Risk Reserve Process Example � t Cram´ er-Lundberg Model: X t = x + 0 c s ds − S t � t Add expense loading: X t = x + 0 c s (1 + ρ s ) ds − S t � t Add interest income: X t = x + 0 [ r s X s + c s (1 + ρ s )] ds − S t Reserve with Investment � t � � X t = x + X s [ r s + � π s , µ s − r s 1 � ] + c s (1 + ρ s ) ds 0 � t � t � + X s � π s , σ s dW s � − f ( s , z ) µ ( dsdz ) , (2) 0 0 R + General Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 11/ 57
Ruin Problems Consider the simplest Cram´ er-Lundberg model: � t X t = x + c s ds − S t , t ≥ 0 . (3) 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 12/ 57
Ruin Problems Consider the simplest Cram´ er-Lundberg model: � t X t = x + c s ds − S t , t ≥ 0 . (3) 0 Ruin Problem Find/estimate the “ruin probabilities”: ψ ( x , T ) = P { X t < 0 : ∃ t ∈ (0 , T ] } ; (Finite horizon) ψ ( x ) = P { X t < 0 : ∃ t > 0 } . (Infinite horizon) . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 12/ 57
Ruin Problems Consider the simplest Cram´ er-Lundberg model: � t X t = x + c s ds − S t , t ≥ 0 . (3) 0 Ruin Problem Find/estimate the “ruin probabilities”: ψ ( x , T ) = P { X t < 0 : ∃ t ∈ (0 , T ] } ; (Finite horizon) ψ ( x ) = P { X t < 0 : ∃ t > 0 } . (Infinite horizon) . Thinking finance? Default probability? Structure model? ... Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 12/ 57
Existing ways/methods of studying ruin probabilities Direct Calculation : (e.g, vi IDE) — Lundberg (’26), Cram´ er (’35), Segerdahi (’42)... Bounds : — Lundberg (’26, 32, 34), Crem´ er (’55), Gerber (’76), Feller (’71) ... u →∞ ψ ( u ) e γ u =? u →∞ ψ ( u , T ) e γ u =?) Asymptotics : (e.g., lim lim — Teugels-Veraverbeke (’73), Djehiche (’93), Asmussen-kl¨ uppelberg (’96)... Approximations (of claim size dist.): — De Vylder (’78), Daley Rolski (’84)... Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 13/ 57
Existing ways/methods of studying ruin probabilities One of most notable discovery in ruin theory is that the ruin probablity satisfies a differential or integro-differential equation. Main Result (Feller (1971), Gerber (1990)) Assume classical Cram´ er-Lundberg model. L ψ ( x ) be the infinite horizon ruin probability with initial capital x , and ϕ ( x ) = 1 − ψ ( x ) be the corresponding non-ruin probability. Then � x λ ϕ ( x − z )¯ ϕ ( x ) = ϕ (0) + F Z ( z ) dz , (4) c (1 + ρ ) 0 where F is the jump size distribution and ¯ F = 1 − F , and λ is the intensity of jump frequency. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 14/ 57
Existing ways/methods of studying ruin probabilities One of most notable discovery in ruin theory is that the ruin probablity satisfies a differential or integro-differential equation. Main Result (Feller (1971), Gerber (1990)) Assume classical Cram´ er-Lundberg model. L ψ ( x ) be the infinite horizon ruin probability with initial capital x , and ϕ ( x ) = 1 − ψ ( x ) be the corresponding non-ruin probability. Then � x λ ϕ ( x − z )¯ ϕ ( x ) = ϕ (0) + F Z ( z ) dz , (4) c (1 + ρ ) 0 where F is the jump size distribution and ¯ F = 1 − F , and λ is the intensity of jump frequency. More general model— Reinhard (1984), Asmusson (1989) (Hidden Markovian), Asmusson-Petersen (1988) (reserve dependent premium) ... Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 14/ 57
Ruin Probility via Differential Equations Assume that the risk reserve satisfies the following SDE: � t � t � X t = x + b ( s , X s ) ds − f ( s , z ) N p ( dzds ) , (5) 0 0 I R + where b : [0 , ∞ ) × R �→ R is some (deterministic!) measurable function (could be Lipschitz..., if you wish). Then X is (strong) Markov. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 15/ 57
Ruin Probility via Differential Equations Assume that the risk reserve satisfies the following SDE: � t � t � X t = x + b ( s , X s ) ds − f ( s , z ) N p ( dzds ) , (5) 0 0 I R + where b : [0 , ∞ ) × R �→ R is some (deterministic!) measurable function (could be Lipschitz..., if you wish). Then X is (strong) Markov. Define τ = inf { t ≥ 0 : X t < 0 } . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 15/ 57
Ruin Probility via Differential Equations Assume that the risk reserve satisfies the following SDE: � t � t � X t = x + b ( s , X s ) ds − f ( s , z ) N p ( dzds ) , (5) 0 0 I R + where b : [0 , ∞ ) × R �→ R is some (deterministic!) measurable function (could be Lipschitz..., if you wish). Then X is (strong) Markov. Define τ = inf { t ≥ 0 : X t < 0 } . Then, ∀ 0 < t < T , 1 { τ< T } = 1 { τ< t } + 1 { t ≤ τ } 1 { inf t ≤ s < T X s < 0 } . (6) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 15/ 57
