On optimal stopping of autoregressive sequences

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On optimal stopping of autoregressive sequences Sören Christensen (joint work with A. Irle, A. Novikov) Mathematisches Seminar, CAU Kiel September 24, 2010 Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 1 / 22

Outline 1 AR(1) processes 2 Basic results on optimal stopping 3 Threshold times as optimal stopping times 4 Phasetype distributions and overshoot 5 Continuous fit condition 6 Summary Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 1 / 22

Outline AR(1) processes 1 AR(1) processes 2 Basic results on optimal stopping 3 Threshold times as optimal stopping times 4 Phasetype distributions and overshoot 5 Continuous fit condition 6 Summary Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 2 / 22

AR(1) processes Setting AR(1) process X n = λx n 1 + Z n, X 0 = x, Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 3 / 22

AR(1) processes Setting AR(1) process X n = λx n 1 + Z n, X 0 = x, λ (0, 1), Z 1, Z 2,... iid. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 3 / 22

AR(1) processes Setting AR(1) process X n = λx n 1 + Z n, X 0 = x, λ (0, 1), Z 1, Z 2,... iid. X n = (1 λ)x n 1 n + L n, L n = n i=1 Z i Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 3 / 22

Setting AR(1) processes AR(1) process X n = λx n 1 + Z n, X 0 = x, λ (0, 1), Z 1, Z 2,... iid. X n = (1 λ)x n 1 n + L n, X n = λ n X 0 + n 1 i=0 λi Z n i. L n = n i=1 Z i Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 3 / 22

AR(1) processes Path of an AR(1) process Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 4 / 22

AR(1) processes Threshold time, overshoot In many applications such as statistical surveillance it is of interest to find the joint distribution of τ a = inf{n N 0 : X n a} X τa a (threshold time) (overshoot). a X τa a τ a Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 5 / 22

Outline Basic results on optimal stopping 1 AR(1) processes 2 Basic results on optimal stopping 3 Threshold times as optimal stopping times 4 Phasetype distributions and overshoot 5 Continuous fit condition 6 Summary Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 6 / 22

Basic results on optimal stopping Optimal stopping of Markov processes Let (X n ) n N0 be a Markov process on E, g : E R, ρ (0, 1). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 7 / 22

Basic results on optimal stopping Optimal stopping of Markov processes Let (X n ) n N0 be a Markov process on E, g : E R, ρ (0, 1). Find a stopping time τ that maximizes E x (ρ τ g(x τ )). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 7 / 22

Basic results on optimal stopping Optimal stopping of Markov processes Let (X n ) n N0 be a Markov process on E, g : E R, ρ (0, 1). Find a stopping time τ that maximizes Solution : E x (ρ τ g(x τ )). v(x) := sup E x (ρ τ g(x τ )) τ S := {x E : v(x) g(x)} Optimal stopping time: value function. optimal stopping set. τ = inf{n N 0 : X n S}. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 7 / 22

Outline Threshold times as optimal stopping times 1 AR(1) processes 2 Basic results on optimal stopping 3 Threshold times as optimal stopping times 4 Phasetype distributions and overshoot 5 Continuous fit condition 6 Summary Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 8 / 22

Threshold times as optimal stopping times Elementary approach to optimal stopping of AR(1) processes 1 Reduce the set of potential optimal stopping times to threshold time. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 9 / 22

Threshold times as optimal stopping times Elementary approach to optimal stopping of AR(1) processes 1 Reduce the set of potential optimal stopping times to threshold time. 2 Find the joint distribution of (τ a, X τa a) to obtain an expression for E x (ρ τa g(x τa )). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 9 / 22

Threshold times as optimal stopping times Elementary approach to optimal stopping of AR(1) processes 1 Reduce the set of potential optimal stopping times to threshold time. 2 Find the joint distribution of (τ a, X τa a) to obtain an expression for E x (ρ τa g(x τa )). 3 Find the optimal boundary a. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 9 / 22

Threshold times as optimal stopping times Novikov-Shiryaev problem Let g(x) = (x + ) α, α > 0. Consider sup τ E x (ρ τ (X τ + ) α ) = sup E(ρ τ ((λ τ x + X τ ) + ) α ). τ Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 10 / 22

Threshold times as optimal stopping times Novikov-Shiryaev problem Let g(x) = (x + ) α, α > 0. Consider sup τ E x (ρ τ (X τ + ) α ) = sup E(ρ τ ((λ τ x + X τ ) + ) α ). τ Proposition If (X n ) n N0 is a AR(1) process, then there exists a such that the optimal stopping set S is given by S = [a, ) Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 10 / 22

Proof Threshold times as optimal stopping times For simplicity: Z 1 0. State-space: [0, ) Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 11 / 22

Proof Threshold times as optimal stopping times For simplicity: Z 1 0. State-space: [0, ) Then v(x) g(x) = sup τ E(ρ τ (λ τ x + X τ ) α ) x α = sup E(ρ τ (λ τ + X τ /x) α ) τ Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 11 / 22

Proof Threshold times as optimal stopping times For simplicity: Z 1 0. State-space: [0, ) Then v(x) g(x) = sup τ E(ρ τ (λ τ x + X τ ) α ) x α = sup E(ρ τ (λ τ + X τ /x) α ) τ Hence S = {x : v(x) g(x)} is of the form [a, ). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 11 / 22

Outline Phasetype distributions and overshoot 1 AR(1) processes 2 Basic results on optimal stopping 3 Threshold times as optimal stopping times 4 Phasetype distributions and overshoot 5 Continuous fit condition 6 Summary Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 12 / 22

