Controlled Risk Processes and Large Claims Hanspeter Schmidli - PowerPoint PPT Presentation

Controlled Risk Processes and Large Claims Hanspeter Schmidli University of Cologne Optimization and Optimal Control Linz, 21st of October

Introduction The Optimisation Problem Asymptotic Properties Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction 1 The Model Large Claims The Optimisation Problem 2 Asymptotic Properties 3 Optimal Reinsuance Optimal Investment Optimal Investment and Reinsurance

Introduction The Optimisation Problem Asymptotic Properties The Model The Classical Risk Model Lundberg (1903) intruduced the model N t � S t = Y i , i =1 X 0 t = x + ct − S t . x : Initial capital. c : Rate of the linear income, premium rate. { N t } : Poisson process with rate λ . { Y i } : iid sequence, distribution function G ( y ), G (0) = 0. { N t } and { Y i } are independent. Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties The Model Proportional Reinsurance The insurer can buy proportional reinsurance, i.e. the insurer pays bY , the reinsurer pays (1 − b ) Y of a claim of size Y . There is a premium at rate c − c ( b ) the insurer has to pay. We assume c ( b ) continuous. c ( b ) increasing. c (1) = c . c (0) < 0. Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties The Model Proportional Reinsurance The insurer can at any time choose the retention level b t ∈ [0 , 1]. Then the surplus process becomes � t N t X b � t = x + c ( b s ) d s − b T i − Y i . 0 i =1 Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties The Model Investment The insurer can invest the surplus into a risky asset (Black-Scholes model) 2 σ 2 ) t + σ W t } . Z t = exp { ( m − 1 { W t } and { S t } are independent. Choosing a strategy { A t } the surplus process becomes d X A X A t = ( c + A t m ) d t + σ A t d W t − d S t , 0 = x Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties The Model Proportional Reinsurance and Investment If the insurer can buy reinsurance and invest the surplus process fulfils d X A , b X A , b = ( c ( b t ) + A t m ) d t + σ A t d W t − b t − d S t , = x t 0 Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties Large Claims Heavy-Tailed Claims We say a distribution function F is heavy-tailed (F ∈ H ) if � ∞ e rx d F ( x ) = ∞ M F ( r ) := 0 for all r > 0. We say a distribution function F is long-tailed (F ∈ L ) if 1 − F ( x + y ) lim = 1 1 − F ( x ) x →∞ for all y ∈ I R . Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties Large Claims Subexponential Distributions A distribution function F ( x ) with F (0) = 0 is called subexponential ( F ∈ S ) if 1 − F ∗ n ( x ) lim = n 1 − F ( x ) x →∞ for some (and therefore all) n ≥ 2. The definition can be interpreted as � n � � I P I X i > x ∼ I I P [max { X 1 , . . . , X n } > x ] . i =1 Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties Large Claims The Class S ∗ A distribution function F ( x ) is in S ∗ if it has finite mean µ F and � x (1 − F ( x − y ))(1 − F ( y )) lim d y = 2 µ F . 1 − F ( x ) x →∞ 0 Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties Large Claims Regularly Varying Tail A distribution function F ( x ) has a regularly varying tail with index − α ( F ∈ R − α ) if 1 − F ( tx ) 1 − F ( x ) = t − α . lim x →∞ R − α ⊂ S ∗ ⊂ S ⊂ L ⊂ H . Moreover, the log-normal and the heavy-tailed Weibull distributions belong to S ∗ . Thus S ∗ contains all heavy-tailed distribution functions of interest. Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties The Optimisation Problem The Optimisation Problem τ Ab := inf { t ≥ 0 : X Ab < 0 } : time of ruin t P [ τ Ab < ∞ ]: ruin probability ψ Ab ( x ) := I I Goal: Minimisation of the ruin probability A , b ψ Ab ( x ) . ψ ( x ) = inf The problem is connected to the Hamilton-Jacobi-Bellman (HJB) equation b ∈ [0 , 1] c ( b ) ψ ′ ( x ) + λ I inf E [ ψ ( x − bY ) − ψ ( x )] = 0 . I Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties The Optimisation Problem Verification Theorem Theorem (Hipp and Plum, S.) Suppose there is an increasing (and twice continuously differentiable) function f ( x ) such that 1 − f ( x ) / f ( ∞ ) solves HJB. Then f ( x ) is bounded and f ( x ) = f ( ∞ )(1 − ψ ( x )) . Moreover, ( A ( X t ) , b ( X t )) is an optimal strategy, where A ( x ) , b ( x ) are the arguments where the minimum in the HJB is taken. Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties The Optimisation Problem Sketch of Proof Proof. Case with investment: The process � t ∧ τ � σ 2 f ( X A 2 A 2 s f ′′ ( X A s ) + ( c + mA s ) f ′ ( X A t ∧ τ ) − s ) 0 � E [ f ( X A s − Y ) − f ( X A + λ I I s )] d s is a local martingale. Using the HJB it follows that f ( X A t ∧ τ ) is a supermartingale. Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties The Optimisation Problem Sketch of Proof Proof (continued). E [ f ( X A Thus f ( x ) ≥ I I t ∧ τ )]. Choosing a strategy for which ruin is not certain shows that f ( x ) is bounded. Thus P [ τ = ∞ ] = f ( ∞ )(1 − ψ A ( x )) . f ( x ) ≥ f ( ∞ ) I I Choosing the optimal strategy gives that f ( X ∗ τ ∧ t ) is a bounded martingale. Thus E [ f ( X ∗ τ )] = f ( ∞ )(1 − ψ ∗ ( x )) . f ( x ) = I I Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties The Optimisation Problem Existence of a Solution Theorem (S.) Suppose G ( x ) is continuous. Then ψ ( x ) solves the HJB. Theorem (Hipp and Plum, S.) Suppose G ( x ) is absolutely continuous with a bounded density. Then ψ ( x ) is twice continuously differentiable and solves the HJB. Proof. Contraction arguments. Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties Optimal Reinsuance Claim Sizes with a Regularly Varying Tail Theorem Suppose that the tail of the claim size distribution is regularly varying with index α . Then λ b α ψ ( x ) lim x (1 − G ( z )) d z = inf ( c ( b ) − λµ b ) + . � ∞ x →∞ b ∈ (0 , 1] If there is a unique value b ∗ for which the infimum is taken we also have convergence of the strategy lim x →∞ b ( x ) = b ∗ . Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties Optimal Reinsuance Proof. Choose b t = b constant such that c ( b ) > λµ b . ψ b ( x ) ψ b ( x ) b α lim = lim � ∞ � ∞ x (1 − G ( z )) d z x (1 − G ( z / b )) d z x →∞ x →∞ λ b α = c ( b ) − λµ b Thus λ b α ψ ( x ) lim sup x (1 − G ( z )) d z ≤ inf ( c ( b ) − λµ b ) + . � ∞ b ∈ (0 , 1] x →∞ Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties Optimal Reinsuance Proof (continued). Let g ( x ) = − ψ ′ ( x ) / (1 − G ( x )). Then for b = b ( x ) �� x g ( z )(1 − G ( z ))(1 − G (( x − z ) / b )) λ d z 1 − G ( x ) 0 + δ (0)1 − G ( x / b ) � − c ( b ) g ( x ) = 0 . 1 − G ( x ) g 0 = lim inf x →∞ g ( x ) > 0. Take a sequence { x n } such that g ( x n ) → g 0 . Take a subsequence such that b ( x n ) → b 0 . Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties Optimal Reinsuance Proof (continued). � x n / 2 g ( z )(1 − G ( z ))1 − G (( x n − z ) / b ( x n )) d z 1 − G ( x n ) 0 � x n / 2 ψ ′ ( z ) d z . ≤ − C 0 � x n / 2 g ( z )(1 − G ( z ))1 − G (( x n − z ) / b ( x n )) d z → b α 0 ψ (0) . 1 − G ( x n ) 0 Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims

Introduction The Optimisation Problem Asymptotic Properties Optimal Reinsuance Proof (continued). The second part of the integral is � x n / 2 g ( x n − z )1 − G ( x n − z ) [1 − G ( z / b ( x n ))] d z . 1 − G ( x n ) 0 ≥ g 0 − ε ≥ 1 lim inf bounded from below by g 0 b 0 µ . λ b α 0 + λµ b 0 g 0 − c ( b 0 ) g 0 ≤ 0 . This shows that c ( b 0 ) − λµ b 0 > 0. Hanspeter Schmidli University of Cologne Controlled Risk Processes and Large Claims λ b α