Robust Detection of Unobservable Disorder Time in Poisson Rate

Transcription

1 Robust Detection of Unobservable Disorder Time in Poisson Rate Angers, Septembre 2015 Nicole El Karoui (LPMA UPMC/Paris 6) Stéphane Loisel and Yahia Salhi (ISFA Actuarial School, Lyon 1) with the financial support of ANR ( project LoLitA) and the Chair Risques financiers, 1/36 NEK, Angers Septembre 2015

2 Plan 1 What is the disorder problem 2 Cusum processes 3 Differential finite variation calculus 4 Optimality result 2/36 NEK, Angers Septembre 2015

3 Motivation The Poisson disorder problem is less formally stated as follows Observe a trajectory of the Poisson process (N t ) whose intensity changes from λ to ρλ at some (unknown) time θ. The problem is to find a rule to detect θ as quickly as possible with a limited number of false alarms 3/36 NEK, Angers Septembre 2015

4 Where do the disorder problems arise? Industrial Problems Quality control and maintenance, fraud and computer intrusion detection... etc. See also other application discussed during this workshop Our Main Motivation Non-life Insurance: Recalculate the premiums for the future sales of insurance policies when the risk structure changes (ρ > 1) Pension funds: Large exposure to change in mortality risk : (ρ < 1) Life insurance: Monitoring and surveillance of mortality dynamics Sequential information on death occurrences Updating mortality assumptions 4/36 NEK, Angers Septembre 2015

5 Overview of change detection The observation process Let N = (N t ) t 0 be a counting process (of claims, or deaths) with intensity λ = (λ t ) t 0. A change in the intensity occurs at an unobservable date θ, from λ to ρλ, ρ > 0. Change of Point θ Random with known prior (Bayesian) Deterministic but unknown (Non-Bayesian but Robust) Known statistics P and P (λ, ρλ) Under P, no change, (λ t ) holds Under P, immediate change, (ρλ t ) holds Under P θ, λ θ t = 1 t<θ λ t + 1 t θ ρλ t 5/36 NEK, Angers Septembre 2015

6 Bayesian setup for random change-point Based on the conditional distribution of the time of change, Formulated as an optimal stopping problem for partially observable process Brownian framework with abrupt change in the drift Page (1954), Shiryaev (1963), Roberts (1966), Beibel (1988), Moustakides (2004), and Dayanik (2006),... Poisson framework with abrupt change in intensity More recent studies : Gal (1971), Gapeev (2005), Bayraktar (2005, 2006), Dayanik (2006) for compound Poisson, Peskir, Shyriaev (2009) and others New methods using particle filters Andrieu, Legland (2004), Zhang (2005) 6/36 NEK, Angers Septembre 2015

7 Robust Detection Problem Robust Detection Non-Bayesian framework, mainly motivated by the lack of any prior on the statistical behavior of the change-point. Concerned with general counting process, in particular inhomogeneous Poisson process Robust Criterium, Lorden (1971) Let T be a stopping time, candidate for the estimation of θ The robust Lorden criterium with worst case detection delay C Lorden (T ) = sup θ [0, ] [ ess sup ω E ] θ (T θ) + F θ Min-Max Robust Optimisation Problem Find T such that C Lorden (T ) = min T C Lorden (T ), s.t. the false alarm constraint E[T ] π /36 NEK, Angers Septembre 2015

8 A min-max problem under constraint Our Lorden-modified criterium Well-adapted to our general framework C(T ) = [ sup ess sup ω E θ (NT N θ ) + ] F θ θ [0, ] with the false alarm constraint E[N T ] π. Stable by time rescaling: Let τ(t) be a time rescaling process, ˆN t = N τ(t) the rescaled counting process, with intensity ˆΛ t = Λ τ(t) = τ(t) 0 λ s ds, If T is an optimal stopping rule for the min-max problem, then τ 1 (T ) is optimal for the modified criterium associated with ˆN It follows that we can only consider the case of constant intensity (Poisson case, i.e. τ(t) = Λ 1 t ) 8/36 NEK, Angers Septembre 2015