Ruin Probility via Differential Equations Assume that the risk reserve satisfies the following SDE: � t � t � X t = x + b ( s , X s ) ds − f ( s , z ) N p ( dzds ) , (5) 0 0 I R + where b : [0 , ∞ ) × R �→ R is some (deterministic!) measurable function (could be Lipschitz..., if you wish). Then X is (strong) Markov. Define τ = inf { t ≥ 0 : X t < 0 } . Then, ∀ 0 < t < T , 1 { τ< T } = 1 { τ< t } + 1 { t ≤ τ } 1 { inf t ≤ s < T X s < 0 } . (6) △ = P { τ < T | F X t } = E { 1 { τ< T } | F X Define M t t } ; and � � � � △ Ψ( t , r ) = P t ≤ s < T X t < 0 inf � X t = r . (7) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 15/ 57
Ruin Probility via Differential Equations Taking conditional expectations E { · | F X t } on both sides of (6) and using the Markovian Property of X : � � � � M t = 1 { τ ≤ t } + 1 { τ> t } P t ≤ s < T X t < 0 inf � X t = 1 { τ< t } + 1 { τ ≥ t } Ψ( t , X t ) . (8) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 16/ 57
Ruin Probility via Differential Equations Taking conditional expectations E { · | F X t } on both sides of (6) and using the Markovian Property of X : � � � � M t = 1 { τ ≤ t } + 1 { τ> t } P t ≤ s < T X t < 0 inf � X t = 1 { τ< t } + 1 { τ ≥ t } Ψ( t , X t ) . (8) Setting t = t ∧ τ in (8), we obtain that M t ∧ τ = Ψ( t ∧ τ, X t ∧ τ ) . (9) Thus by Optional Sampling t �→ Ψ( t ∧ τ, X t ∧ τ ) is an (UI) F X -mg! Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 16/ 57
Ruin Probility via Differential Equations Taking conditional expectations E { · | F X t } on both sides of (6) and using the Markovian Property of X : � � � � M t = 1 { τ ≤ t } + 1 { τ> t } P t ≤ s < T X t < 0 inf � X t = 1 { τ< t } + 1 { τ ≥ t } Ψ( t , X t ) . (8) Setting t = t ∧ τ in (8), we obtain that M t ∧ τ = Ψ( t ∧ τ, X t ∧ τ ) . (9) Thus by Optional Sampling t �→ Ψ( t ∧ τ, X t ∧ τ ) is an (UI) F X -mg! Now denote Φ( t , r ) = 1 − Ψ( t , r ) ( non-ruin probability ), and assume that Φ( · , · ) ∈ C 1 , 1 . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 16/ 57
Ruin Probility via Differential Equations Applying Itˆ o (BV version) to get Φ( t ∧ τ, X t ∧ τ ) − Φ(0 , x ) � t ∧ τ � t ∧ τ = ∂ t Φ( s , X s ) ds + ∂ r Φ( s , X s ) b ( s , X s ) ds 0 0 � t ∧ τ � + [Φ( s , X s − − f ( s , z )) − Φ( s , X s − )] N p ( dzds ) 0 R + � t ∧ τ � t ∧ τ = ∂ t Φ( s , X s ) ds + ∂ r Φ( s , X s ) b ( s , X s ) ds 0 0 � t ∧ τ � [Φ( s , X s − − f ( s , z )) − Φ( s , X s − )] ν ( dz ) ds + M ∗ + t ∧ τ , 0 R + where � t ∧ τ � M ∗ [Φ( s , X s − − f ( s , z )) − Φ( s , X s − )] � t = N p ( dzds ) 0 R + is an martingale with zero mean. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 17/ 57
Ruin Probility via Differential Equations Thus � t ∧ τ � t ∧ τ ∂ t Φ( s , X s ) ds + ∂ r Φ( s , X s ) b ( s , X s ) ds 0 0 � t ∧ τ � + [Φ( s , X s − − f ( s , z )) − Φ( s , X s − )] ν ( dz ) ds 0 R + Φ( t ∧ τ, X t ∧ τ ) − Φ(0 , x ) − M ∗ = t ∧ τ Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 18/ 57
Ruin Probility via Differential Equations Thus � t ∧ τ � t ∧ τ ∂ t Φ( s , X s ) ds + ∂ r Φ( s , X s ) b ( s , X s ) ds 0 0 � t ∧ τ � + [Φ( s , X s − − f ( s , z )) − Φ( s , X s − )] ν ( dz ) ds 0 R + Φ( t ∧ τ, X t ∧ τ ) − Φ(0 , x ) − M ∗ = t ∧ τ = 0 . (It is a continuous (local) martingale with zero mean and with bounded variation paths = ⇒ it is a zero martingale!) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 18/ 57
Ruin Probility via Differential Equations Thus � t ∧ τ � t ∧ τ ∂ t Φ( s , X s ) ds + ∂ r Φ( s , X s ) b ( s , X s ) ds 0 0 � t ∧ τ � + [Φ( s , X s − − f ( s , z )) − Φ( s , X s − )] ν ( dz ) ds 0 R + Φ( t ∧ τ, X t ∧ τ ) − Φ(0 , x ) − M ∗ = t ∧ τ = 0 . (It is a continuous (local) martingale with zero mean and with bounded variation paths = ⇒ it is a zero martingale!) Similarly, for any t ′ ∈ [0 , T ) and τ ′ = inf { t ≥ t ′ | X t < 0 } , one shows that � t ∧ τ ′ � t ∧ τ ′ ∂ t Φ( s , X s ) ds + ∂ r Φ( s , X s ) b ( s , X s ) ds (10) t ′ t ′ � t ∧ τ ′ � = [Φ( s , X s − ) − Φ( s , X s − − f ( s , z ))] ν ( dz ) ds . t ′ I R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 18/ 57
Ruin Probility via Differential Equations Since t ′ is arbitrary and τ ′ ≥ t ′ , we can “ differentiating ” (10) to get the following IPDE: � [ ∂ t Φ + ∂ r Φ b ]( t , r ) = [Φ( t , r ) − Φ( t , r − f ( t , z ))] ν ( dz ) . (11) R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 19/ 57