Motivation Phasetype distributions and overshoot If (X n ) n N0 is a random walk (or an AR(1) process) with Exp(β)-distributed jumps. Then τ a, X τa a are independent and X τa a d = Exp(β). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 13 / 22

Phasetype distributions and overshoot Distributions of phasetype (J t ) t 0 : Markov chain in continuous time on {1,..., m} { }. 1,..., m: transient, absorbing. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 14 / 22

Phasetype distributions and overshoot Distributions of phasetype (J t ) t 0 : Markov chain in continuous time on {1,..., m} { }. 1,..., m: transient, absorbing. ( ) Generator: ˆQ 1 Q q =, Q R 0 0 m m, q = Q... 1 ˆα = (α 1,..., α m, 0) initial distribution Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 14 / 22

Phasetype distributions and overshoot Distributions of phasetype (J t ) t 0 : Markov chain in continuous time on {1,..., m} { }. 1,..., m: transient, absorbing. ( ) Generator: ˆQ 1 Q q =, Q R 0 0 m m, q = Q... 1 ˆα = (α 1,..., α m, 0) initial distribution η := inf{t 0 : J t = } PH(Q, α) := P ηˆα is called phasetype distribution with parameters Q, α. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 14 / 22

Phasetype distributions and overshoot Properties of phasetype distributions density: f (t) = αe Qt q Laplace transform: Eˆα (e sη ) = α( si Q) 1 q, s C with R(s) < δ. closed under mixture and convolution. Phasetype distributions are dense in all distributions on (0, ). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 15 / 22

Phasetype distributions and overshoot Properties of phasetype distributions density: f (t) = αe Qt q Laplace transform: Eˆα (e sη ) = α( si Q) 1 q, s C with R(s) < δ. closed under mixture and convolution. Phasetype distributions are dense in all distributions on (0, ). Ex: m = 1, Q = ( λ) PH(Q, (1)) := Exp(λ). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 15 / 22

Phasetype distributions and overshoot Phasetype distribution and overshoot Generalization: If (X n ) n N0 is AR(1) process with innovations Z n = S n T n, S n, T n 0 independent, S n PH(Q, α), then E x (ρ τa g(x τa )) = m E x (ρ τa 1 Ki )E(g(a + R i )), i=1 R i PH(Q, e i ), K i events. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 16 / 22

Phasetype distributions and overshoot Phasetype distribution and overshoot Generalization: If (X n ) n N0 is AR(1) process with innovations Z n = S n T n, S n, T n 0 independent, S n PH(Q, α), then E x (ρ τa g(x τa )) = m E x (ρ τa 1 Ki )E(g(a + R i )), i=1 R i PH(Q, e i ), K i events. How to find E x (ρ τa 1 Ki )? Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 16 / 22

Phasetype distributions and overshoot Phasetype distribution and overshoot Generalization: If (X n ) n N0 is AR(1) process with innovations Z n = S n T n, S n, T n 0 independent, S n PH(Q, α), then E x (ρ τa g(x τa )) = m E x (ρ τa 1 Ki )E(g(a + R i )), i=1 R i PH(Q, e i ), K i events. How to find E x (ρ τa 1 Ki )? Answer: Use the stationary distribution, complex martingales and complex analysis. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 16 / 22

Outline Continuous fit condition 1 AR(1) processes 2 Basic results on optimal stopping 3 Threshold times as optimal stopping times 4 Phasetype distributions and overshoot 5 Continuous fit condition 6 Summary Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 17 / 22

Continuous fit condition How to find the find the optimal threshold a? For each a R we consider v a (x) = E x (ρ τa g(x τa )). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 18 / 22

Continuous fit condition How to find the find the optimal threshold a? For each a R we consider v a (x) = E x (ρ τa g(x τa )). Choose a such that v a (a ) = g(a ). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 18 / 22

Continuous fit Continuous fit condition a 0.7 is the unique solution to the transcendental equation g(a ) = v a (a ). Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 19 / 22

Continuous fit condition Smooth and continuous fit condition Principle for continuous time processes Let g be differentiable, then the following can be expected: 1 If the process X enters the interior of the optimal stopping set S immediately after starting on a S, then the value function v is smooth at a. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 20 / 22

Continuous fit condition Smooth and continuous fit condition Principle for continuous time processes Let g be differentiable, then the following can be expected: 1 If the process X enters the interior of the optimal stopping set S immediately after starting on a S, then the value function v is smooth at a. 2 If the process X doesn t enter the interior of the optimal stopping set S immediately after starting on a S, then the value function v is continuous at a Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 20 / 22

Continuous fit condition Smooth and continuous fit condition Principle for continuous time processes Let g be differentiable, then the following can be expected: 1 If the process X enters the interior of the optimal stopping set S immediately after starting on a S, then the value function v is smooth at a. 2 If the process X doesn t enter the interior of the optimal stopping set S immediately after starting on a S, then the value function v is continuous at a The second principle holds in our situation. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 20 / 22

Outline Summary 1 AR(1) processes 2 Basic results on optimal stopping 3 Threshold times as optimal stopping times 4 Phasetype distributions and overshoot 5 Continuous fit condition 6 Summary Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 21 / 22

Summary Summary We studied optimal stopping problems driven by autoregressive processes. Elementary arguments reduces the problem to finding an optimal threshold time. The joint distribution of (τ a, X τa a) can be obtained for phasetype innovations. The optimal threshold can be found using the continuous fit principle. Sören Christensen (CAU Kiel) Optimal stopping of AR(1) sequences 20.09.10 22 / 22

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