9 Likelihood formulation Conditional probability ratio between P =no change, P= immediate change. The conditional probability ratio process of P w.r to P is d P/dP = E t = exp ( log(ρ)(n t (ρ 1)Λ t ) Put U ρ t = N t β(ρ)λ t, so ρ Uρ t is a P-martingale. where β(ρ) = ρ 1 log(ρ), with β(ρ) = 1 0 ρu du.= Laplace transform of U[0, 1]. Put β(ρ) = β(1/ρ) = β(ρ)/ρ, and ρ = 1/ρ. Then the CPR of P w.r to P is : ρ Uρ t = (1/ρ) Uρ t /36 NEK, Angers Septembre 2015

10 Sequential CPR, and CUSUM Sequential conditional probability ratio between P θ w.r to P is dp θ /dp = Et θ with Et θ = exp ( log(ρ)(n t θ N θ ) (ρ 1) t θ θ λ s ds ) = ρ Uρ t θ Uρ θ Useful in test hypothesis on intensity λ The Cumulative Sum rule (cusum)= Max in time of Likelihood Based on max s t (ρ Ut Us ), ( sign of ln(ρ)) Cusum stopping time, τ m = inf{t ; cusum t m} (cusum t defined later) Main question: The optimality of τ m, if the false alarm constraint is achieved, E(N τm ) = E(N T ) = π. 10/36 NEK, Angers Septembre 2015

11 Connection to insurance theory Surplus process (Abundant literature) premium rate (ρ 1)λ t and constant size of claims log ρ Surplus by claim is X t (z) = z N t + β(ρ)λ t = z U t Ruin problem and Viability condition β > 1 (ρ > 1) is called the security loading condition, and E(X t ) = z + (β(ρ) 1)E(Λ t ) drifts to P(sup t U t m) = ū(m) is finite and of main interest. Well known for a long time (Feller 1971) as scale function Equity process, U t (z) or dual risk model premia viewed as costs, and claims as profits coming suddently as in RD Viability condition E(U 1 ) > 0 or β < 1 11/36 NEK, Angers Septembre 2015

12 Plan 1 What is the disorder problem 2 Cusum processes 3 Differential finite variation calculus 4 Optimality result 12/36 NEK, Angers Septembre 2015

13 Reflected Counting process with drift Running supremum Running supremum Z t = sup s t Z s Put X t = U t = N t βλ t. So, X t is continuous. Ū t is not continuous and increases only at jumps of N, such that Ū t = U t Reflected processes and Cusum processes Cusum rule based on max s t ρ Ut Us ρ > 1: Reflected process X at its maximum, V t = sup θ t (U t U θ ) = X t X t ρ < 1: Reflected process U at its maximum: Y t = sup θ t (X t X θ ) = Ūt U t, 3/36 NEK, Angers Septembre 2015

14 Typical paths for ρ > 1 and ρ < 1 V t Y t Y 0 V X t U t (a) ρ > 1 (b) ρ < 1 Figure: Sample paths of the processes V, U, X and Y when λ is time-homogeneous. 14/36 NEK, Angers Septembre 2015

15 Markovian definition of reflected processes Definitions with initial conditions V t (Z 0 ) = U t + sup{z 0, X t } = U t (Z 0 ) + ( X t Z 0 ) +, Y t (Z 0 ) = X t + sup{z 0, Ūt} = X t (Z 0 ) + (Ūt Z 0 ) +, X t ad = ( X t Z 0 ) +, Ūt ad = (Ū t Z 0 ) +. Differential point of view of reflected processes (j(y) = y 1) V and Y are solutions of the ODE s driven by N, dv t = dn t β 1 (0, ) (V t )dλ t, d dy t = j(y t )dn t + βdλ t, Exit times: For any cadlag process Z ad X t dūt ad = β1 {Vt=0}dΛ t τ Z m = inf{t : Z t m} and σ Z b = inf{t : Z t b} = 1 {Yt=0}(1 Y t )dn t ρ > 1: ū(m) = P(τ U m = + ) and ū(m U t ) is a martingale. 15/36 NEK, Angers Septembre 2015