Ruin Probility via Differential Equations Since t ′ is arbitrary and τ ′ ≥ t ′ , we can “ differentiating ” (10) to get the following IPDE: � [ ∂ t Φ + ∂ r Φ b ]( t , r ) = [Φ( t , r ) − Φ( t , r − f ( t , z ))] ν ( dz ) . (11) R + Remark Since Φ( t , X t ) = 0 for X t < 0, the RHS in (11) is actually � [Φ( t , r ) − Φ( t , r − f ( t , z ))] ν ( dz ) . { r ≥ f ( t , z ) } Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 19/ 57
Ruin Probility via Differential Equations Since t ′ is arbitrary and τ ′ ≥ t ′ , we can “ differentiating ” (10) to get the following IPDE: � [ ∂ t Φ + ∂ r Φ b ]( t , r ) = [Φ( t , r ) − Φ( t , r − f ( t , z ))] ν ( dz ) . (11) R + Remark Since Φ( t , X t ) = 0 for X t < 0, the RHS in (11) is actually � [Φ( t , r ) − Φ( t , r − f ( t , z ))] ν ( dz ) . { r ≥ f ( t , z ) } In the compound Poisson case f ( t , z ) ≡ z , ν ( dz ) = λ F Z ( dz ), where Z is the jump size. Thus (11) becomes � [ ∂ t Φ + ∂ r Φ b ]( t , r ) = Φ( t , r ) λ − λ Φ( t , r − z ) F Z ( dz ) . { r ≥ z } Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 19/ 57
Special Cases Infinite horizon case Assume b ( t , r ) = b ( r ). Denote ψ ( r ) = lim t →∞ Ψ( t , r ) and ϕ ( r ) = 1 − ψ ( r ). Then � ϕ ′ ( r ) b ( r ) = ϕ ( r ) λ − λ ϕ ( r − z ) F Z ( dz ) . (12) { r ≥ z } Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 20/ 57
Special Cases Infinite horizon case Assume b ( t , r ) = b ( r ). Denote ψ ( r ) = lim t →∞ Ψ( t , r ) and ϕ ( r ) = 1 − ψ ( r ). Then � ϕ ′ ( r ) b ( r ) = ϕ ( r ) λ − λ ϕ ( r − z ) F Z ( dz ) . (12) { r ≥ z } Example △ If b ( r ) = c (1 + ρ ) = β and Z ∼ exp { δ } Then (12) becomes � � r � ϕ ′ ( r ) β = λ ϕ ( r ) − e − δ r ϕ ( z ) δ e δ z dz . (13) 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 20/ 57
Special Cases Infinite horizon case Assume b ( t , r ) = b ( r ). Denote ψ ( r ) = lim t →∞ Ψ( t , r ) and ϕ ( r ) = 1 − ψ ( r ). Then � ϕ ′ ( r ) b ( r ) = ϕ ( r ) λ − λ ϕ ( r − z ) F Z ( dz ) . (12) { r ≥ z } Example △ If b ( r ) = c (1 + ρ ) = β and Z ∼ exp { δ } Then (12) becomes � � r � ϕ ′ ( r ) β = λ ϕ ( r ) − e − δ r ϕ ( z ) δ e δ z dz . (13) 0 Differentiating: ϕ ′′ ( r ) β = ( λ − δβ ) ϕ ′ ( r ). Solving: ϕ ( r ) = c 1 − c 2 e − ( δ − λ/β ) r , where c 1 , c 2 ∈ R . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 20/ 57
An Integral Equation Denoting β = c (1 + ρ ) again, and integrate (13) from 0 to x : � x β β ϕ ′ ( r ) dr λ ( ϕ ( x ) − ϕ (0)) = λ 0 � x � x � u = ϕ ( r ) dr − ϕ ( u − z ) F Z ( dz ) du 0 0 0 = · · · · · · � x � x � x − u = ϕ ( r ) dr − F Z ( dz ) ϕ ( u ) du 0 0 0 � x = [1 − F Z ( x − u )] ϕ ( u ) du . 0 � x ϕ ( x ) = ϕ (0) + λ ϕ ( x − z )¯ = ⇒ F Z ( z ) dz . (14) β 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 21/ 57
Lundberg bounds An Evidence Recall IDE (14). By Expected Value Principle c = dE [ S t ] = λµ , � x dt denoting F I ( x ) = µ − 1 ¯ F ( z ) dz (14) becomes 0 1 ϕ ( x ) = ϕ (0) + (1 + ρ ) ϕ ∗ F I ( x ) , (15) where ∗ means convolution. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 22/ 57
Lundberg bounds An Evidence Recall IDE (14). By Expected Value Principle c = dE [ S t ] = λµ , � x dt denoting F I ( x ) = µ − 1 ¯ F ( z ) dz (14) becomes 0 1 ϕ ( x ) = ϕ (0) + (1 + ρ ) ϕ ∗ F I ( x ) , (15) where ∗ means convolution. Solving (15) by Laplace transforms and using the initial value ρ ϕ (0) = 1+ ρ we have � � n ∞ � ρ 1 F n ∗ ϕ ( x ) = I ( x ) . (16) 1 + ρ 1 + ρ n =0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 22/ 57
Lundberg Bounds Example If Z ∼ exp( δ ), then we see that � � 1 ρ ≤ e − Rx . ψ ( x ) = 1 − ϕ ( x ) = 1 + ρ exp − δ (1 + ρ ) x Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 23/ 57
Lundberg Bounds Example If Z ∼ exp( δ ), then we see that � � 1 ρ ≤ e − Rx . ψ ( x ) = 1 − ϕ ( x ) = 1 + ρ exp − δ (1 + ρ ) x Remark A primitive method for the Lundberg bound is to consider ψ n ( x ), the ruin probability up to ( n + 1)-st claim. By an inductional argument one proves that, there exists an R > 0 such that ψ n ( x ) ≤ e − Rx , ∀ n . (17) Letting n → ∞ one derives the (upper) bound for (infinite horizon) ruin probability ψ ( x ). The constant R is called “ Lundberg coefficient ” or “ adjustment coefficients ”. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 23/ 57
Exponential Martingale Approach (Gerber, (1973)) Consider the classical model X t = x + ct − S t , where ct = E [ S t ] = λµ t . Denote Q t = ct − S t ( profit process ). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 24/ 57
Exponential Martingale Approach (Gerber, (1973)) Consider the classical model X t = x + ct − S t , where ct = E [ S t ] = λµ t . Denote Q t = ct − S t ( profit process ). For any given x and r > 0, consider the F p -adapted process = e − r ( x + Q t ) △ M x , t ≥ 0 , (18) t e t θ ( r ) where θ ( · ) is a function to be determined. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 24/ 57
