Some multivariate risk indicators; minimization by using stochastic algorithms

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1 Some multivariate risk indicators; minimization by using stochastic algorithms Véronique Maume-Deschamps, université Lyon 1 - ISFA, Joint work with P. Cénac and C. Prieur. AST&Risk (ANR Project) 1 / 51

2 Multivariate risk process Consider a vectorial risk process X n = X 1 n., X d n Xn k = gains of the kth branch of a compagny, during the nth period, i.e. Xn k = Gn k L k n where G n k is the gain and Lk n the loss. There may be dependence: vectorial (with respect to k) and/or temporal (with respect to n). 2 / 51

3 Multivariate risk process Consider a vectorial risk process X n = X 1 n., X d n Xn k = gains of the kth branch of a compagny, during the nth period, i.e. Xn k = Gn k L k n where G n k is the gain and Lk n the loss. There may be dependence: vectorial (with respect to k) and/or temporal (with respect to n). u > 0: total capital of the compagny. How to allocate u = u u d between the d branches in an optimal way? (u k is allocated to the kth branch). 3 / 51

4 Solvency 2 rules Introduction New european rules Solvency 2: insurance companies have to better take into account dependencies in order to compute their solvency margin. Standard formula vs Internal models. Once the main risk drivers for the overall company have been identied and the global solvency capital requierement has been computed, it is necessary to split it into marginal capitals for each line of business, avoid as far as possible that some lines of business become insolvent too often. Minimize a risk indicator. 4 / 51

5 Ruin probability Introduction Consider Y k j periods and R k j = j p=1 X k p the total gain of the kth branch on j = u k + Y k j. The kth branch is insolvent at time j if R k j < 0. Ruin probability widely studied in dimension 1 both for continuous and discrete models (Cramer, Lundberg, Gerber, De Finetti,...): j Ψ(u) = P( j = 1,..., n X p + u < 0). p=1 5 / 51

6 Ruin probability Introduction Consider Y k j periods and R k j = j p=1 X k p the total gain of the kth branch on j = u k + Y k j. The kth branch is insolvent at time j if R k j < 0. Ruin probability widely studied in dimension 1 both for continuous and discrete models (Cramer, Lundberg, Gerber, De Finetti,...): j Ψ(u) = P( j = 1,..., n X p + u < 0). p=1 In higher dimension (d), we may consider the probability that one branch fails: Ψ(u 1,..., u d ) = P( k = 1,..., d; j = 1,..., n R k j < 0), not easy to manipulate (not convex). 6 / 51

7 Multivariate risk indicators (1) S. Loisel (2004) introduced the two following risk indicators (continuous time: Ruin cost: A(u 1,..., u d ) = d n E R k p 11 {R k <0}, p k=1 p=1 Does not take into account the dependence structure. 7 / 51

8 Multivariate risk indicators (1) S. Loisel (2004) introduced the two following risk indicators (continuous time: Ruin cost: A(u 1,..., u d ) = d n E R k p 11 {R k <0}, p k=1 p=1 Does not take into account the dependence structure. The following indicator takes into account the dependence structure: d n B(u 1,..., u d ) = E 11 {R k <0}11 p { d. k=1 p=1 l=1 Rl p >0} not convex and does not take into account the ruin cost. 8 / 51

9 Multivariate risk indicators (2) We consider: I (u 1,..., u d ) = d n E g k (R k p )11 {R k <0}11 p { d k=1 p=1 l=1 Rl p>0} g k : R R is a C 1 convex function, with g k (0) = 0, g k (x) 0 for x 0 and g k (x) 0 for x 0, k = 1,..., d it is a penalty function to the kth branch when it becomes insolvable.. 9 / 51

10 Multivariate risk indicators (2) We consider: I (u 1,..., u d ) = d n E g k (R k p )11 {R k <0}11 p { d k=1 p=1 l=1 Rl p>0} g k : R R is a C 1 convex function, with g k (0) = 0, g k (x) 0 for x 0 and g k (x) 0 for x 0, k = 1,..., d it is a penalty function to the kth branch when it becomes insolvable. Remark: g k (x) = x is a possible choice (then we consider the ruin amount). I is the expected sum of penalties that each line of business would have to pay due to its temporary potential insolvency (orange area).. 10 / 51

11 R R2 1 u = u1 + u2 R 2 1 u2 R 1 1 u time d k=1 R k p R 2 p Local insolvency Global insolvency R 1 p The red area has been studied by R. Biard, S. Loisel, C. Macci and N. Veraverbeke (asymptotic properties as u ) for a continuous model. 11 / 51

