Finance, Insurance, and Stochastic Control (III) Jin Ma Spring - PowerPoint PPT Presentation

Finance, Insurance, and Stochastic Control (III) Jin Ma Spring School on “Stochastic Control in Finance” Roscoff, France, March 7-17, 2010 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 1/ 58

Outline Reinsurance and Stochastic Control Problems 1 Proportional Reinsurance with Diffusion Models 2 General Reinsurance Problems 3 Admissibility of Strategies 4 Existence of Admissible Strategies 5 Utility Optimization 6 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 2/ 58

Reinsurance Problem Basic Idea An insurance company may choose to “cede” some of its risk to a reinsurer by paying a premium. Thus the reserve may look like � t � t � c h X t = x + s (1 + ρ s ) ds − h ( s , x ) µ ( dxds ) , 0 0 R + where h is the “retention function” Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 3/ 58

Reinsurance Problem Basic Idea An insurance company may choose to “cede” some of its risk to a reinsurer by paying a premium. Thus the reserve may look like � t � t � c h X t = x + s (1 + ρ s ) ds − h ( s , x ) µ ( dxds ) , 0 0 R + where h is the “retention function” Common types of retention functions: h ( x ) = α x , 0 ≤ α ≤ 1 — Proportional Reinsurance h ( x ) = α ∧ x , α > 0 — Stop-loss Reinsurance Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 3/ 58

Reinsurance Problem Basic Idea An insurance company may choose to “cede” some of its risk to a reinsurer by paying a premium. Thus the reserve may look like � t � t � c h X t = x + s (1 + ρ s ) ds − h ( s , x ) µ ( dxds ) , 0 0 R + where h is the “retention function” Common types of retention functions: h ( x ) = α x , 0 ≤ α ≤ 1 — Proportional Reinsurance h ( x ) = α ∧ x , α > 0 — Stop-loss Reinsurance Purpose Determine the “reasonable” reinsurance premium, find the ”best” reinsurance policy,..., etc. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 3/ 58

Generalized Cram´ er-Lundberg model (Ω , F , P ) — a complete probability space W = { W t } t ≥ 0 — a d -dimensional Brownian Motion p = { p t } t ≥ 0 — stationary Poisson point process, ⊥ ⊥ W N p ( dtdz ) — counting measure of p on (0 , ∞ ) × R + ˆ N p ( dtdz ) = E ( N p ( dtdz )) = ν ( dz ) dt F = F W ⊗ F p , � T △ � F q = { ϕ : F p -predi’ble, E R + | ϕ | q d ν ds < ∞ , q ≥ 1 } p 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 4/ 58

Generalized Cram´ er-Lundberg model (Ω , F , P ) — a complete probability space W = { W t } t ≥ 0 — a d -dimensional Brownian Motion p = { p t } t ≥ 0 — stationary Poisson point process, ⊥ ⊥ W N p ( dtdz ) — counting measure of p on (0 , ∞ ) × R + ˆ N p ( dtdz ) = E ( N p ( dtdz )) = ν ( dz ) dt F = F W ⊗ F p , � T △ � F q = { ϕ : F p -predi’ble, E R + | ϕ | q d ν ds < ∞ , q ≥ 1 } p 0 Claim Process � t + � S t = f ( s , x , ω ) N p ( dsdx ) , t ≥ 0 , f ∈ F p . (1) 0 R + Compound Poisson Case: f ( t , z ) ≡ z , ν ( R + ) = λ . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 4/ 58

Profit Margin Principle A “Counting Principle” for Reinsurance Premiums ρ — original safety loading of the cedent company ρ r — safety loading of the reinsurance company ρ α — modified safety loading of the cedent company (after reinsurance) If the claim size is U , then the “ profit margin principle ” states = (1 + ρ r ) E [ U − h ( U )] + (1 + ρ α ) E [ h ( U )] (1 + ρ ) E [ U ] . (2) � �� � � �� � � �� � original premium premium to the modified premium reinsurance company ρ r = ρ α = ρ — “Cheap” Reinsurance ρ r � = ρ α — “Non-cheap” Reinsurance Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 5/ 58

Existing Literature Stop-Loss Reinsurance (e.g., Sondermann (1991), Mnif-Sulem (2001), Azcue-Muler (2005), ...) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 6/ 58

Existing Literature Stop-Loss Reinsurance (e.g., Sondermann (1991), Mnif-Sulem (2001), Azcue-Muler (2005), ...) Proportional Reinsurance Diffusion approximation: dX t = µα t dt + σα t dW t , X 0 = x (e.g., Asmussen-Hojgaard-Taksar (2000), ...) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 6/ 58

Existing Literature Stop-Loss Reinsurance (e.g., Sondermann (1991), Mnif-Sulem (2001), Azcue-Muler (2005), ...) Proportional Reinsurance Diffusion approximation: dX t = µα t dt + σα t dW t , X 0 = x (e.g., Asmussen-Hojgaard-Taksar (2000), ...) General reserve models (Liu-M. 2009, ...) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 6/ 58

Proportional Reinsurance with Diffusion Models The following case study is based on Hojgaard-Taksar (1997). Consider the reserve with “proportional reinsurance” : � t X t = x + α c (1 + ρ s ) ds − α S t . 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 7/ 58