Exponential Martingale Approach (Gerber, (1973)) Consider the classical model X t = x + ct − S t , where ct = E [ S t ] = λµ t . Denote Q t = ct − S t ( profit process ). For any given x and r > 0, consider the F p -adapted process = e − r ( x + Q t ) △ M x , t ≥ 0 , (18) t e t θ ( r ) where θ ( · ) is a function to be determined. Suppose that { M x t } is an F p -martingale(!) Then, by optional sampling, for any given time t 0 > 0 and stopping △ time τ x = inf { t ≥ 0 : X t = x + Q t < 0 } , one has � � � � � � e − rx � F p M x M x M x = 0 = E = E (19) t 0 ∧ τ x 0 t 0 ∧ τ x � � � � M x ≥ E � τ x ≤ t 0 P { τ x ≤ t 0 } . τ x Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 24/ 57
Exponential Martingale Approach (Gerber, (1973)) But on the set { τ x ≤ t 0 } one must have X τ x = x + Q τ x ≤ 0. Thus e − rx e − rx P { τ x ≤ t 0 } ≤ τ x | τ x ≤ t 0 } ≤ E { M x E { e − τ x θ ( r ) | τ x ≤ t 0 } e − rx e t θ ( r ) . ≤ sup 0 ≤ t ≤ t 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 25/ 57
Exponential Martingale Approach (Gerber, (1973)) But on the set { τ x ≤ t 0 } one must have X τ x = x + Q τ x ≤ 0. Thus e − rx e − rx P { τ x ≤ t 0 } ≤ τ x | τ x ≤ t 0 } ≤ E { M x E { e − τ x θ ( r ) | τ x ≤ t 0 } e − rx e t θ ( r ) . ≤ sup 0 ≤ t ≤ t 0 Letting t 0 → ∞ we obtain that ψ ( x ) ≤ e − rx sup e t θ ( r ) . (20) t ≥ 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 25/ 57
Exponential Martingale Approach (Gerber, (1973)) But on the set { τ x ≤ t 0 } one must have X τ x = x + Q τ x ≤ 0. Thus e − rx e − rx P { τ x ≤ t 0 } ≤ τ x | τ x ≤ t 0 } ≤ E { M x E { e − τ x θ ( r ) | τ x ≤ t 0 } e − rx e t θ ( r ) . ≤ sup 0 ≤ t ≤ t 0 Letting t 0 → ∞ we obtain that ψ ( x ) ≤ e − rx sup e t θ ( r ) . (20) t ≥ 0 Question How to determine θ ? Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 25/ 57
Exponential Martingale Approach (Gerber, (1973)) Analysis � ∞ Denote ˆ 0 e − sx dF ( x ) = E [ e − sU 1 ]. Then f ( s ) = � � e sS t � ∞ � � � e s � Nt � k =1 U k E = E � N t = n P ( N t = n ) n =0 ∞ � f n ( − s )( λ t ) n e − λ t = e λ (ˆ ˆ f ( − s ) − 1) t = n ! n =0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 26/ 57
Exponential Martingale Approach (Gerber, (1973)) Analysis � ∞ Denote ˆ 0 e − sx dF ( x ) = E [ e − sU 1 ]. Then f ( s ) = � � e sS t � ∞ � � � e s � Nt � k =1 U k E = E � N t = n P ( N t = n ) n =0 ∞ � f n ( − s )( λ t ) n e − λ t = e λ (ˆ ˆ f ( − s ) − 1) t = n ! n =0 Thus to make M x a martingale, one need only choose � e − sQ t � � e sS t � f ( − s ) − 1] t △ = e − sct + λ [ˆ = e − sct E = e t θ ( s ) , (21) E △ = λ [ˆ where θ ( s ) f ( − s ) − 1] − sc . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 26/ 57
Exponential Martingale Approach (Gerber, (1973)) With this choice of θ , and using (21) and the fact that Q has independent increments, we have � � � � e − r ( Q t − Q s ) � E [ M x t | F p s ] = M x � F p = M x s E s . � s e ( t − s ) θ ( r ) ⇒ M x is a F p -martingale! = Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 27/ 57
Exponential Martingale Approach (Gerber, (1973)) With this choice of θ , and using (21) and the fact that Q has independent increments, we have � � � � e − r ( Q t − Q s ) � E [ M x t | F p s ] = M x � F p = M x s E s . � s e ( t − s ) θ ( r ) ⇒ M x is a F p -martingale! = Recall (20). Clearly the sharp estimate of ruin probability is obtained by minimizing the RHS w.r.t. r . Namely, choosing r ∗ △ = sup { r : θ ( r ) ≤ 0 } would give the best estimate ψ ( x ) ≤ e − r ∗ t . (22) r ∗ is thus called Lundberg coefficient . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 27/ 57
Another look at Exponential Martingales Consider the more general model: � t � t � X t = x + b ( s , X s ) ds − f ( s , z ) N p ( dsdz ) . (23) 0 0 R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 28/ 57
Another look at Exponential Martingales Consider the more general model: � t � t � X t = x + b ( s , X s ) ds − f ( s , z ) N p ( dsdz ) . (23) 0 0 R + For any g ∈ C 1 , 1 ([0 , T ] × R ), applying Itˆ o’s formula to get � t g ( t , X t ) = g (0 , x ) + { ∂ t g + ∂ x gb } ( s , X s ) ds 0 � t � . + [ g ( s , X s − − f ( s , z )) − g ( s , X s − )] ν ( dz ) ds + mg 0 R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 28/ 57
