✩ Applications Deficit | R ( τ x ) | : ◮ If we were able to readily compute F | R ( τ x ) | we would have a family of distributions indexed by the initial reserve level. ◮ Ruin-based risk measures could then be constructed.
✩ Applications Deficit | R ( τ x ) | : ◮ If we were able to readily compute F | R ( τ x ) | we would have a family of distributions indexed by the initial reserve level. ◮ Ruin-based risk measures could then be constructed. VaR x α
✩ Applications Deficit | R ( τ x ) | : ◮ If we were able to readily compute F | R ( τ x ) | we would have a family of distributions indexed by the initial reserve level. ◮ Ruin-based risk measures could then be constructed. VaR x α The smallest deficit in the top 5% worst case scenarios.
✩ Applications Deficit | R ( τ x ) | : ◮ If we were able to readily compute F | R ( τ x ) | we would have a family of distributions indexed by the initial reserve level. ◮ Ruin-based risk measures could then be constructed. VaR x α The smallest deficit in the top 5% worst case scenarios. ◮ P ( | R ( τ x ) | > VaR x α ) = α .
Applications Deficit | R ( τ x ) | : ◮ If we were able to readily compute F | R ( τ x ) | we would have a family of distributions indexed by the initial reserve level. ◮ Ruin-based risk measures could then be constructed. VaR x α The smallest deficit in the top 5% worst case scenarios. ◮ P ( | R ( τ x ) | > VaR x α ) = α . ◮ VaR x 0 . 05 . If ruin occurs, we can expect to observe (five times out of a hundred) a deficit of at least ✩ VaR x 0 . 05 when we start off with a level x .
Applications Deficit | R ( τ x ) | : ◮ If we were able to readily compute F | R ( τ x ) | we would have a family of distributions indexed by the initial reserve level. ◮ Ruin-based risk measures could then be constructed. VaR x α The smallest deficit in the top 5% worst case scenarios. ◮ P ( | R ( τ x ) | > VaR x α ) = α . ◮ VaR x 0 . 05 . If ruin occurs, we can expect to observe (five times out of a hundred) a deficit of at least ✩ VaR x 0 . 05 when we start off with a level x . ◮ It gives a solvency argument to set an appropriate initial reserve x .
✩ ✩ Applications
✩ ✩ Applications Last minimum R ( τ − ) :
✩ ✩ Applications Last minimum R ( τ − ) : ◮ If we were able to readily compute F R ( τ − ) we would have a family of distributions indexed by the initial reserve level.
✩ ✩ Applications Last minimum R ( τ − ) : ◮ If we were able to readily compute F R ( τ − ) we would have a family of distributions indexed by the initial reserve level. ◮ Due to its non-local nature at ruin, ruin-based risk measures could be used to set warning levels.
✩ ✩ Applications Last minimum R ( τ − ) : ◮ If we were able to readily compute F R ( τ − ) we would have a family of distributions indexed by the initial reserve level. ◮ Due to its non-local nature at ruin, ruin-based risk measures could be used to set warning levels. VaR x α
✩ ✩ Applications Last minimum R ( τ − ) : ◮ If we were able to readily compute F R ( τ − ) we would have a family of distributions indexed by the initial reserve level. ◮ Due to its non-local nature at ruin, ruin-based risk measures could be used to set warning levels. VaR x α The smallest last minimum in the top 5% worst case scenarios.
✩ ✩ Applications Last minimum R ( τ − ) : ◮ If we were able to readily compute F R ( τ − ) we would have a family of distributions indexed by the initial reserve level. ◮ Due to its non-local nature at ruin, ruin-based risk measures could be used to set warning levels. VaR x α The smallest last minimum in the top 5% worst case scenarios. ◮ P ( R ( τ − ) > VaR x α ) = α .
✩ Applications Last minimum R ( τ − ) : ◮ If we were able to readily compute F R ( τ − ) we would have a family of distributions indexed by the initial reserve level. ◮ Due to its non-local nature at ruin, ruin-based risk measures could be used to set warning levels. VaR x α The smallest last minimum in the top 5% worst case scenarios. ◮ P ( R ( τ − ) > VaR x α ) = α . ◮ VaR x 0 . 05 . In those cases when ruin occurs, the last minimum will be observed to be (ninety five times out of a hundred) smaller than ✩ VaR x 0 . 05 when starting off with a level x .
✩ Applications Last minimum R ( τ − ) : ◮ If we were able to readily compute F R ( τ − ) we would have a family of distributions indexed by the initial reserve level. ◮ Due to its non-local nature at ruin, ruin-based risk measures could be used to set warning levels. VaR x α The smallest last minimum in the top 5% worst case scenarios. ◮ P ( R ( τ − ) > VaR x α ) = α . ◮ VaR x 0 . 05 . In those cases when ruin occurs, the last minimum will be observed to be (ninety five times out of a hundred) smaller than ✩ VaR x 0 . 05 when starting off with a level x . ◮ Does it give a warning level?
