends in Time Series Analysis Nonlinear Time Series, High Dimensional Inference and Beyond
More informationf X (x) = λe λx, , x 0, k 0, λ > 0 Γ (k) f X (u)f X (z u)du
11 COLLECTED PROBLEMS Do the following problems for coursework 1. Problems 11.4 and 11.5 constitute one exercise leading you through the basic ruin arguments. 2. Problems 11.1 through to 11.13 but excluding
More informationNonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1
Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium
More information1 Poisson processes, and Compound (batch) Poisson processes
Copyright c 2007 by Karl Sigman 1 Poisson processes, and Compound (batch) Poisson processes 1.1 Point Processes Definition 1.1 A simple point process ψ = {t n : n 1} is a sequence of strictly increasing
More informationAssessing Dependence in Extreme Values
02/09/2016 1 Motivation Motivation 2 Comparison 3 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations 4 Idea Results Motivation Given historical flood levels, how high should
More informationStochastic volatility models: tails and memory
: tails and memory Rafa l Kulik and Philippe Soulier Conference in honour of Prof. Murad Taqqu 19 April 2012 Rafa l Kulik and Philippe Soulier Plan Model assumptions; Limit theorems for partial sums and
More informationBeyond the color of the noise: what is memory in random phenomena?
Beyond the color of the noise: what is memory in random phenomena? Gennady Samorodnitsky Cornell University September 19, 2014 Randomness means lack of pattern or predictability in events according to
More informationThe exponential distribution and the Poisson process
The exponential distribution and the Poisson process 1-1 Exponential Distribution: Basic Facts PDF f(t) = { λe λt, t 0 0, t < 0 CDF Pr{T t) = 0 t λe λu du = 1 e λt (t 0) Mean E[T] = 1 λ Variance Var[T]
More informationQingwu Gao and Yang Yang
Bull. Korean Math. Soc. 50 2013, No. 2, pp. 611 626 http://dx.doi.org/10.4134/bkms.2013.50.2.611 UNIFORM ASYMPTOTICS FOR THE FINITE-TIME RUIN PROBABILITY IN A GENERAL RISK MODEL WITH PAIRWISE QUASI-ASYMPTOTICALLY
More informationErgodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R.
Ergodic Theorems Samy Tindel Purdue University Probability Theory 2 - MA 539 Taken from Probability: Theory and examples by R. Durrett Samy T. Ergodic theorems Probability Theory 1 / 92 Outline 1 Definitions
More informationA SHORT INTRODUCTION TO BANACH LATTICES AND
CHAPTER A SHORT INTRODUCTION TO BANACH LATTICES AND POSITIVE OPERATORS In tis capter we give a brief introduction to Banac lattices and positive operators. Most results of tis capter can be found, e.g.,
More informationShape of the return probability density function and extreme value statistics
Shape of the return probability density function and extreme value statistics 13/09/03 Int. Workshop on Risk and Regulation, Budapest Overview I aim to elucidate a relation between one field of research
More informationOperational Risk and Pareto Lévy Copulas
Operational Risk and Pareto Lévy Copulas Claudia Klüppelberg Technische Universität München email: cklu@ma.tum.de http://www-m4.ma.tum.de References: - Böcker, K. and Klüppelberg, C. (25) Operational VaR
More informationLIMITS FOR QUEUES AS THE WAITING ROOM GROWS. Bell Communications Research AT&T Bell Laboratories Red Bank, NJ Murray Hill, NJ 07974
LIMITS FOR QUEUES AS THE WAITING ROOM GROWS by Daniel P. Heyman Ward Whitt Bell Communications Research AT&T Bell Laboratories Red Bank, NJ 07701 Murray Hill, NJ 07974 May 11, 1988 ABSTRACT We study the
More informationOn pathwise stochastic integration
On pathwise stochastic integration Rafa l Marcin Lochowski Afican Institute for Mathematical Sciences, Warsaw School of Economics UWC seminar Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic
More informationTail Dependence of Multivariate Pareto Distributions
!#"%$ & ' ") * +!-,#. /10 243537698:6 ;=@?A BCDBFEHGIBJEHKLB MONQP RS?UTV=XW>YZ=eda gihjlknmcoqprj stmfovuxw yy z {} ~ ƒ }ˆŠ ~Œ~Ž f ˆ ` š œžÿ~ ~Ÿ œ } ƒ œ ˆŠ~ œ
More informationWeak Limits for Multivariate Random Sums
Journal of Multivariate Analysis 67, 398413 (1998) Article No. MV981768 Weak Limits for Multivariate Random Sums Tomasz J. Kozubowski The University of Tennessee at Chattanooga and Anna K. Panorska - The
More informationLimiting Distributions
Limiting Distributions We introduce the mode of convergence for a sequence of random variables, and discuss the convergence in probability and in distribution. The concept of convergence leads us to the
More informationClassical Extreme Value Theory - An Introduction
Chapter 1 Classical Extreme Value Theory - An Introduction 1.1 Introduction Asymptotic theory of functions of random variables plays a very important role in modern statistics. The objective of the asymptotic
More informationTrivariate polynomial approximation on Lissajous curves 1
Trivariate polynomial approximation on Lissajous curves 1 Stefano De Marchi DMV2015, Minisymposium on Mathematical Methods for MPI 22 September 2015 1 Joint work with Len Bos (Verona) and Marco Vianello
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More information5 Introduction to the Theory of Order Statistics and Rank Statistics
5 Introduction to the Theory of Order Statistics and Rank Statistics This section will contain a summary of important definitions and theorems that will be useful for understanding the theory of order
More informationPoisson Processes. Particles arriving over time at a particle detector. Several ways to describe most common model.
Poisson Processes Particles arriving over time at a particle detector. Several ways to describe most common model. Approach 1: a) numbers of particles arriving in an interval has Poisson distribution,
More informationExtreme Value Theory and Applications
Extreme Value Theory and Deauville - 04/10/2013 Extreme Value Theory and Introduction Asymptotic behavior of the Sum Extreme (from Latin exter, exterus, being on the outside) : Exceeding the ordinary,
More informationLecture 5. If we interpret the index n 0 as time, then a Markov chain simply requires that the future depends only on the present and not on the past.
1 Markov chain: definition Lecture 5 Definition 1.1 Markov chain] A sequence of random variables (X n ) n 0 taking values in a measurable state space (S, S) is called a (discrete time) Markov chain, if
More informationMathematical Economics: Lecture 9
Mathematical Economics: Lecture 9 Yu Ren WISE, Xiamen University October 17, 2011 Outline 1 Chapter 14: Calculus of Several Variables New Section Chapter 14: Calculus of Several Variables Partial Derivatives
More informationReduced-load equivalence for queues with Gaussian input
Reduced-load equivalence for queues with Gaussian input A. B. Dieker CWI P.O. Box 94079 1090 GB Amsterdam, the Netherlands and University of Twente Faculty of Mathematical Sciences P.O. Box 17 7500 AE
More informationA Dynamic Contagion Process with Applications to Finance & Insurance
A Dynamic Contagion Process with Applications to Finance & Insurance Angelos Dassios Department of Statistics London School of Economics Angelos Dassios, Hongbiao Zhao (LSE) A Dynamic Contagion Process
More informationCharacterizations on Heavy-tailed Distributions by Means of Hazard Rate
Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 1 (23) 135 142 Characterizations on Heavy-tailed Distributions by Means of Hazard Rate Chun Su 1, Qi-he Tang 2 1 Department of Statistics
More informationELEMENTS OF PROBABILITY THEORY
ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable
More information