16 Typical paths with change of regime at date 3 t t Vt N t V t Nt Yt N t Y t Nt t t (a) Processes N and V t (b) Processes N and Y t Figure: Sample paths, for ρ = 1.5, of the cusum processes N, V ρ (left) and N, Y ρ t for ρ = 0.5 (right ). 16/36 NEK, Angers Septembre 2015

17 Performance analysis for ρ > 1, and τ V m Performance functions of the V -cusum rule The performance of the cusum stopping is based on ] Γ m t (x) = [1 Ẽx τ Vm t(n τ N Vm t ) F θ = h m (V t (x)) P a.s. Ht (x, m) = h m (V t ) + N t is a P x -local martingale on [0, τ V m ), Similar definition under P x, with h m (x) = E x (N τ V m ) and H t (x, m) = h m (V t ) + N t is a P x -local martingale on [0, τ m ) Performance functions of the Y -cusum rule The performance of the cusum stopping is based on ] Γ m t (x) = [1 Ẽx τ Ym t(n τ N Ym t ) F θ = g m (Y t (x)) P a.s. Gt (x, m) = g m (Y t ) + N t is a P x -local martingale on [0, τ m ) Similar definition under P x, with g m (x) = E x (N τ Y m ) and G t (x, m) = g m (Y t ) + N t is a P x -local martingale on [0, τ m ) 7/36 NEK, Angers Septembre 2015

18 Toward optimality Two problems: Extension of the martingale property for any T Computation of the functions h n, h m, g m, g m 18/36 NEK, Angers Septembre 2015

19 Plan 1 What is the disorder problem 2 Cusum processes 3 Differential finite variation calculus 4 Optimality result 19/36 NEK, Angers Septembre 2015

20 Stochastic differential calculus Typical stochastic processes and Itô s formula for continuous functions dz t = σ(z t )dn t + b(z t )dλ t dφ(z t ) = (φ(z t + σ(z t )) φ(z t )) dn t + φ (Z t ) b(z t ) dλ t. Example with the ū function, ρ > 1 ū(x) = P(Ū x) and ū(m U t ) is the martingale on [0, τm U ] dū(m U t ) = (ū(m U t 1) ū(m U t )dn s + βū (m U t )dλ t if βū (x) = ū(x) ū(x 1)= delayed equation Extension to monotonic functions with one jumps, δφ(m) Jt d,z = number of down-crossings of m continuously dφ(z t ) = standartpart + (φ(m) φ(m ))djt d,z 20/36 NEK, Angers Septembre 2015

21 Other martingales Example with V -performance functions dht m = dh m (V t ) + dnt m h m (m )djt d,v is the martingale = (h m (V t + 1) h m (V t ))dn t h m(v t )β1 (0, ) (V t )dλ t if βh m(x) = h m (x + 1) h m (x) + 1, x (0, m), h (0) = 0, Example with Y -performance functions dgt m = dg m (Y t ) + dnt Y,m is the martingale = (g m ((Y t 1) + ) g m (Y t ))dn t g m(y t )βdλ t if βg m(x) = g m (x) g m ((x 1) + ) 1, x (0, m) Same result for the tilded functions with β 1/36 NEK, Angers Septembre 2015

22 A system of ODE s with delay Delayed equation Delayed equation with continuous solution on (0, ) (0 for x < 0) with only one jump at 0. Delayed equation properties βu (x) = u(x) u(x 1), β > 0. If β = β(ρ), then ρ x u(x) is solution of DDE with β(ρ) = β(1/ρ) û(x) = x 0 u(z)dz is solution of DDE βû (x) = û(x) û(x 1) + βu(0) β > 0, u(0) = û (0). ū(x) = P(Ū x) is solution of the DDE by martingale property. 22/36 NEK, Angers Septembre 2015