Applications Last minimum R ( τ − ) : ◮ If we were able to readily compute F R ( τ − ) we would have a family of distributions indexed by the initial reserve level. ◮ Due to its non-local nature at ruin, ruin-based risk measures could be used to set warning levels. VaR x α The smallest last minimum in the top 5% worst case scenarios. ◮ P ( R ( τ − ) > VaR x α ) = α . ◮ VaR x 0 . 05 . In those cases when ruin occurs, the last minimum will be observed to be (ninety five times out of a hundred) smaller than ✩ VaR x 0 . 05 when starting off with a level x . ◮ Does it give a warning level? ◮ Do you want to be below a reserve level of ✩ VaR x 0 . 05 !!!!
Computing the EDPF
Computing the EDPF Theorem (Biffis and Morales (2010)) Let φ δ G denote the Generalized EDPF. Moreover, let K denote the exponential distribution with mean σ 2 / 2 c and density k. Then, φ G is given by w 0 e − ρ x (1 − K ( x )) + H G ( x ) φ δ � g ∗ ( n ) ( x ) , � � G ( x ) = ∗ x � 0 . n � 0 (7)
Computing the EDPF
Computing the EDPF Functions involved are
Computing the EDPF Functions involved are ◮ The function g is given by � y �� + ∞ g ( y ) = 1 � e − ρ ( y − s ) k ( y − s ) e − ρ ( x − s ) ν S ( dx ) + G ρ ( s ) ds , c 0 s (8) with the function G ρ defined through its Laplace transform � + ∞ e − ξ x G ρ ( x ) dx = Ψ � J ( ξ ) − Ψ � J ( ρ ) , ξ � 0 , (9) ρ − ξ 0 and ρ the unique non-negative solution of the generalized Lundberg equation cr + Ψ S − Z ( r ) = δ .
Computing the EDPF
Computing the EDPF ◮ The function H G is given by � u � + ∞ H G ( u ) = 1 e − ρ ( u − s ) k ( u − s ) e − ρ ( x − s ) χ G ( x , s ) dx ds , c 0 s (10) where, for x , s > 0, the function χ G is defined as � + ∞ χ G ( x , s ) = w ( y − x , x , s ) ν S − Z ( dy ) . (11) x +
Three examples
Three examples
Three examples ◮ θ -process with parameter λ = 3 / 2 1 2 σ 2 z 2 + µ z − c � � � � ψ X ( z ) = α + z /β coth π α + z /β + c √ α coth π √ α � � ,
Three examples ◮ θ -process with parameter λ = 3 / 2 1 2 σ 2 z 2 + µ z − c � � � � ψ X ( z ) = α + z /β coth π α + z /β + c √ α coth π √ α � � , ◮ θ -process with parameter λ = 5 / 2 1 2 σ 2 z 2 + µ z + c ( α + z /β ) 3 � � 2 coth � ψ X ( z ) = π α + z /β π √ α 3 2 coth � � − c α ,
Three examples ◮ θ -process with parameter λ = 3 / 2 1 2 σ 2 z 2 + µ z − c � � � � ψ X ( z ) = α + z /β coth π α + z /β + c √ α coth π √ α � � , ◮ θ -process with parameter λ = 5 / 2 1 2 σ 2 z 2 + µ z + c ( α + z /β ) 3 � � 2 coth � ψ X ( z ) = π α + z /β π √ α 3 2 coth � � − c α , ◮ β -process with parameter λ ∈ (0 , 3) \ { 1 , 2 } 1 2 σ 2 z 2 + µ z + c B(1 + α + z /β, 1 − λ ) ψ X ( z ) = − c B(1 + α, 1 − λ ) . where B( x , y ) = Γ( x )Γ( y ) / Γ( x + y ) is the Beta function.
Three examples
Three examples These were introduced in Kuznetsov (2009) with first-passage times problems in mind. Features
Three examples These were introduced in Kuznetsov (2009) with first-passage times problems in mind. Features ◮ Good risk models equivalent to GIG, IG and Gamma.
Three examples These were introduced in Kuznetsov (2009) with first-passage times problems in mind. Features ◮ Good risk models equivalent to GIG, IG and Gamma. ◮ as x → 0 − , | x | − λ , π ( x ) ∼ e β (1+ α ) x , π ( x ) ∼ as x → −∞ .
Three examples These were introduced in Kuznetsov (2009) with first-passage times problems in mind. Features ◮ Good risk models equivalent to GIG, IG and Gamma. ◮ as x → 0 − , | x | − λ , π ( x ) ∼ e β (1+ α ) x , π ( x ) ∼ as x → −∞ . ◮ No closed-form densities
Three examples These were introduced in Kuznetsov (2009) with first-passage times problems in mind. Features ◮ Good risk models equivalent to GIG, IG and Gamma. ◮ as x → 0 − , | x | − λ , π ( x ) ∼ e β (1+ α ) x , π ( x ) ∼ as x → −∞ . ◮ No closed-form densities ◮ Infinite series expressions for the L´ evy measures b m e ρ m x . � π ( x ) = m ≥ 1
Three examples These were introduced in Kuznetsov (2009) with first-passage times problems in mind. Features ◮ Good risk models equivalent to GIG, IG and Gamma. ◮ as x → 0 − , | x | − λ , π ( x ) ∼ e β (1+ α ) x , π ( x ) ∼ as x → −∞ . ◮ No closed-form densities ◮ Infinite series expressions for the L´ evy measures b m e ρ m x . � π ( x ) = m ≥ 1 ◮ Quasi-closed form expressions for the EPDF in both infinite- and finite- time horizon!!!!
