Ruin Probabilities in a Diffusion Environment Jan Grandell & - PowerPoint PPT Presentation

Ruin Probabilities in a Diffusion Environment Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne New Frontiers in Applied Probability, Sandbjerg, 2nd of August

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes Cox Models 1 Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes Cox Models 1 Ornstein–Uhlenbeck Intensities 2 Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes Cox Models 1 Ornstein–Uhlenbeck Intensities 2 Subexponential Claim Sizes 3 Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes Cox Models 1 Ornstein–Uhlenbeck Intensities 2 Subexponential Claim Sizes 3 Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Risk Model N t � X t = x + ct − Y i i =1 x : initial capital Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Risk Model N t � X t = x + ct − Y i i =1 x : initial capital c : premium rate Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Risk Model N t � X t = x + ct − Y i i =1 x : initial capital c : premium rate { N t } : A single point process Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Risk Model N t � X t = x + ct − Y i i =1 x : initial capital c : premium rate { N t } : A single point process { Y i } : iid, Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Risk Model N t � X t = x + ct − Y i i =1 x : initial capital c : premium rate { N t } : A single point process { Y i } : iid, independent of { N t } Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Risk Model N t � X t = x + ct − Y i i =1 x : initial capital c : premium rate { N t } : A single point process { Y i } : iid, independent of { N t } G ( y ): distribution function of Y i , G (0) = 0 Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Risk Model N t � X t = x + ct − Y i i =1 x : initial capital c : premium rate { N t } : A single point process { Y i } : iid, independent of { N t } G ( y ): distribution function of Y i , G (0) = 0 E [ e rY − 1]. E [ Y n µ n = I i ], µ = µ 1 , h ( r ) = I I I Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes Diffusion Intensities Let { Z t } be a diffusion process following the stochastic differential equation d Z t = b ( Z t ) d W t + a ( Z t ) d t for some Brownian motion { W t } . We assume that there is a strong solution. Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes Diffusion Intensities Let { Z t } be a diffusion process following the stochastic differential equation d Z t = b ( Z t ) d W t + a ( Z t ) d t for some Brownian motion { W t } . We assume that there is a strong solution. � t Let Λ( t ) = 0 ℓ ( Z s ) d s for some function ℓ . We define N ( t ) = ˜ N (Λ( t )) , where { ˜ N t } is a Poisson process with rate 1. Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes Diffusion Intensities Let { Z t } be a diffusion process following the stochastic differential equation d Z t = b ( Z t ) d W t + a ( Z t ) d t for some Brownian motion { W t } . We assume that there is a strong solution. � t Let Λ( t ) = 0 ℓ ( Z s ) d s for some function ℓ . We define N ( t ) = ˜ N (Λ( t )) , where { ˜ N t } is a Poisson process with rate 1. Thus, given { Z t } , the claim number process { N t } is conditionally an inhomogeneous Poisson process with rate { ℓ ( Z t ) } . Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Martingale The process M = { g ( Z t ) e − r ( X t − x ) − θ ( r ) t } is a martingale if 1 2 b 2 ( z ) g ′′ ( z ) + a ( z ) g ′ ( z ) + [ ℓ ( z ) h ( r ) − θ − cr − ] g ( z ) = 0 . Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Martingale The process M = { g ( Z t ) e − r ( X t − x ) − θ ( r ) t } is a martingale if 1 2 b 2 ( z ) g ′′ ( z ) + a ( z ) g ′ ( z ) + [ ℓ ( z ) h ( r ) − θ − cr − ] g ( z ) = 0 . Suppose we found a solution with a positive g ( z ). This is only possible if θ = θ ( r ) depends on the important parameter r . Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Martingale The process M = { g ( Z t ) e − r ( X t − x ) − θ ( r ) t } is a martingale if 1 2 b 2 ( z ) g ′′ ( z ) + a ( z ) g ′ ( z ) + [ ℓ ( z ) h ( r ) − θ − cr − ] g ( z ) = 0 . Suppose we found a solution with a positive g ( z ). This is only possible if θ = θ ( r ) depends on the important parameter r . We norm g , such that lim t →∞ I E [ g ( Z t )] = 1. I Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Change of Measure Consider the measure E [ g ( Z T ) e − r ( X T − x ) − θ ( r ) T ; A ] Q [ A ] = I I . E [ g ( Z 0 )] I I Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Change of Measure Consider the measure E [ g ( Z T ) e − r ( X T − x ) − θ ( r ) T ; A ] Q [ A ] = I I . E [ g ( Z 0 )] I I The process ( { X t , Z t ) } remains a Cox model with claim size distribution � x e ry d G ( y ) , Q [ Y ≤ x ] = ( h ( r ) + 1) − 1 0 Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Change of Measure Consider the measure E [ g ( Z T ) e − r ( X T − x ) − θ ( r ) T ; A ] Q [ A ] = I I . E [ g ( Z 0 )] I I The process ( { X t , Z t ) } remains a Cox model with claim size distribution � x e ry d G ( y ) , Q [ Y ≤ x ] = ( h ( r ) + 1) − 1 0 claim intensity ℓ ( Z t )( h ( r ) + 1), Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Change of Measure Consider the measure E [ g ( Z T ) e − r ( X T − x ) − θ ( r ) T ; A ] Q [ A ] = I I . E [ g ( Z 0 )] I I The process ( { X t , Z t ) } remains a Cox model with claim size distribution � x e ry d G ( y ) , Q [ Y ≤ x ] = ( h ( r ) + 1) − 1 0 claim intensity ℓ ( Z t )( h ( r ) + 1), and generator of the diffusion A f = ga + b 2 g ′ f ′ + 1 2 b 2 f ′′ . ˜ g Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment

Cox Models Ornstein–Uhlenbeck Intensities Subexponential Claim Sizes The Change of Measure Typically, the function θ ( r ) will be convex. Since P [ X t g ( Z t ) e − rX t e − θ ( r ) t ] I I E Q [ X t ] = I I E II we have d P [ g ( Z t ) e − rX t e − θ ( r ) t ] 0 = d r I E II I �� d � e − rX t e − θ ( r ) t � E Q [ X t ] − t θ ′ ( r ) . = d r g ( Z t ) − I I I E II I P Jan Grandell & Hanspeter Schmidli KTH Stockholm & University of Cologne Ruin Probabilities in a Diffusion Environment