On a class of dependent Sparre Andersen risk models with - PowerPoint PPT Presentation

Model and Notation The fixed point equation Two applications On a class of dependent Sparre Andersen risk models with application. F.Avram, A.Badescu, M.Pistorius and L.Rabehasaina Laboratory of Mathematics, Besan¸ con, University of Franche Comt´ e, France. MAM Conference, Budapest, 28th-30th June 2016 1 / 16

Model and Notation The fixed point equation Two applications Content Model and Notation The fixed point equation Two applications 2 / 16

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Model and Notation The fixed point equation Two applications Risk process { X ( t ) , t ≥ 0 } N ( t ) � X ( t ) = u + ct − J k , t ≥ 0 . i =1 N ( t ) = max { n ∈ N : � n k =1 T k ≤ t } number of claims up to time t , T k interclaim, J k claim size, u ≥ 0 initial capital, c > 0 premium rate, c E [ T 1 ] > E [ J 1 ], { ( T k , J k ) , k ∈ N } i.i.d. with dependence structure, defined by P ( T k ∈ d t , J k ∈ d x ) = α ( d t ) e Rx r d x t , x ∈ R + , where α ( d t ) ∈ R 1 × m , is a 1 × m distribution vector, R ∈ R m × m sub-generator matrix, r = ( − R )1. 4 / 16

Model and Notation The fixed point equation Two applications Ruin probability We let τ := { t ≥ 0 , X ( t ) < 0 } the ruin time and its Laplace Transform ˆ e − q τ � � ψ ( q , u ) := E u , q ≥ 0 , u ≥ 0 . → Goal : Compute ˆ − ψ ( q , u ) with efficient algorithm, with LT � ∞ e − qt α ( d t ) ∈ R 1 × m , q ∈ R + , available. α ( − q ) := ˆ 0 Notation : If Q ∈ R m × m negative-definite, we extend definition of LT : � ∞ α ( d t ) e Qt ∈ R 1 × m . α ( Q ) := ˆ 0 α ( − qI ) available for all q ∈ R + , but ˆ ˆ α ( Q ) not explicitly computable in practice for general Q ! 5 / 16

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Model and Notation The fixed point equation Two applications Fixed point equation Theorem Laplace transform ˆ ψ ( q , u ) verifies ˆ ρ ( q ) e [ R + r ˆ ρ ( q )] u 1 , ψ ( q , u ) = ˆ u ≥ 0 , q ≥ 0 , (1) where ˆ ρ ( q ) is a 1 × m sub-probability vector satisfying the fixed point equation ρ ( q ) = ˆ ˆ α ( cR + c r ˆ ρ ( q ) − qI ) , q > 0 . (2) If q = 0 there exists a 1 × m sub-probability vector ˆ ρ (0) verifying (2) such that expression (1) holds for ˆ ψ (0 , u ) . 7 / 16

Model and Notation The fixed point equation Two applications Example and issues Main issue is solving (2), i.e. ρ ( q ) = ˆ ˆ α ( cR + c r ˆ ρ ( q ) − qI ) , ρ ( q ) ∈ R 1 × m subprobability vector. E.g. with unkwown ˆ α ( d t ) ∈ R scalar, J k exponentially distributed : Malinovskii (1998), ˆ ρ ( q ) scalar, α ( d t ) ∈ R scalar, J k ∼ PH ( r , R ) : Asmussen and Albrecher (2010), ˆ ρ ( q ) scalar. Issues here : (2) does not necessarily have a unique solution, 1 α ( d t ) vector, 2 need to be able to compute ˆ α ( M ) where M is a matrix : no 3 explicit form . 8 / 16

Model and Notation The fixed point equation Two applications Algorithm for fixed point equation ρ N ( q ), N ∈ N , solution to Idea : Approximating ˆ ρ ( q ) by ˆ ρ N ( q ) = ˆ α N ( cR + c r ˆ ρ N ( q ) − qI ) , ˆ N M k ( δ )( Q + δ I ) k α N ( Q ) := � where ˆ , δ > 0 large enough, and k ! k =0 � ∞ t k e − δ t α ( d t ) ∈ R 1 × m . M k ( δ ) := 0 9 / 16

Model and Notation The fixed point equation Two applications Algorithm for fixed point equation Advantages : α N ( Q ) computable if the M k ( δ ) ’s, k ∈ N , are computable , − → ˆ − → Convergence : Theorem ρ N ( q ) − One has ˆ → ˆ ρ ( q ) as N → ∞ for all q ≥ 0 . Besides, for q large enough : � N � | M k ( δ ) | m � � ρ N ( q ) � δ k � ˆ ρ ( q ) − ˆ m ≤ C α (0) . 1 − ˆ � � k ! � k =0 with explicit C, and ˆ α (0) explicit. 10 / 16

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Model and Notation The fixed point equation Two applications Bailout problem U 1 ( t ) u 1 Replenishment at level 0 t ζ (1) ζ (1) 1 2 U 0 ( t ) k 1 ζ (1) (prop.cost) 1 u 0 K (1) (fixed cost) 1 k 1 ζ (1) 2 K (1) 2 t Ruin time τ of CB Figure : Sample path with proportional and fixed cost. 12 / 16

Model and Notation The fixed point equation Two applications Bailout problem Goal : Determine LT of ruin time τ of { U 0 ( t ) , t ≥ 0 } ( Central Branch ) starting from u 0 ≥ 0, when claims and interclaims for { U 1 ( t ) , t ≥ 0 } ( subsidiary ) are PH distributed. Step 1 : Identify dependence structure α ( d t ) and matrix R : α ( d t ) ∼ ruin time distribution of τ 1 jointly to phase at ruin , R ∼ same as claims of U 1 ( t ) + independent PH ( k , K ) , − → ˆ α ( − q ), q ∈ R + , available. Step 2 : Compute the M k ( δ )’s, k ∈ N : Ren and Stanford (2012). Step 3 : Run the algorithm. 13 / 16

Model and Notation The fixed point equation Two applications Queues and flushes (in progress) U 1 ( t ) Flush U 0 ( t ) Server Figure : Flush from queue 1 to 0. 14 / 16

Model and Notation The fixed point equation Two applications Queues and flushes Fluid queues { U 0 ( t ) , t ≥ 0 } and { U 1 ( t ) , t ≥ 0 } fluid queues, fed at constant rate c 0 and c 1 . U 1 ( t ) served with priority over U 0 ( t ), instantaneously, according to PH services, Content of U 1 ( t ) is occasionally flushed into U 0 ( t ) at time according to a Poisson process. Goal : Determine LT of ruin time τ of U 0 ( t ) = busy period of U 0 ( t ). Steps : Identify dependence structure α ( d t ) and matrix R , and compute the M k ( δ )’s, k ∈ N . 15 / 16

Model and Notation The fixed point equation Two applications Thank you ! 16 / 16