Main Results: Infinite-time Horizon
Main Results: Infinite-time Horizon The discounted joint density of all three quantities under these three models is given in the following result.
Main Results: Infinite-time Horizon The discounted joint density of all three quantities under these three models is given in the following result. Theorem For δ ≥ 0 , x > 0 , y > 0 , z > 0 and u ∈ (0 , z ∧ x ) � � e − δτ x I ( | R τ x | < y ; R τ x − < z ; R τ x − < u ) I { τ x < ∞} | R 0 = x = E � σ 2 b m (1 − e − ρ m y ) Φ( δ ) � c n ζ n e − ζ n x � 2 + δ ρ m (Φ( δ ) + ρ m ) n ≥ 1 m ≥ 1 � e ( ζ n − ρ m ) u − 1 − e − (Φ( δ )+ ρ m ) z × e (Φ( δ )+ ζ n ) u − 1 � � × , ζ n − ρ m Φ( δ ) + ζ n where Φ( δ ) as the unique positive solution to ψ X ( z ) = δ (generalized Lundberg equation).
Non-ruin quantities
Non-ruin quantities Let us define, D t = X t − X t , where X t is the running supremum process X t = sup s ∈ [0 , t ] X s . We are interested primarily in the following stopping-times: τ a = inf { t > 0 | D t > a } , ρ = sup { t ∈ [0 , τ a ] | X t = X t } , for some predetermined value a > 0.
Non-ruin quantities Let us define, D t = X t − X t , where X t is the running supremum process X t = sup s ∈ [0 , t ] X s . We are interested primarily in the following stopping-times: τ a = inf { t > 0 | D t > a } , ρ = sup { t ∈ [0 , τ a ] | X t = X t } , for some predetermined value a > 0. These are the times of the first drawdown larger than a and the last time that the reserve was at its supremum before the a -drawdown.
Non-ruin quantities
Non-ruin quantities Related quantities are:
Non-ruin quantities Related quantities are: ◮ D τ a size of drawdown ,
Non-ruin quantities Related quantities are: ◮ D τ a size of drawdown , ◮ τ a − ρ speed of depletion ,
Non-ruin quantities Related quantities are: ◮ D τ a size of drawdown , ◮ τ a − ρ speed of depletion , ◮ X τ a the maximum of X at the first-passage time,
Non-ruin quantities Related quantities are: ◮ D τ a size of drawdown , ◮ τ a − ρ speed of depletion , ◮ X τ a the maximum of X at the first-passage time, ◮ X τ a the minimum of X at the first-passage time,
Non-ruin quantities Related quantities are: ◮ D τ a size of drawdown , ◮ τ a − ρ speed of depletion , ◮ X τ a the maximum of X at the first-passage time, ◮ X τ a the minimum of X at the first-passage time, ◮ D τ a − drawdown size just before it crosses the level a ,
Non-ruin quantities Related quantities are: ◮ D τ a size of drawdown , ◮ τ a − ρ speed of depletion , ◮ X τ a the maximum of X at the first-passage time, ◮ X τ a the minimum of X at the first-passage time, ◮ D τ a − drawdown size just before it crosses the level a , ◮ D τ a − a the overshoot of the drawdown process over the level a .
Non-ruin quantities Related quantities are: ◮ D τ a size of drawdown , ◮ τ a − ρ speed of depletion , ◮ X τ a the maximum of X at the first-passage time, ◮ X τ a the minimum of X at the first-passage time, ◮ D τ a − drawdown size just before it crosses the level a , ◮ D τ a − a the overshoot of the drawdown process over the level a .
Illustration for drawdown related variables
Illustration for drawdown related variables
Expressions
Expressions Work is not complete but a very advance stage.
Expressions Work is not complete but a very advance stage. Key issues:
Expressions Work is not complete but a very advance stage. Key issues: ◮ All expressions are given in terms of scale functions ,
Expressions Work is not complete but a very advance stage. Key issues: ◮ All expressions are given in terms of scale functions , ◮ Expressions are tractable for exponential jumps,
Expressions Work is not complete but a very advance stage. Key issues: ◮ All expressions are given in terms of scale functions , ◮ Expressions are tractable for exponential jumps, ◮ And potentially some classes of subordinators,
Expressions Work is not complete but a very advance stage. Key issues: ◮ All expressions are given in terms of scale functions , ◮ Expressions are tractable for exponential jumps, ◮ And potentially some classes of subordinators, ◮ Expressions for the speed of depletion seems to be the most complicated of all.
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