Proportional Reinsurance with Diffusion Models The following case study is based on Hojgaard-Taksar (1997). Consider the reserve with “proportional reinsurance” : � t X t = x + α c (1 + ρ s ) ds − α S t . 0 Replacing this by the following “Diffusion Model”: � t � t X t = x + µα t dt + σα t dW t , t ≥ 0 , (3) 0 0 where µ > 0, σ > 0, and α t ∈ [0 , 1] is a stochastic process representing the fraction of the incoming claim that the insurance company retains to itself. We call it a “ admissible reinsurance policy ” if it is F W -adapted. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 7/ 58

Proportional Reinsurance with Diffusion Models “ Return Function ”: � τ △ e − ct X x ,α J ( x ; α ) = E dt , t 0 where τ = τ x ,α = inf { t ≥ 0 : X x ,α = 0 } is the ruin time and t c > 0 is the “ discount factor ”. “ Value Function ”: V ( x ) = sup J ( x ; α ) α ∈ A Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 8/ 58

Proportional Reinsurance with Diffusion Models “ Return Function ”: � τ △ e − ct X x ,α J ( x ; α ) = E dt , t 0 where τ = τ x ,α = inf { t ≥ 0 : X x ,α = 0 } is the ruin time and t c > 0 is the “ discount factor ”. “ Value Function ”: V ( x ) = sup J ( x ; α ) α ∈ A Note For any α ∈ A and x > 0, define ˆ α t = α t 1 { t ≤ τ x ,α } . Then α = τ x ,α = τ x , ˆ ⇒ J ( x , ˆ α ) = J ( x , α ). we can work on △ A ′ ( x ) = { α ∈ A : α t = 0 for all t > τ x ,α } and � ∞ △ J ′ ( x ; α ) e − ct X x ,α α ∈ A ′ ( x ) . = E dt , t 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 8/ 58

The HJB Equation 1. The Concavity of V . For any x 1 , x 2 > 0 and λ ∈ (0 , 1), let α i ∈ A ( x i ), i = 1 , 2. △ △ = λ x 1 + (1 − λ ) x 2 , α = λα 1 + (1 − λ ) α 2 . Define ξ Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 9/ 58

The HJB Equation 1. The Concavity of V . For any x 1 , x 2 > 0 and λ ∈ (0 , 1), let α i ∈ A ( x i ), i = 1 , 2. △ △ = λ x 1 + (1 − λ ) x 2 , α = λα 1 + (1 − λ ) α 2 . Define ξ Denote X i = X x i ,α i and τ i = τ x i ,α i , i = 1 , 2. Then by the linearity of the reserve equation (3) one has △ = τ ξ,α = τ 1 ∨ τ 2 . △ = X ξ,α = λ X 1 t + (1 − λ ) X 2 X t t , and τ t ⇒ J ( ξ, α ) = λ J ( x 1 , α 1 ) + (1 − λ ) J ( x 2 , α 2 ) . = Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 9/ 58

The HJB Equation 1. The Concavity of V . For any x 1 , x 2 > 0 and λ ∈ (0 , 1), let α i ∈ A ( x i ), i = 1 , 2. △ △ = λ x 1 + (1 − λ ) x 2 , α = λα 1 + (1 − λ ) α 2 . Define ξ Denote X i = X x i ,α i and τ i = τ x i ,α i , i = 1 , 2. Then by the linearity of the reserve equation (3) one has △ = τ ξ,α = τ 1 ∨ τ 2 . △ = X ξ,α = λ X 1 t + (1 − λ ) X 2 X t t , and τ t ⇒ J ( ξ, α ) = λ J ( x 1 , α 1 ) + (1 − λ ) J ( x 2 , α 2 ) . = ∀ ε > 0, choose α i , s.t. J ( x i , α i ) ≥ V ( x i ) − ε/ 2, i = 1 , 2. ⇒ J ( ξ, α ) = λ J ( x 1 , α 1 ) + (1 − λ ) J ( x 2 , α 2 ) = ≥ λ V ( x 1 ) + (1 − λ ) V ( x 2 ) − ε ⇒ V ( ξ ) ≥ λ V ( x 1 ) + (1 − λ ) V ( x 2 ) − ε = = ⇒ Done! Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 9/ 58

The HJB Equation 2. The HJB Equation. Let τ be any F -stopping time. By “ Bellman Principle ” �� τ α ∧ τ � e − ct X x , ˆ dt + e − c ( τ α ∧ τ ) V ( X x , ˆ α α V ( x ) = sup E τ α ∧ τ ) . t α ∈ A ( x ) 0 △ ∀ α ∈ A and h > 0 let τ h = τ h = h ∧ inf { t : | X α t − x | > h } . α Then τ h < ∞ , a.s. and τ h → 0, as h → 0, a.s. Assume V ∈ C 2 . For any a ∈ [0 , 1], define α ≡ a ∈ A . Then for any h < x , we have τ h < τ α . Letting τ = τ h in (4) and o (to F ( t , x ) = e − ct V ( x )) we deduce applying Itˆ �� τ h � e − ct X x ,α dt + e − ct [ L a V ]( X x ,α 0 ≥ E ) dt , t t 0 △ = σ 2 a 2 where [ L a g ]( x ) 2 g ′′ ( x ) + µ ag ′ ( x ) − cg ( x ). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 10/ 58

The HJB Equation Letting h → 0, one has 0 ≥ x + [ L a V ]( x ) . 0 ≥ x + max a ∈ [0 , 1] [ L a V ]( x ), since a is arbitrary. = ⇒ Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 11/ 58