Parisian ruin for fluid flow risk processes Oscar Peralta Gutirrez 1 - PowerPoint PPT Presentation

Fluid flow risk processes. Parisian ruin. Parisian ruin for fluid flow risk processes Oscar Peralta Gutiérrez 1 , Mogens Bladt 2 , Bo Friis Nielsen 1 . 1 Technical University of Denmark Department of Applied Mathematics and Compute Science. 2 Autonomous National University of Mexico. Budapest, Hungary, June 2016. Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Fluid flow model. Consider a fluid flow model with Brownian noise (started at level u ∈ R ) on the form � t � t ( t ≥ 0) , V t = u + r J s ds + σ J s dB s , (1) 0 0 where Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Fluid flow model. Consider a fluid flow model with Brownian noise (started at level u ∈ R ) on the form � t � t ( t ≥ 0) , V t = u + r J s ds + σ J s dB s , (1) 0 0 where { J t } t ≥ 0 is a Markov jump process with finite state space E and intensity matrix Λ Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Fluid flow model. Consider a fluid flow model with Brownian noise (started at level u ∈ R ) on the form � t � t ( t ≥ 0) , V t = u + r J s ds + σ J s dB s , (1) 0 0 where { J t } t ≥ 0 is a Markov jump process with finite state space E and intensity matrix Λ { B t } t ≥ 0 is an independent standard Brownian motion, and Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Fluid flow model. Consider a fluid flow model with Brownian noise (started at level u ∈ R ) on the form � t � t ( t ≥ 0) , V t = u + r J s ds + σ J s dB s , (1) 0 0 where { J t } t ≥ 0 is a Markov jump process with finite state space E and intensity matrix Λ { B t } t ≥ 0 is an independent standard Brownian motion, and for every i ∈ E , r i ∈ R \ { 0 } and σ i ≥ 0. Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Fluid flow model. Consider a fluid flow model with Brownian noise (started at level u ∈ R ) on the form � t � t ( t ≥ 0) , V t = u + r J s ds + σ J s dB s , (1) 0 0 where { J t } t ≥ 0 is a Markov jump process with finite state space E and intensity matrix Λ { B t } t ≥ 0 is an independent standard Brownian motion, and for every i ∈ E , r i ∈ R \ { 0 } and σ i ≥ 0. Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Fluid flow model. Consider a fluid flow model with Brownian noise (started at level u ∈ R ) on the form � t � t ( t ≥ 0) , V t = u + r J s ds + σ J s dB s , (1) 0 0 where { J t } t ≥ 0 is a Markov jump process with finite state space E and intensity matrix Λ { B t } t ≥ 0 is an independent standard Brownian motion, and for every i ∈ E , r i ∈ R \ { 0 } and σ i ≥ 0. Suppose that V t → + ∞ as t → ∞ a.s.. Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Notation for the fluid flow model. E is partitioned and ordered into E σ := { i ∈ E : σ i > 0 } , E + := { i ∈ E : σ i = 0 , r i > 0 } , and E − := { i ∈ E : σ i = 0 , r i < 0 } . Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Notation for the fluid flow model. E is partitioned and ordered into E σ := { i ∈ E : σ i > 0 } , E + := { i ∈ E : σ i = 0 , r i > 0 } , and E − := { i ∈ E : σ i = 0 , r i < 0 } . The infinitesimal generator of { J t } t ≥ 0 is written as  Λ σσ Λ σ + Λ σ −   , Λ + σ Λ ++ Λ + − Λ = (2)  Λ − σ Λ − + Λ −− Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Notation for the fluid flow model. E is partitioned and ordered into E σ := { i ∈ E : σ i > 0 } , E + := { i ∈ E : σ i = 0 , r i > 0 } , and E − := { i ∈ E : σ i = 0 , r i < 0 } . The infinitesimal generator of { J t } t ≥ 0 is written as  Λ σσ Λ σ + Λ σ −   , Λ + σ Λ ++ Λ + − Λ = (2)  Λ − σ Λ − + Λ −− Define the row vectors r σ := { r i : i ∈ E σ } , r + := { r i : i ∈ E + } , r − := { r i : i ∈ E − } , σ := { σ i : i ∈ E σ } . Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Which is the connection with risk theory? We define the fluid flow risk process { R t } t ≥ 0 by regarding the linear downward segments of { V t } t ≥ 0 as downward jumps of the same height. Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Which is the connection with risk theory? We define the fluid flow risk process { R t } t ≥ 0 by regarding the linear downward segments of { V t } t ≥ 0 as downward jumps of the same height. Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Which is the connection with risk theory? We define the fluid flow risk process { R t } t ≥ 0 by regarding the linear downward segments of { V t } t ≥ 0 as downward jumps of the same height. Classic task: Compute ψ ( u ) := P (inf s ≥ 0 R s < 0 | R 0 = u ) , the classic probability of ruin . Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Classic risk processes as fluid flow risk processes. Example (Cramér-Lundberg process) The classic Cramér-Lundberg process with linear drift p > 0, Poisson arrival rate β , and PH( α , S )-distributed claims can be represented as a fluid flow risk process { R t } t ≥ 0 with characteristics E σ = ∅ , E + = { 1 } , E − = { 2 , 3 , . . . , m + 1 } , r + = ( p ), r − = ( − 1 , . . . , − 1) and � − β � β α Λ = . − Se S Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Classic risk processes as fluid flow risk processes. Example (Cramér-Lundberg process) The classic Cramér-Lundberg process with linear drift p > 0, Poisson arrival rate β , and PH( α , S )-distributed claims can be represented as a fluid flow risk process { R t } t ≥ 0 with characteristics E σ = ∅ , E + = { 1 } , E − = { 2 , 3 , . . . , m + 1 } , r + = ( p ), r − = ( − 1 , . . . , − 1) and � − β � β α Λ = . − Se S Remark Instead of having upward linear segments, we can opt to have a Brownian component, so that Lévy risk processes with phase-type jumps are an example of a fluid flow risk process. Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Classic risk processes as fluid flow risk processes. Example (Sparre-Andersen process) The Sparre-Andersen process with PH( α , S )-distributed claims and PH( π , T )-distributed interarrival times can be represented as a fluid flow risk process { R t } t ≥ 0 with characteristics E σ = ∅ , E + = { 1 , . . . , n } , E − = { n + 1 , . . . , n + m } , r + = (1 , . . . , 1), r − = ( − 1 , . . . , − 1) and � � T − Te α Λ = . − Se π S Technical University of Denmark Parisian ruin for fluid flow risk processes

