1/37 Threshold Strategies for Risk Processes and their Relation to Queueing Theory Onno Boxma, David Perry, Andreas Löpker CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
This talk 2/37 I. Setting / Literature II. Risk and queues / Dualities III. Ruin probability IV. More general ideas (V. If time allows: Ruin time) CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
I 3/37 I. Setting / Literature Risk process with threshold dividend strategy CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Setting I 4/37 We consider a simple insurance risk model (Cramèr-Lundberg regime), where R t denotes the surplus of an insurance com- pany at time t . Some of the income is re-distributed as dividends: whenever R t is larger than some threshold b , a fraction of γ is paid out as dividends. ⇒ Threshold strategy (refracting barrier) CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Setting I 5/37 Risk process: dR t = r ( R t ) dt + dS t , with aggregated claims S t = ∑ N t i = 1 X i , i.i.d. claims ( X i ) i = 1,2,... , distribution G , ❊ [ X 1 ] = 1/ λ , Poisson claim number process N t with rate µ , "plowback rate" r ( x ) = 1 − γ 1 ( x ≥ b ) , dividend process D t = γ � t 0 1 ( R s > b ) ds , We let ρ = µ / λ and ψ ( x ) = P x ( τ < ∞ ) . CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Setting I 6/37 Three scenarios (I) (II) (III) 1 − γ < ρ < 1, ρ < 1 − γ , ρ > 1, R t → − ∞ R t → ∞ R t pos. recurrent ψ ( x ) = 1 ψ ( x ) = 1 ψ ( x ) < 1 We omit the cases ρ = 1 and ρ = 1 − γ . CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Literature I 7/37 CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
II 8/37 II. Risk and queues Dualities CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Duality I II 9/37 Costruct dual process V t as follows: fix a time T , use the same jump sizes and inter-jump times, but in re- versed order and reversed direction, set dV t dt = 0 for V t ≤ 0 and dV t dt = − r ( V t ) else. V t is the workload process of a M/G/1 with service time dis- tribution G , arrival rate µ and server speed 1 − γ 1 ( V t ≥ b ) . CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Duality I II 10/37 Then surprisingly (A smussen & P etersen (1988)) P x ( τ ≤ T ) = P ( V T > x | V 0 = 0 ) and, if ρ < 1 − γ , ψ ( x ) = P x ( τ = ∞ ) = F ( x ) , where F is the stationary distribution of the dual queue. CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Duality II II 11/37 There is another (more obvious) duality: Risk process Workload G/M/1 ⇔ Service mean 1 − γ Arrival rate µ µ Claim size mean 1 Arrival rate λ λ CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
III 12/37 III. Ruin probability Case ρ < 1 − γ (upward drift) The survival probability is positive: ψ ( x ) = P x ( τ = ∞ ) > 0 CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Ruin probability III 13/37 It follows from duality I that ψ ( x ) = F ( x ) , where F is the stationary distribution of an M/G/1 queue with server speed 1 − γ 1 ( V t > b ) , service distribution G and arrival rate µ . The Laplace transform of F has been derived by G aver & M iller (1962) (context: storage processes). Aim: express ψ in terms of F ( γ ) and F ( 0 ) (stationary distribu- tion of the standard queue), where F ( γ ) denotes the stationary distribution of an M/G/1 queue with server speed 1 − γ . CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Ruin probability III 14/37 (A) For x < b : ψ ( x ) = θ ( x , b ) ψ ( b ) , where θ ( x , b ) = P x ( R t = b for some t < τ ) . The same formula holds in the γ = 0 case, too, hence θ ( x , b ) = F ( 0 ) ( x ) F ( 0 ) ( b ) . by duality I. CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Ruin probability III 15/37 (B) For x ≥ b we have τ = ∞ if either 1. R t never reaches b , the probability being y → ∞ θ ( x − b , y ) = F ( γ ) ( x − b ) . lim 2. R s < b for some s > 0, but never jumps below 0, the proba- bility being � b F ( γ ) ( x − b ) 0 θ ( b − u , b ) dH x − b ( u ) ψ ( b ) , where H x − b ( u ) is the p.d.f. of the excess. CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Ruin probability III 16/37 We obtain: θ ( x , b ) ψ ( b ) ; x < b F ( γ ) ( x − b ) ψ ( x ) = + F ( γ ) ( x − b ) ψ ( b ) � b 0 θ ( b − u , b ) H x − b ( du ) ; x ≥ b Still to determine: i) excess distribution H x − b ( u ) and ii) survival probability ψ ( b ) . CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Ruin probability III 17/37 i) c.f. Figure. Excesses U 1 , U 2 , . . . form a re- newal process. H x − b ( u ) is the distribution of the forward recurrence time at time x − b of a renewal process with inter-occurrence time distribution H 0 ( u ) . The distribution H 0 ( u ) equals the distribution of the idle pe- riod of a transient G/M/1 queue (or deficit at ruin): � u H 0 ( u ) = λ 0 ( 1 − G ( t )) dt . (e.g. P rabhu (1997) - apparently well known in risk theory: B owers , G erber , H ickmann , J ones & N esbitt (1987)) CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Ruin probability III 18/37 ii) ψ ( b ) Two possibilites 1. Process never reaches b again, the probability being 1 − ρ γ = 1 − ρ / ( 1 − γ ) . (e.g. by duality I: steady state idle probability) 2. Process jumps below b , but τ = ∞ , the probability being � b 0 θ ( b − u , b ) dH 0 ( u ) ψ ( b ) . ρ γ CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Ruin probability III 19/37 It follows that � b ψ ( b ) = 1 − ρ γ + ρ γ 0 θ ( b − u , b ) dH 0 ( u ) ψ ( b ) . After rearranging we obtain 1 − ρ γ ψ ( b ) = , � b 1 − ρ γ 0 θ ( b − u , b ) dH 0 ( u ) 1 − ρ γ ψ ( b ) = . � b ρ γ 0 F ( 0 ) ( b − u ) dH 0 ( u ) 1 − F ( 0 ) ( b ) The term � b 0 F ( 0 ) ( b − u ) dH 0 ( u ) looks familiar... CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Ruin probability III 20/37 C ohen (1982): For the G/G/1 queue with V ∞ =steady-state workload, W ∞ =steady-state waiting time, W r =residual waiting time, V ∞ | ( V ∞ > 0 ) d ∼ W ∞ + W r . With Poisson arrivals and PASTA we obtain � y F ( 0 ) ( y ) = 1 − ρ + ρ 0 F ( 0 ) ( y − u ) dH 0 ( u ) . (follows also by integration of the well known integro-differential equation for F ). CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Ruin probability III 21/37 Inserting into equation for ψ ( b ) yields 1 − ρ − γ ψ ( b ) = F ( 0 ) ( b ) 1 − ρ − γ F ( 0 ) ( b ) . (1) Theorem: The survival probability is given by θ ( x , b ) ψ ( b ) ; x < b F ( γ ) ( x − b ) ψ ( x ) = + F ( γ ) ( x − b ) ψ ( b ) � b 0 θ ( b − u , b ) H x − b ( du ) ; x ≥ b with ψ ( b ) given in (1) and H x − b ( u ) the distribution of the forward recurrence time at time x − b of a renewal process with renewal distribution H 0 ( u ) . (c.f. L in & P avlova (2006)). CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
IV 22/37 IV. Some general ideas Case P x ( τ < ∞ ) = 1. CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Gerber-Shiu discounted penalty function IV 23/37 For the dividend-free risk process ( γ = 0) the functional Φ α , w ( x ) = ❊ x [ e − ατ w ( R τ − , | R τ | )] was introduced by G erber & S hiu (1998). On can show that � ∞ Φ ′ α , w ( x ) = ( µ + α ) Φ α , w ( x ) − µ w ( x , y − x ) dG ( y ) x � x − µ 0 ϕ α ( x − y ) dG ( y ) . Gerber-Shiu-ism: Extensive literature about solutions. Approaches available for the usual suspects: exponential G , Erlang G , phase-type G . CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Generalization IV 24/37 More generally one could investigate the two functionals � τ 0 e − � t 0 v ( R s ) ds β ( R t ) dt ] . Ψ v , β ( x ) = ❊ x [ and Φ v , w ( x ) = ❊ x [ e − � τ 0 v ( R s ) ds w ( R τ − , | R τ | )] , ( β , v , w non-negative and bounded) for a risk process dR t = r ( R t ) dt + dS t with general ("plowback"-)rate r : R → ( 0, 1 ] . CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
Generalization IV 25/37 Why is � τ 0 e − � t 0 v ( R s ) ds β ( R t ) dt ] Ψ v , β ( x ) = ❊ x [ useful? Expected present value of the discounted dividends: � τ 0 e − δ t ( 1 − r ( R t )) dt ] . Ψ δ ,1 − r ( x ) = ❊ x [ Expected value of the total dividends (undiscounted): � τ Ψ 0,1 − r ( x ) = ❊ x [ 0 ( 1 − r ( R t )) dt ] . CONFERENCE IN HONOUR OF SØREN ASMUSSEN - NEW FRONTIERS IN APPLIED PROBABILITY
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