Another look at Exponential Martingales Consider the more general model: � t � t � X t = x + b ( s , X s ) ds − f ( s , z ) N p ( dsdz ) . (23) 0 0 R + For any g ∈ C 1 , 1 ([0 , T ] × R ), applying Itˆ o’s formula to get � t g ( t , X t ) = g (0 , x ) + { ∂ t g + ∂ x gb } ( s , X s ) ds 0 � t � . + [ g ( s , X s − − f ( s , z )) − g ( s , X s − )] ν ( dz ) ds + mg 0 R + △ Thus M t = g ( t , X t ) is a mg (or local mg) ⇐ ⇒ g satisfies � ∂ t g + ∂ x gb + [ g ( t , x − f ( t , z )) − g ( t , x )] ν ( dz ) = 0 . (24) R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 28/ 57
Another look at Exponential Martingales In the compound Poisson case b ( t , x ) = β , f ≡ z , and ν ( dz ) = λ F U ( dz ). The equation (24) becomes �� � [ ∂ t g + ∂ x g ] β + λ [ g ( t , x − z ) − g ( t , x )] F U ( dz ) = 0 . R + If g = g ( x ), then �� � g ′ ( x ) β + λ g ( x − z ) F U ( dz ) − g ( x ) = 0 . (25) R + Setting g ( x ) = ϕ ( x ) for x ≥ 0 and g ( x ) = 0 for x < 0 we see that � x the integral becomes 0 g ( x − z ) F U ( dz ) and we recover (14) for the infinite horizon ruin probability. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 29/ 57
Finite Horizon Case Assume g ( t , x ) = e − sx − θ t , where s and θ are parameters. Then (25) reads �� � [ e sz F U ( dz ) − 1] g ( t , x ) [ − θ − β s ] g ( t , x ) + λ = 0 . R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 30/ 57
Finite Horizon Case Assume g ( t , x ) = e − sx − θ t , where s and θ are parameters. Then (25) reads �� � [ e sz F U ( dz ) − 1] g ( t , x ) [ − θ − β s ] g ( t , x ) + λ = 0 . R + � R + e sz F U ( dz ), then the above becomes Denoting ˆ m U ( s ) = {− θ − β s + λ [ ˆ m U ( s ) − 1] } g ( t , x ) = 0 . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 30/ 57
Finite Horizon Case Assume g ( t , x ) = e − sx − θ t , where s and θ are parameters. Then (25) reads �� � [ e sz F U ( dz ) − 1] g ( t , x ) [ − θ − β s ] g ( t , x ) + λ = 0 . R + � R + e sz F U ( dz ), then the above becomes Denoting ˆ m U ( s ) = {− θ − β s + λ [ ˆ m U ( s ) − 1] } g ( t , x ) = 0 . Thus (since g ( t , x ) > 0!) θ = θ ( s ) = − β s + λ [ ˆ m U ( s ) − 1] . (26) We obtain the adjustment coefficient θ = θ ( s ), and M t = g ( t , X t ) = exp {− sX t − θ ( s ) t } is a martingale! Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 30/ 57
Risk Reserve with Interests Consider the reserve equation with interst: X 0 = x � dX t = [ r t X t + c t (1 + ρ t )] dt − f ( t , z ) N p ( dzdt ) . R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 31/ 57
Risk Reserve with Interests Consider the reserve equation with interst: X 0 = x � dX t = [ r t X t + c t (1 + ρ t )] dt − f ( t , z ) N p ( dzdt ) . R + � t △ 0 r s ds , and � X t = Γ t X t . Then � = e − Denote Γ t X satisfies � t � t � � X t = x + Γ s c s (1 + ρ s ) ds − Γ s f ( s , z ) N p ( dzds ) . 0 0 R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 31/ 57
Risk Reserve with Interests Consider the reserve equation with interst: X 0 = x � dX t = [ r t X t + c t (1 + ρ t )] dt − f ( t , z ) N p ( dzdt ) . R + � t △ 0 r s ds , and � X t = Γ t X t . Then � = e − Denote Γ t X satisfies � t � t � � X t = x + Γ s c s (1 + ρ s ) ds − Γ s f ( s , z ) N p ( dzds ) . 0 0 R + Assume β = c (1 + ρ ) is constant, and r t is deterministic, Then for g ∈ C 1 , 1 ( R + × R ), we have � t g ( t , � [ ∂ t g + ∂ x g Γ s β ]( s , � X t ) = g (0 , x ) + X s ) ds 0 � t � [ g ( · , · − Γ s f ) − g ]( s , � . + X s − ) ν ( dz ) ds + mg 0 R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 31/ 57
Risk Reserve with Interests Thus M t = g ( t , � X t ) is a martingale if and only if � [ ∂ t g + ∂ x g β Γ t ] + [ g ( t , x − Γ t f ) − g ( t , x )] ν ( dz ) = 0 . R + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 32/ 57
Risk Reserve with Interests Thus M t = g ( t , � X t ) is a martingale if and only if � [ ∂ t g + ∂ x g β Γ t ] + [ g ( t , x − Γ t f ) − g ( t , x )] ν ( dz ) = 0 . R + Assume that g ( t , x ) = a ( t ) e − sx , a ( t ) > 0 to be determined, and f ≡ z and ν ( dz ) = λ F U ( dz ), then the above becomes 0 = a ′ ( t ) e − sx + {− β s Γ t + λ [ ˆ m ( s Γ t ) − 1] } g ( t , x ) � � a ′ ( t ) − θ ( s Γ t ) a ( t ) e − sx . = Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 32/ 57
Risk Reserve with Interests Thus M t = g ( t , � X t ) is a martingale if and only if � [ ∂ t g + ∂ x g β Γ t ] + [ g ( t , x − Γ t f ) − g ( t , x )] ν ( dz ) = 0 . R + Assume that g ( t , x ) = a ( t ) e − sx , a ( t ) > 0 to be determined, and f ≡ z and ν ( dz ) = λ F U ( dz ), then the above becomes 0 = a ′ ( t ) e − sx + {− β s Γ t + λ [ ˆ m ( s Γ t ) − 1] } g ( t , x ) � � a ′ ( t ) − θ ( s Γ t ) a ( t ) e − sx . = Assume a (0) = 1. We can solve the ODE a ′ ( t ) + θ ( s Γ t ) a ( t ) = 0 , t ≥ 0 � t to get a ( t ) = e − 0 θ ( s Γ u ) du . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 32/ 57
