Erlang(n) risk models with risky investments Corina Constantinescu - PowerPoint PPT Presentation

Erlang(n) risk models with risky investments Corina Constantinescu Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences joint work with H. Albrecher, University of Linz & RICAM E. Thomann, Oregon State University Special Semester on Stochastics with emphasis in Finance RICAM, Linz, December 2nd, 2008

U t = u + ct − � N ( t ) k =1 X k • u : initial surplus • c : premium rate 3 • N ( t ) = number of claims 2.5 2 up to time t (Poisson/renewal) 1.5 1 • X k : claim size 0.5 (light/heavy) 0 • T k : time of claim −0.5 • τ n = T n − T n − 1 : −1 inter-arrival time ( τ 0 = 0) −1.5 • X , τ are independent −2 0 2 4 6 8 10 12 14 16 • the net profit condition c − λµ > 0 holds • Time of ruin T u = inf t ≥ 0 { t : U t < 0 | U 0 = u } • Probability of ruin in finite time Ψ( u , t ) = P ( T u < t ) • Probability of ruin Ψ( u ) = P ( T u < ∞ ) .

Gerber-Shiu function    e − δ T u w ( U ( T u − ) m δ ( u ) = E , | U ( T u ) | )1( T u < ∞ ) | U (0) = u  � �� � � �� � surpl.im.bef.ruin deficit at ruin • w = 1 (LT of the time of ruin) � � e − δ T u 1( T u < ∞ ) | U (0) = u m δ ( u ) = E = φ δ ( u ) � ∞ e − δ t Ψ( u , t ) dt = φ δ ( u ) δ 0 • w = 1, δ = 0 (Probability of ruin) m 0 ( u ) = E (1( T u < ∞ ) | U (0) = u ) = Ψ( u )

Gerber-Shiu function on the Cramer-Lundberg model By conditioning on the time and size of the first claim... • Integral equation. � ∞ � u + ct λ e − ( δ + λ ) t m δ ( u ) = m δ ( u + ct − y ) dF ( y ) dt 0 0 � ∞ � ∞ λ e − ( δ + λ ) t + w ( u + ct , y − u − ct ) dF ( y ) dt 0 u + ct with boundary condition, lim u →∞ m δ ( u ) = 0 .

Through integration by parts... Integro-differential equation. • Gerber-Shiu function: � u ( − c d du + λ + δ ) m δ ( u ) = λ m δ ( u − x ) f X ( x ) dx 0 � ∞ + w ( u , x − u ) f X ( x ) dx λ u � �� � = ω ( u ) � lim u →∞ m δ ( u ) = 0 with boundary conditions, m δ (0) = λ c ˆ ω ( ρ )

For special penalties... • Gerber-Shiu function � u ( − c d du + λ + δ ) m δ ( u ) = λ m δ ( u − x ) f X ( x ) dx + λω ( u ) 0 • Laplace transform of the time of ruin � u ( − c d du + λ + δ ) φ δ ( u ) = λ φ δ ( u − x ) f X ( x ) dx + λ F X ( u ) 0 • Probability of ruin � u ( − c d du + λ )Ψ( u ) = λ Ψ( u − x ) f X ( x ) dx + λ F X ( u ) 0

Classical results-Probability of ruin Z u ( − c d du + λ )Ψ( u ) = λ Ψ( u − x ) f X ( x ) dx + λ F X ( u ) 0 u →∞ Ψ( u ) = 0 lim • If claim sizes are exponentially bounded (light claims) then c − λµ e − Ru , u → ∞ Ψ( u ) ∼ − λ ˆ f ′ X ( − R ) − c Cramer(1930) • If claims sizes are heavy-tailed (heavy claims) then Ψ( u ) ∼ kF I ( u ) , u → ∞ Embrechts et al(1997)

Classical results-LT of finite-time ruin probability Z ∞ „ − c d « du + λ + δ φ δ ( u ) = λ φ δ ( u − x ) f X ( x ) dx + λ F X ( u ) 0 u →∞ φ δ ( u ) = 0 lim • If light claims then • Laplace transform of time of ruin � 1 � R + 1 δ e − Ru , φ δ ( u ) ∼ u → ∞ , − λ ˆ ρ X ( − R ) − c f ′ δ → 0 φ δ ( u ) = Ψ( u ) lim • (single) Laplace transform of the finite-time ruin probability � 1 � ∞ � 1 R + 1 e − δ t Ψ( u , t ) dt ∼ e − Ru , u → ∞ . − λ ˆ ρ f ′ X ( − R ) − c 0 (Gerber & Shiu, 1998)

Classical results-Gerber-Shiu functions Z u ( − c d du + λ + δ ) m δ ( u ) = λ m δ ( u − x ) f X ( x ) dx + λω ( u ) 0 u →∞ m δ ( u ) = 0 lim • If light claims then � ∞ � ∞ w ( x , y )( e Rx − e − ρ x ) f X ( x + y ) dxdy m δ ( u ) ∼ λ 0 0 e − Ru − λ ˆ f ′ X ( − R ) − c Gerber&Shiu(1998)

Sparre Andersen model with investments • The claim number process N ( t ) is a renewal process • We allow an additional non-traditional feature: investments in a risky asset with returns modeled by a stochastic process Z t , described by an SDE • Denote U k := U ( T k ). The model U k = Z U k − 1 − X k τ k is a discrete Markov process. We refer to this process as renewal jump-diffusion process .

Renewal jump-diffusion process 7 6 5 4 3 2 1 0 −1 0 2 4 6 8 10 12 We assume that the company invests all its money continuously in a risky asset with the price modeled by a geometric Brownian motion.