Fluid flow risk processes. Model definition. Parisian ruin. Examples. Classic risk processes as fluid flow risk processes. Example (Sparre-Andersen process) The Sparre-Andersen process with PH( α , S )-distributed claims and PH( π , T )-distributed interarrival times can be represented as a fluid flow risk process { R t } t ≥ 0 with characteristics E σ = ∅ , E + = { 1 , . . . , n } , E − = { n + 1 , . . . , n + m } , r + = (1 , . . . , 1), r − = ( − 1 , . . . , − 1) and � � T − Te α Λ = . − Se π S Remark We can represent risk processes with MAP arrivals and phase type jumps as fluid flow risk processes. This processes are basically Markov additive risk processes with phase-type jumps . Technical University of Denmark Parisian ruin for fluid flow risk processes

Definition. The associated Markov chain. Fluid flow risk processes. Main result. Parisian ruin. Main result. Numerical example. Parisian ruin (when E σ = ∅ ). Definition Suppose that E σ = ∅ , let { L i } i ≥ 1 be i.i.d. clocks and associate each L i to the (possible) i-th excursion below zero of { R t } t ≥ 0 . Technical University of Denmark Parisian ruin for fluid flow risk processes

Definition. The associated Markov chain. Fluid flow risk processes. Main result. Parisian ruin. Main result. Numerical example. Parisian ruin (when E σ = ∅ ). Definition Suppose that E σ = ∅ , let { L i } i ≥ 1 be i.i.d. clocks and associate each L i to the (possible) i-th excursion below zero of { R t } t ≥ 0 . Technical University of Denmark Parisian ruin for fluid flow risk processes