Risk Reserve with Interests Thus M t = g ( t , � X t ) is a martingale if and only if � [ ∂ t g + ∂ x g β Γ t ] + [ g ( t , x − Γ t f ) − g ( t , x )] ν ( dz ) = 0 . R + Assume that g ( t , x ) = a ( t ) e − sx , a ( t ) > 0 to be determined, and f ≡ z and ν ( dz ) = λ F U ( dz ), then the above becomes 0 = a ′ ( t ) e − sx + {− β s Γ t + λ [ ˆ m ( s Γ t ) − 1] } g ( t , x ) � � a ′ ( t ) − θ ( s Γ t ) a ( t ) e − sx . = Assume a (0) = 1. We can solve the ODE a ′ ( t ) + θ ( s Γ t ) a ( t ) = 0 , t ≥ 0 � t to get a ( t ) = e − 0 θ ( s Γ u ) du . � t △ Thus ˜ = g ( t , � X t ) = exp {− s � M t X t − 0 θ ( s Γ u ) du } is a mg. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 32/ 57
Lundberg Bounds for General Models Question: Can we find an exponential martingale that leads to the Lundberg bound for the general reserve model (2)? Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 33/ 57
Lundberg Bounds for General Models Question: Can we find an exponential martingale that leads to the Lundberg bound for the general reserve model (2)? Recall the exponential martingale � � � △ � = exp {− I s ( t , X t ) − K s M t = exp − s Γ t X t − θ ( s Γ u ) du t } . R + � △ = sx Γ t and K s where I s ( t , x ) t = R + θ ( s Γ u ) du . Define � t β t = − 0 r s ds , t ≥ 0 � t △ 0 r s ds = δ x Γ t = δ xe β t , δ ∈ R . = δ xe − I δ ( t , x ) � X t = e β t X t = Γ t X t (discounted risk reserve). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 33/ 57
Lundberg Bounds for General Models In general, we replace s by a parameter δ , and look for a possible exponential mg M δ = exp { I δ + K δ } , where I δ ( t , X t ) = δ � X t , and � X satisfies: d ˜ Γ t ( � b ( t , β t , � X t = X t ) + η t ) dt + � ˆ σ t , dW t � � − R + Γ t f ( t , x ) N p ( dtdx ) , where � b ( t , β t , � X t ) = b ( t , e − β t � X t )) = b ( t , X t ). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 34/ 57
Lundberg Bounds for General Models In general, we replace s by a parameter δ , and look for a possible exponential mg M δ = exp { I δ + K δ } , where I δ ( t , X t ) = δ � X t , and � X satisfies: d ˜ Γ t ( � b ( t , β t , � X t = X t ) + η t ) dt + � ˆ σ t , dW t � � − R + Γ t f ( t , x ) N p ( dtdx ) , where � b ( t , β t , � X t ) = b ( t , e − β t � X t )) = b ( t , X t ). � △ R + [ e γ f ( t , z ) − 1] ν ( dz ). To “decompose K δ , define m f t ( γ ) = Then m f ( γ ) is increasing in γ and integrable for all γ ≤ δ 0 . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 34/ 57
Lundberg Bounds for General Models In general, we replace s by a parameter δ , and look for a possible exponential mg M δ = exp { I δ + K δ } , where I δ ( t , X t ) = δ � X t , and � X satisfies: d ˜ Γ t ( � b ( t , β t , � X t = X t ) + η t ) dt + � ˆ σ t , dW t � � − R + Γ t f ( t , x ) N p ( dtdx ) , where � b ( t , β t , � X t ) = b ( t , e − β t � X t )) = b ( t , X t ). � △ R + [ e γ f ( t , z ) − 1] ν ( dz ). To “decompose K δ , define m f t ( γ ) = Then m f ( γ ) is increasing in γ and integrable for all γ ≤ δ 0 . In compound Poisson case, f ≡ z and ν ( dz ) = λ F U ( dz ), then � △ R + [ e γ z − 1] F U ( dz ) = λ ( ˆ m f t ( γ ) = λ m U ( γ ) − 1), again. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 34/ 57
Lundberg Bounds for General Models Now define K δ t = − V δ t + 1 2 Y δ t + Z δ t , where � t V δ e β s [ � b ( s , β s , � t = δ X s ) + η s ] ds ; 0 � t � t △ Y δ t = δ 2 e 2 β s | ˆ σ s | 2 ds ; Z δ m f s ( δ e β s ) ds . = t 0 0 � t △ Define also Z δ, 0 m f = s ( δ ) ds , and t 0 � D = { δ ≥ 0 : Z δ t < ∞ , P -a.s., ∀ t ≥ 0 } ; D 0 = { δ ≥ 0 : Z δ, 0 < ∞ , P -a.s., ∀ t ≥ 0 } . t Since γ ≥ 0 and β s ≤ 0, the monotonicity of m f ( · ) shows that D 0 ⊆ D . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 35/ 57
Main Results Theorem (M. Sun (02)) △ = exp {− δ � The process M δ X t − K δ t } , t ≥ 0 , enjoys the following t properties: For every δ ∈ D , { M δ t : t ≥ 0 } is an F -local martingale. If the processes π , σ , µ , and r are all bounded and F W -adapted, and that f ( · , · , · ) is deterministic, then for every δ ∈ D 0 , { M δ t : t ≥ 0 } is an F -martingale. If r is also deterministic, then (ii) holds for all δ ∈ D . If π is allowed to be F -adapted, then (ii) and (iii) hold for all δ such that 2 δ ∈ D and D 0 , respectively. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 36/ 57