23 Pollaczek-Khintchine formula Old result, Feller (1971), β > 1 The derivative is a solution of the convolution equation, u (x) = (1/β) 1 [0,1) (x)u(0) + (1/β) 1 0 u (x z)dz When β > 1, let S n be sum of i.i.d. unif on [0, 1], S n, and ν an indep. geometric r.v. with, P(ν = j) = (1 1/β)β j. u (x) = u(0)/(β 1)P(S ν [x 1, x)), ū(x) = P(Ū x) is equal to P(S ν x) 1 (β 1)ū(x) is a scale function W (x) (Bertoin (1996)). 23/36 NEK, Angers Septembre 2015

24 Scale functions Scale functions For ρ > 1, W (x) = 1 (β 1) P(S ν x) and W (x) = ρ x W (x) For ρ < 1, W (x) = ρ x W (x) and W (x) = 1 P(Ū ρ( β 1) ρ x). Y -and V performance functions Y -performance, ρ < 1 g m (y) = m y W (z)dz, g m(y) = m y ρ W (z)dz, y [0, m]. V -performance, ρ > 1 h m (m ) = W (0) W (m) W (m), h m (m ) = ρw (0) W (m) W (m). h m (x) = W (m x) W (m) W (m) m x 0 W (y)dy h m (x) = ρ( W (m x) W (m) W m x W (y)dy (m) 0 ) 24/36 NEK, Angers Septembre 2015

25 Plan 1 What is the disorder problem 2 Cusum processes 3 Differential finite variation calculus 4 Optimality result 25/36 NEK, Angers Septembre 2015

26 Modified Lorden Criterium Using similar arguments of Shiryaev (1996) and Moustakides (2004). Integration by parts Γ T t := Ẽx( T t dn s F t ), and ( Z t ) an increasing process By integration of Γ T t with respect to ρ Z t [ T ] [ Ẽ x Γ T t α dρ Z α Ft T ] = Ex (ρ Z s ρ Z t )dn Ft t s. Applications to lower bounds for ρ > 1, ρ E [ T t ρ V ] ( ) s dn s F t C(T ) E ρ V T F t for ρ < 1, ρ E [ T t ρ Y s dn s Ft ] C(T ) E ( ρ Y T Ft ) Applications to performance functions ρ x ( h m (x) h m (0)) = ρ E x ( τm 0 ρ V s dn s ) h m (0) E x (ρ Vτm ), ρ y ( g m (y) g m (0)) = ρ E y ( τm 0 ρ Ys dn s ) gm (0) E y (ρ Yτm ). 26/36 NEK, Angers Septembre 2015

27 On the use of false alarm constraint Example based on the process V False alarm (with the notation Nt m,v = t 0 1 [0,m)(V s )dn s ): E(N T ) = E(N τ V m ) = h m (0) = E(N m,v T h m (m )J d,v T + h m (V T )), so that E(h m (V T )) = E( T 0 1 [m, )(V s )dn s + h m (m )J d,v T )( ), By the martingale property E( T 0 ρvs dns m,v h m (0)ρ V T ) = E(ρ m h m (m )J d,v T h m (V T ))ρ V T ) So the problem is reduced to show that, given ( ) E( T 0 ρvs 1 [m, ) (V s )dn s + ρ m hm (m )J d,v T h m (V T ))ρ V T ] 0 We eliminate J d,v T in this inequality by multiplying the false alarm ( ) by ρ m hm (m )/h m (m ) 27/36 NEK, Angers Septembre 2015

28 Argument for the proofs Reduction to functions comparison E( T 0 ρvs 1 [m, ) (V s )dn s ) ρ m h m(m ) h m(m ) E( T 0 1 [m, )(V s )dn s ). E ( ρ m h m(m ) h m(m ) h m(v T ) ρ V T h(vt ) ) 0 Example based on the process Y Similar arguments apply but without the discontinuities, and the comparison reduces to E( T 0 ρ Y s 1 [m, ) (Y s )dn s ) ρ m E( T 0 1 [m, )(Y s )dn s ). True if E(ρρ m g m (Y T ) ρ Y T g m (Y T )) 0 Comparison based on the scale function representation ψ(y) = ρ (m y) g m (y) g m (y)/ρ is positive if ρ < 1. φ m (m z) = h m(m ) h m(m ) ρm z h m (z) h m (z) is positive for ρ > 1 and h m(x) = ρ m x h m(m ) h m(m ) h m(x) 28/36 NEK, Angers Septembre 2015

