On the analysis of a multi-threshold On the analysis of a multi-threshold Markovian risk modelMarkovian risk model

Andrei Badescu – University of TorontoAndrei Badescu – University of Toronto

Steve Drekic – University of WaterlooSteve Drekic – University of Waterloo

David Landriault – University of WaterlooDavid Landriault – University of Waterloo

IME 2007, University of Piraeus, Piraeus, GreeceIME 2007, University of Piraeus, Piraeus, Greece

The authors gratefully acknowledge the support provided by NSERCThe authors gratefully acknowledge the support provided by NSERC

OutlineOutline

Introduction : Fluid flow process vs surplus processIntroduction : Fluid flow process vs surplus process

A multi-level threshold-type risk model with Markovian A multi-level threshold-type risk model with Markovian claim arrivals (MAP)claim arrivals (MAP)

Analysis of the expected discounted dividend paymentsAnalysis of the expected discounted dividend payments

Numerical illustrationNumerical illustration

Connection between fluid flow process Connection between fluid flow process and surplus processand surplus process

Asmussen (1995)Asmussen (1995)

Badescu, Breuer, Da Silva Soares, Latouche, Remiche Badescu, Breuer, Da Silva Soares, Latouche, Remiche and Stanford (2005a) and Stanford (2005a)

Badescu, Breuer, Drekic, Latouche and Stanford (2005b)Badescu, Breuer, Drekic, Latouche and Stanford (2005b)

Ahn, Badescu and Ramaswami (2006)Ahn, Badescu and Ramaswami (2006)

Ahn and Ramaswami (2004, 2005)Ahn and Ramaswami (2004, 2005)

Ramaswami (2007)Ramaswami (2007)

A fluid flow processA fluid flow process

A bivariate Markov process: whereA bivariate Markov process: where - : the level of the fluid buffer - : the level of the fluid buffer - : a CTMC that describes the states- : a CTMC that describes the states

of the environmental processof the environmental process

The fluid level is such thatThe fluid level is such that For , the fluid level increases at rate c(i) > 0For , the fluid level increases at rate c(i) > 0 For , the fluid level decreases at rate c(i) > 0For , the fluid level decreases at rate c(i) > 0

The finite state space The finite state space

The infinitesimal generatorThe infinitesimal generator

A surplus processA surplus process

An insurer’s surplusAn insurer’s surplus

wherewhere

- : initial capital- : initial capital

- : premium rate- : premium rate

- : number of claims by time t- : number of claims by time t

- : claim sizes - : claim sizes

Fluid flow process vs surplus Fluid flow process vs surplus processprocess

Claim number process : Markov Arrival Process Claim number process : Markov Arrival Process

of order mof order m

: the initial state probability vector: the initial state probability vector : transition rates among states without an arrival: transition rates among states without an arrival : transition rates among states at the time of an arrival: transition rates among states at the time of an arrival

Claim sizes : a transition from Claim sizes : a transition from to to at at the the time of a claim yields a claim size of time of a claim yields a claim size of

distribution distribution of order n of order n

A risk model with Markovian arrivals

Equivalent fluid flow representationEquivalent fluid flow representation

- the ascending phases - of order - the ascending phases - of order

- the descending phases - of order - the descending phases - of order

the infinitesimal generator of such a process:the infinitesimal generator of such a process:

withwith

A risk model with Markovian arrivals

A threshold-type risk model with MAPA threshold-type risk model with MAP

IdeaIdea : Use the connection between fluid flow processes : Use the connection between fluid flow processes and risk processes to analyze threshold-type and risk processes to analyze threshold-type

risk risk models defined in a Markovian environment models defined in a Markovian environment

Generalizes the class of risk models studied in the Generalizes the class of risk models studied in the context of a threshold-type dividend strategy bycontext of a threshold-type dividend strategy by

Lin and Sendova (2007)Lin and Sendova (2007) Albrecher and Hartinger (2007)Albrecher and Hartinger (2007) Zhou (2006)Zhou (2006)

Cramer-Lundberg risk model

A multi-threshold risk model with MAPA multi-threshold risk model with MAP

Insurer’s surplus:Insurer’s surplus:

A multi-threshold risk model with MAPA multi-threshold risk model with MAP

Expected discounted dividend paymentsExpected discounted dividend payments

Objective Objective : : analysis of the expected discounted dividend payments analysis of the expected discounted dividend payments

MethodologyMethodology sample path analysissample path analysis recursive calculation : adding a surplus layer at each iterationrecursive calculation : adding a surplus layer at each iteration

Idea Idea starting point : barrier-free risk model (known) starting point : barrier-free risk model (known) proceed recursively by adding the next top layerproceed recursively by adding the next top layer

: : expected discounted dividends (with initial surplus expected discounted dividends (with initial surplus uu)) for for the risk model the risk model

A multi-threshold risk model with MAPA multi-threshold risk model with MAP

Risk process constructed by ignoring the first (i-1) layersRisk process constructed by ignoring the first (i-1) layers

Expected discounted dividend paymentsExpected discounted dividend payments

First term : expected discounted dividend from time 0 to the timeFirst term : expected discounted dividend from time 0 to the time that the surplus process reaches level b that the surplus process reaches level b i i or any ruin or any ruin

level for the first timelevel for the first time

Second term : expected discounted dividend received thereafterSecond term : expected discounted dividend received thereafter

Expected discounted dividend paymentsExpected discounted dividend payments

First term : expected discounted dividend from time 0 to the time First term : expected discounted dividend from time 0 to the time that the surplus level is less than b that the surplus level is less than b i i for the first timefor the first time

Second term : expected discounted dividend received during the Second term : expected discounted dividend received during the first sojourn of the surplus level in the bottom layer first sojourn of the surplus level in the bottom layer

Third term : expected discounted dividend received thereafterThird term : expected discounted dividend received thereafter

Expected discounted dividend paymentsExpected discounted dividend payments

or equivalently or equivalently

A numerical illustrationA numerical illustration

MAP contagion model exampleMAP contagion model example** see Badescu, Breuer, Da Silva Soares, Latouche, Remiche and Stanford (2005) see Badescu, Breuer, Da Silva Soares, Latouche, Remiche and Stanford (2005)

Dependence structure between the claim sizes and the interclaim timesDependence structure between the claim sizes and the interclaim times

Two environments:Two environments: First environment (i.e. standard environment) – only First environment (i.e. standard environment) – only smallsmall claims claims Second environment (i.e. infectious environment) – Second environment (i.e. infectious environment) – smallsmall and and largelarge claims claims

A numerical illustrationA numerical illustration

With a gross premium rate of With a gross premium rate of cc = 2.5 and = 2.5 and BB = (20, 40, 60), = (20, 40, 60),

consider the following 4 dividend strategies:consider the following 4 dividend strategies:

Expected discounted dividend payments prior to ruin (Expected discounted dividend payments prior to ruin (δδ = 0.001) = 0.001)

RatesRates Strategy 1Strategy 1 Strategy 2Strategy 2 Strategy 3Strategy 3 Strategy 4Strategy 4

PremiumPremium (2, 2, 2, 2)(2, 2, 2, 2) (2, 1.5, 1.5, 1.5)(2, 1.5, 1.5, 1.5) (2, 1.5, 1, 1)(2, 1.5, 1, 1) (2, 1.5, 1, 0.5)(2, 1.5, 1, 0.5)

DividendDividend (0.5, 0.5, 0.5, 0.5)(0.5, 0.5, 0.5, 0.5) (0.5, 1, 1, 1)(0.5, 1, 1, 1) (0.5, 1, 1.5, 1.5)(0.5, 1, 1.5, 1.5) (0.5, 1, 1.5, 2)(0.5, 1, 1.5, 2)

Initial surplusInitial surplus Strategy 1Strategy 1 Strategy 2Strategy 2 Strategy 3Strategy 3 Strategy 4Strategy 4

uu = 10 = 10 251.48251.48 442.80442.80 503.00503.00 318.58318.58

uu = 30 = 30 340.48340.48 608.82608.82 692.60692.60 435.94435.94

uu = 50 = 50 397.60397.60 727.93727.93 842.07842.07 521.89521.89

uu = 70 = 70 434.27434.27 810.78810.78 968.82968.82 597.38597.38