Main Results Theorem (M. Sun (02)) △ = exp {− δ � The process M δ X t − K δ t } , t ≥ 0 , enjoys the following t properties: For every δ ∈ D , { M δ t : t ≥ 0 } is an F -local martingale. If the processes π , σ , µ , and r are all bounded and F W -adapted, and that f ( · , · , · ) is deterministic, then for every δ ∈ D 0 , { M δ t : t ≥ 0 } is an F -martingale. If r is also deterministic, then (ii) holds for all δ ∈ D . If π is allowed to be F -adapted, then (ii) and (iii) hold for all δ such that 2 δ ∈ D and D 0 , respectively. △ = exp( − δ x + v − 1 Proof: Define F δ ( x , v , y , z ) 2 y − z ), and o’s formula to F δ ( � X t , V δ t , Y δ t , Z δ applying Itˆ t )... Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 36/ 57
Main Results Example Classical Model π t ≡ 0 , r t ≡ 0 , ρ ≡ 0 , µ t ≡ 0 , σ t ≡ 0, S t is Compound Poisson � ∞ 0 ( e δ x − 1) λ F ( dx ) − c δ ) (= θ ( δ ) t !) K δ t = t ( � δ = sup { δ : θ ( δ ) ≤ 0 } = r ∗ Lundberg Exponent Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 37/ 57
Main Results Example Classical Model π t ≡ 0 , r t ≡ 0 , ρ ≡ 0 , µ t ≡ 0 , σ t ≡ 0, S t is Compound Poisson � ∞ 0 ( e δ x − 1) λ F ( dx ) − c δ ) (= θ ( δ ) t !) K δ t = t ( � δ = sup { δ : θ ( δ ) ≤ 0 } = r ∗ Lundberg Exponent Discounted Risk Reserve π t = ρ t = µ t = σ t ≡ 0, r > 0 S t is Compound Poisson � t � ∞ K δ 0 [exp( δ e − rs x ) − 1] λ F ( dx ) − ce − r s } ds t = 0 { � δ = sup { δ ≥ 0 : sup t ≥ 0 K δ t < ∞} Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 37/ 57
Main Results Example Classical Model π t ≡ 0 , r t ≡ 0 , ρ ≡ 0 , µ t ≡ 0 , σ t ≡ 0, S t is Compound Poisson � ∞ 0 ( e δ x − 1) λ F ( dx ) − c δ ) (= θ ( δ ) t !) K δ t = t ( � δ = sup { δ : θ ( δ ) ≤ 0 } = r ∗ Lundberg Exponent Discounted Risk Reserve π t = ρ t = µ t = σ t ≡ 0, r > 0 S t is Compound Poisson � t � ∞ K δ 0 [exp( δ e − rs x ) − 1] λ F ( dx ) − ce − r s } ds t = 0 { � δ = sup { δ ≥ 0 : sup t ≥ 0 K δ t < ∞} Perturbed risk reserve π t ≡ 1, ρ t = r t = µ t ≡ 0, σ t ≡ ε , X t = x + ct + ε W t − S t � ∞ 2 δ 2 ε 2 + 0 ( e δ x − 1) λ F ( dx )) △ K δ t = t ( − c δ + 1 = k ( δ ) t △ � δ = sup { δ > 0 : k ( δ ) = 0 } (Delbaen-Haezendonck (1987), ...) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 37/ 57
Ruin Probability via “Rate Functions” Extending the idea of the function I δ ( t , x ) = δ x β t , we can consider a more general “rate function”: I ∈ C 1 , 2 ( R + × R ). Define t + 1 △ △ M I = exp {− I ( t , X t ) − K I t } , K I = − V I 2 Y I t + Z I t , and t t � t � △ Z I = R + [exp { I ( s , X s ) − I ( s , X s − f ( s , x )) } − 1] v ( dx ) ds t 0 � t △ V I = { ∂ x I ( s , X s ) b ( s , X s ) + ∂ t I ( s , X s ) } ds t 0 � t △ { ( ∂ x I ( s , X s )) 2 − ∂ 2 Y I σ s | 2 ds = xx I ( s , X s ) }| ˆ t 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 38/ 57
Ruin Probability via “Rate Functions” Extending the idea of the function I δ ( t , x ) = δ x β t , we can consider a more general “rate function”: I ∈ C 1 , 2 ( R + × R ). Define t + 1 △ △ M I = exp {− I ( t , X t ) − K I t } , K I = − V I 2 Y I t + Z I t , and t t � t � △ Z I = R + [exp { I ( s , X s ) − I ( s , X s − f ( s , x )) } − 1] v ( dx ) ds t 0 � t △ V I = { ∂ x I ( s , X s ) b ( s , X s ) + ∂ t I ( s , X s ) } ds t 0 � t △ { ( ∂ x I ( s , X s )) 2 − ∂ 2 Y I σ s | 2 ds = xx I ( s , X s ) }| ˆ t 0 Definition A function I ∈ C 1 , 2 ( R + × R ) is called a “rate function” if Z I t < ∞ , ∀ t ≥ 0 , P-almost surely. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 38/ 57
Analysis Suppose that we can find I such that M I is a local martingale, and that I ( t , x ) ≤ 0, for all t and x ≤ 0. △ Let τ = inf { t , X t < 0 } , and apply Optional Sampling to supermartingale (nonnegative loc mg) M I t : E { e − I ( τ, X τ ) − K I e − I (0 , x ) τ | τ < T } P { τ < T } ≥ � � 0 ≤ t ≤ T e − K I ≥ E inf ψ ( x , T ) . t Applying Jensen’s inequality we have � � e − I (0 , x ) e K I � ≤ e − I (0 , x ) E � ψ ( x , T ) ≤ sup . t 0 ≤ t ≤ T e − K I E inf 0 ≤ t ≤ T t One can let T → ∞ to obtain the bound for ψ ( x ). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 39/ 57
Ruin Probability via “Rate Functions” Theorem For any rate function I, { M I t : t ≥ 0 } is an F -local martingale. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 40/ 57
Ruin Probability via “Rate Functions” Theorem For any rate function I, { M I t : t ≥ 0 } is an F -local martingale. (Lundberg Bounds) If the rate function I satisfies I ( t , x ) ≤ 0 , for all t and x ≤ 0 . Then, it holds that e − I (0 , x ) E exp( K I ψ ( x , T ) ≤ sup t ) , 0 ≤ t ≤ T e − I (0 , x ) E sup exp( K I ψ ( x ) ≤ t ) . t ≥ 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 40/ 57