Question: When we invest everything in a risky asset, do the ruin probabilities have a faster decay than when there is no investment? Answer: When investments in an asset whose price follows a GBM, the ruin probabilities have a power decay Ψ( u ) ∼ Cu − k , as u → ∞ , where k depends on the parameters of the investments or on those of the claim sizes.

Objective • Analyze the asymptotic behavior of the ruin probability Ψ( u ) and the Laplace transform of the time of ruin φ δ ( u ) (implicitly the Laplace transform of the finite-time ruin probability) as the initial capital (surplus) u → ∞ . • Determine a general integro-differential equation for m δ ( u ) . Main tools • integration by parts • regular variation theory

Assumptions (equation) • Inter-arrival times { τ k } k ≥ 0 have densities f τ satisfying an ODE with constant coefficients L ( d dt ) f τ ( t ) = 0 with homogeneous or non-homogeneous boundary conditions. (example: f τ ( t ) = λ e − λ t = ⇒ ( d dt + λ ) f τ ( t ) = 0) • The price of the investments Z u t up to time t starting with an initial capital u is modeled by a non-negative stochastic process with an infinitesimal generator A

Assumptions (asymptotic behavior) We identify two cases: • Light claims. Claim sizes { X k } k ≥ 0 have well-behaved distributions F X with exponentially bounded tails 1 − F X ( x ) ≤ ce − α x , α, c ∈ R , ∀ x ≥ 0 • Heavy-tailed claims. Claim sizes { X k } k ≥ 0 have regularly varying distribution 1 − F X ( x ) ∼ Cx − α l ( x ) , as x → ∞ (Notation: 1 − F X ( x ) ∈ R ( − α )) where C is a positive constant and l ( x ) is a slowly varying function.

U k = Z U k − 1 − X k τ k Theorem. Let h be a sufficiently smooth function of the risk process such that E ( h ( U 1 ) | U 0 = u ) = h ( u ). If f τ satisfies the ODE of order n , with constant coefficients � d � d � n � k � L f τ ( t ) = f τ ( t ) = 0 α k dt dt k =0 and homogeneous boundary conditions, then �� u � L ∗ ( A ) h ( u ) = α 0 h ( u − x ) f X ( x ) dx + ω ( u ) 0 The proof uses semigroup theory, Kolmogorov backward equation and integration by parts.

Probability of ruin • Since the probability of non-ruin Φ( u ) satisfies the hypothesis and then the IDE. • As a consequence the probability of ruin also satisfies this IDE �� u � L ∗ ( A )Ψ( u ) = α 0 Ψ( u − x ) f X ( x ) dx + F X ( u ) 0 � Ψ( u ) = 1 if u < 0 lim u →∞ Ψ( u ) = 0 Recall: • A : infinitesimal generator of the investment process, • F X claim sizes distribution � d dt ) = � n � k • L ∗ ( d dt ) is adjoint to L ( d k =0 α k dt

More IDEs Laplace transform of the time of ruin �� u � L ∗ ( A − δ ) φ δ ( u ) = α 0 φ δ ( u − x ) f X ( x ) dx + F X ( u ) 0 Gerber-Shiu function �� u � L ∗ ( A − δ ) m δ ( u ) = α 0 m δ ( u − x ) f X ( x ) dx + ω ( u ) 0 Recall: • A infinitesimal generator of the investment process, • F X claim sizes distribution � d � k dt ) = � n • L ∗ ( d dt ) is adjoint to L ( d k =0 α k dt

Classical Cramer-Lundberg model • The surplus model: N ( t ) X U ( t ) = u + ct − X k . k =0 • The ODE satisfied by the exponential inter-arrival times L ( d dt ) f τ ( t ) = ( d ⇒ L ∗ ( d dt ) = ( − d dt + λ ) f τ ( t ) = 0 = dt + λ ) • The SDE satisfied by the investment process t = cdt ; A = c d dZ u du • Then the IDE for Gerber-Shiu function � u ( − c d du + δ + λ ) m δ ( u ) = λ m δ ( u − x ) f X ( x ) dx + λω ( u ) 0 � �� � L ∗ ( A − δ )

Cramer-Lundberg model with investments • The surplus model: Z t Z t N ( t ) X U ( t ) = u + ct + a U ( s ) ds + σ U ( s ) dW S − X k . 0 0 k =0 • The ODE satisfied by the exponential inter-arrival times L ( d dt ) f τ ( t ) = ( d ⇒ L ∗ ( d dt ) = ( − d dt + λ ) f τ ( t ) = 0 = dt + λ ) • The SDE satisfied by the investment process t dW t ; A = σ 2 u 2 d 2 du 2 + ( c + au ) d dZ u t = ( c + aZ u t ) dt + σ Z u 2 du • Then the IDE satisfied by the probability of ruin � u ( − A + λ )Ψ( u ) = λ Ψ( u − y ) dF X ( y ) dy + λ F X ( u ) 0

Asymptotic behavior of the probability of ruin Z u − σ 2 u 2 d 2 „ « du 2 − ( c + au ) d du + λ Ψ( u ) = λ Ψ( u − y ) dF X ( y ) dy + λ F X ( u ) 2 0 • For small volatility ( 2 a σ 2 > 1): Ψ( u ) ∼ Cu − k , u → ∞ • If claim sizes are exponentially bounded (light claims) (Norberg&Kalashnikov(2002), Frolova et.al(2002), C.&Thomann(2005)) k = 2 a σ 2 − 1 • If claims sizes are regularly varying (heavy-tailed claims) (Paulsen(2002)) � � α, 2 a k = max σ 2 − 1 • For large volatility ( 2 a σ 2 < 1): Ψ( u ) = 1 , ∀ u > 0 . (Norberg&Kalashnikov(2002), Frolova et.al(2002))