29 Optimality results Optimality of the cusum rule Bounds of the cusum rule hm (0) and g m (0) are respectively the cusum bounds of the stopping times τ V m (ρ > 1) and τ Y m (ρ < 1). Optimality for a decrease in intensity Let T be a stopping times with finite cusum bound, such that E(N T ) = E(N τ Y m ) = g m (0)., then E ( T 0 ρ1 Y s dn s ) h m (0) E ( ρ Y T ), Optimality for an increase in intensity Let T be a stopping times with finite cusum bound,such that E(N T ) = E(N τ V m ) = h m (0)., then E ( T 0 ρvs dn s ) hm (0) E ( ρ V T ), 29/36 NEK, Angers Septembre 2015

30 Numerical instability of h m hm(0) ρ Figure: Function h m (0) for different values of ρ. 30/36 NEK, Angers Septembre 2015

31 Conclusion The cusum detection rule is optimal in the case of non-homogeneous Poisson process with a modified Lorden criterion Very easy to implement Non asymptotic criterium Based on fine properties of scale functions, easy to extend to Lévy processes The proof provide lower bound for some conditional ratio (Basseville) Applications in non-life insurance and life insurance Further research on possible extensions to Lévy process 31/36 NEK, Angers Septembre 2015

32 Detection Procedure Algorithm tep 1: Fix the input parameters: The post-change intensity through the specification of ρ and the false alarm constraint π. tep 2: Determine the threshold m as the solution of the equation E [N τm ] = π. tep 3: For each new observation at time t compute the value of the CUSUM process V given by the iterative relation V t+1 = (V t 1 + U t ) +. tep 4: Compare the current value of V to the threshold m and stop the procedure once V t m and sound an alarm. Hence τ m (0) = t. 32/36 NEK, Angers Septembre 2015

33 Applications to simulated population Evolution of the global population λ t = a ( 1 + exp( (t b)/c) ) 1, t 0, where a, b and c are some constant parameters given in a = 13, 80 b = 11, 65 c = 26, 40, d = simulations for ρ = 1.1, 1.5, 2 ρ m = 5 h m (0) () h m (0) () () m = 10 h m (0) () () h m (0) () () /36 NEK, Angers Septembre 2015

34 Thank You! 34/36 NEK, Angers Septembre 2015

35 References I 1 M. Basseville and I.V. Nikiforov. Detection of Abrupt Changes in Signals and Dynamics Systems. Springer- Verlag, E.Bayraktar,S.Dayanik,andI.Karatzas.The standard Poisson disorder problem revisited. Stochastic processes and their applications, 115(9): , J. DeLucia and H.V. Poor. Performance analysis of sequential tests between Poisson processes. IEEE Transactions on Information Theory, 43(1): , Y. Mei, S.W. Han, and K.-L. Tsui. Early detection of a change in Poisson rate after accounting for population size effects. Statistica Sinica, 21(2):597, G.V. Moustakides. Optimal stopping times for detecting changes in distributions. The Annals of Statistics, 14(4): , G.V. Moustakides. Performance of CUSUM tests for detecting changes in continuous time processes. In Proceedings of the IEEE International Symposium on Information Theory, page 186. IEEE, G.V. Moustakides. Optimality of the CUSUM procedure in continuous time. The Annals of Statistics, 32(1): , E.S. Page. Continuous inspection schemes. Biometrika, 41(1/2): , /36 NEK, Angers Septembre 2015

36 References II 9 M.R. Pistorius. On exit and ergodicity of the spectrally one-sided L?evy process reflected at its infimum. Journal of Theoretical Probability, 17(1): , H.V. Poor and O. Hadjiliadis. Quickest detection. Cambridge Univ. Press, A.N. Shiryaev. Minimax optimality of the method of cumulative sums (CUSUM) in the case of continuous time. Russian Mathematical Surveys, 51(4): , /36 NEK, Angers Septembre 2015