Ruin Probability via “Rate Functions” Theorem For any rate function I, { M I t : t ≥ 0 } is an F -local martingale. (Lundberg Bounds) If the rate function I satisfies I ( t , x ) ≤ 0 , for all t and x ≤ 0 . Then, it holds that e − I (0 , x ) E exp( K I ψ ( x , T ) ≤ sup t ) , 0 ≤ t ≤ T e − I (0 , x ) E sup exp( K I ψ ( x ) ≤ t ) . t ≥ 0 In the Lundberg bounds above the process K I ( X ) can be △ replaced by K I ( X + ) , where X + = X s ∨ 0 . s Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 40/ 57
Ruin Probability via “Rate Functions” Theorem For any rate function I, { M I t : t ≥ 0 } is an F -local martingale. (Lundberg Bounds) If the rate function I satisfies I ( t , x ) ≤ 0 , for all t and x ≤ 0 . Then, it holds that e − I (0 , x ) E exp( K I ψ ( x , T ) ≤ sup t ) , 0 ≤ t ≤ T e − I (0 , x ) E sup exp( K I ψ ( x ) ≤ t ) . t ≥ 0 In the Lundberg bounds above the process K I ( X ) can be △ replaced by K I ( X + ) , where X + = X s ∨ 0 . s Question: How to find a rate function? Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 40/ 57
Asmussen-Nielsen Bound Assume Compound Poisson ( f ( t , x ) = x , and v ( dx ) = λ F ( dx )), and π ≡ 0, µ ≡ 0, σ ≡ 0, r t = r (constant), ρ ( t , x ) ≡ ρ ( x ) is an increasing function in x . Then � t � t � X t = x + p ( X s ) ds + R + x µ ( dxds ), t ≥ 0, 0 0 △ where p ( x ) = rx + c (1 + ρ ( x )). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 41/ 57
Asmussen-Nielsen Bound Assume Compound Poisson ( f ( t , x ) = x , and v ( dx ) = λ F ( dx )), and π ≡ 0, µ ≡ 0, σ ≡ 0, r t = r (constant), ρ ( t , x ) ≡ ρ ( x ) is an increasing function in x . Then � t � t � X t = x + p ( X s ) ds + R + x µ ( dxds ), t ≥ 0, 0 0 △ where p ( x ) = rx + c (1 + ρ ( x )). � x Consider the Rate function of the form: I ( x ) = γ ( y ) dy , 0 x ≥ 0, γ ( · ) > 0, increasing. Then � t � � X + � � � � s − x γ ( y ) dy − 1 s K I − [ γ p ]( X + X + t = s ) + e λ F ( dx ) ds 0 R + � t � � � s ) x − 1] λ F ( dx ) R + [ e γ ( X + − [ γ p ]( X + ≤ s ) + ds . 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 41/ 57
Asmussen-Nielsen Bound Let γ be the non-decreasing solution to the Lundberg equation: � R + [ e γ x − 1] λ F ( dx ) = 0 , − γ p ( y ) + y ≥ 0 . (such solution exists if the so-called net profit condition: inf x ≥ 0 p ( x ) > λ E [ U 1 ] holds and ρ is monotone.) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 42/ 57
Asmussen-Nielsen Bound Let γ be the non-decreasing solution to the Lundberg equation: � R + [ e γ x − 1] λ F ( dx ) = 0 , − γ p ( y ) + y ≥ 0 . (such solution exists if the so-called net profit condition: inf x ≥ 0 p ( x ) > λ E [ U 1 ] holds and ρ is monotone.) One can show that if p ( · ) ∈ C 1 , then I can be extended so that I ( · ) ∈ C 2 ( R ), I (0) = 0, and I ( x ) ≤ 0 for x < 0. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 42/ 57
Asmussen-Nielsen Bound Let γ be the non-decreasing solution to the Lundberg equation: � R + [ e γ x − 1] λ F ( dx ) = 0 , − γ p ( y ) + y ≥ 0 . (such solution exists if the so-called net profit condition: inf x ≥ 0 p ( x ) > λ E [ U 1 ] holds and ρ is monotone.) One can show that if p ( · ) ∈ C 1 , then I can be extended so that I ( · ) ∈ C 2 ( R ), I (0) = 0, and I ( x ) ≤ 0 for x < 0. t ( X + ) ≤ 0, ∀ t ≥ 0, and we have Thus K I ψ ( x , T ) ≤ e − I ( x ) ψ ( x ) ≤ e − I ( x ) . and This is the Asmussen and Nielsen bound (1995). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 42/ 57
Can We Do Better? Assume now ρ ( x ) ≡ 0, and F ( x ) = 1 − e − θ x , x ≥ 0. Then the Asmussen-Nielsen bound tells us: ψ ( x ) ≤ e − θ x � � λ 1 + r r , c x x ≥ 0 . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 43/ 57
Can We Do Better? Assume now ρ ( x ) ≡ 0, and F ( x ) = 1 − e − θ x , x ≥ 0. Then the Asmussen-Nielsen bound tells us: ψ ( x ) ≤ e − θ x � � λ 1 + r r , c x x ≥ 0 . Let us consider a new rate function: for b ∈ C 2 , I ( y ) = − log b ( y ) 1 [0 , ∞ ) ( y ) , Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 43/ 57
Can We Do Better? Assume now ρ ( x ) ≡ 0, and F ( x ) = 1 − e − θ x , x ≥ 0. Then the Asmussen-Nielsen bound tells us: ψ ( x ) ≤ e − θ x � � λ 1 + r r , c x x ≥ 0 . Let us consider a new rate function: for b ∈ C 2 , I ( y ) = − log b ( y ) 1 [0 , ∞ ) ( y ) , � t Denote K I ( X + ) = 0 L [ I ]( X + s ) ds , where L is an ID operator: � ∞ △ [ e I ( y ) − I ( y − x ) − 1] λθ e − θ x dx . = − I ′ ( y )[ ry + c ] + L [ I ]( y ) 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 43/ 57
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