12 Minimization of I Problem: nd the minimum u R d + v v d = u: with constraint I (u ) = inf I (v), v v v d =u Rd +. Tool: a Kiefer-Wolfowitz version of the stochastic mirror algorithm. Avantages of the stochastic algorithms approach: no parametric hypothesis on the law of the X i 's, dependence allowed over one period, high dimension (d ) allowed. 12 / 51

13 Convexity of I In what follows, it is assumed that for all i, k, (Y k i, S i ) admits a d density, S i = : cumulated gain for the ith period. Property l=1 Y l i g k are dierentiable and convex, g k : R R with g k (0) = 0, g k (x) 0 for x < 0, g k (x) 0 for x > 0. Then the risk indicator I is convex on the convex set U u = {(v 1,..., v d ) (R + ) d / v v d = u}. 13 / 51

14 Classical algorithms Mirror descent algorithm A deterministic version of the Robbins-Monro algorithm x n+1 = x n γ n f (x n ). slope 1 γ 0 slope 1 γ 1 x x2 x1 x0 14 / 51

15 Robbins-Monro algorithm Classical algorithms Mirror descent algorithm Often, we have access only to a perturbation of f. Robbins-Monro type algorithms: χ n+1 = χ n γ n+1 ξ n+1, where ξ n+1 = f (χ n ) + ε n+1, (ε n ) n is an centred i.i.d. sequence, with ε n+1 independent of σ(χ 0,..., χ n ) = F n. Wide literature on this algorithm and its variants: from Robbins-Monro (1951), Duo, Küchner and Yin,... convergence of the algorithm a.e. with TCL... under various hypothesis. 15 / 51

16 Kiefer-Wolfowitz algorithm Classical algorithms Mirror descent algorithm Aim: Minimizing a (strictly) convex C 1 function f = zero of f. f is usually unknown Example f (x) = E(F (x, ξ)) F is approximated by D cn = F (χ n + c n, ξ 1 n) F (χ n c n, ξ 2 n) 2c n, with (ξ 1 n) and (ξ 2 n) two independent i.i.d. sequences of random variables of law ξ. D cn is seen as a perturbation of f. Kifer Wolfowitz algorithm: consider χ n+1 = χ n γ n F (χ n + c n, ξ 1 n) F (X n c n, ξ 2 n) 2c n. Under standard conditions, the algorithm converges a.e. + TCL / 51

17 Why can't we use K-W algorithm? Classical algorithms Mirror descent algorithm Linear constraint : u u d = u Lagrange multipliers (= ane part) = bad convergence properties. 17 / 51

18 Mirror algorithm Introduction Classical algorithms Mirror descent algorithm Deterministic mirror descent algorithm introduced by Nemirovski and Yudin (1983) Stochastic version of this algorithm proposed by C. Tauvel (2008) in her thesis. Optimization problem: f a C 1 convex function, min f (x), C is a compact and convex set of E. x C Observations: we have a noisy observation of f ψ(x) = f (x) + η(x), with a martingale dierence hypothesis for η. 18 / 51

19 Classical algorithms Mirror descent algorithm Auxiliary functions for the mirror descent algorithm Let us x x 0 an initial point, δ a strongly convex function on C, which is dierentiable on x 0 Construct an auxiliary function V : V (x) = δ(x) δ(x 0 ) x 0 x, δ (x 0 ), which will be used to push the trajectory into C. W β denotes the Fenchel-Legendre transform of βv : for z E W β (z) = sup { z, x βv (x)}. x C (γ n ) n and (β n ) n are two positive sequences, (β n ) n is non decreasing. Important property of W β : W β (ξ) C. 19 / 51

20 Classical algorithms Mirror descent algorithm Description of the stochastic mirror algorithm The algorithm constructs two random sequences: (χ n ) in C (ξ n ) in the dual space E Algorithm Initialisation: ξ 0 = 0 E, χ 0 C Update: for n = 1 to N do ξ n = ξ n 1 γ n ψ(χ n 1 ) χ n = W βn (ξ n ) Ouput: S N = N n=1 γ nχ n 1 N n=1 γ n 20 / 51

21 Mirror descent algorithm Classical algorithms Mirror descent algorithm C compact and convex Wβ mirror descent χ n χ n 1 ξ ξ n ξ n 1 21 / 51

22 Classical algorithms Mirror descent algorithm Convergence of the stochastic miror algorithm Under the hypothesis 1 martingale dierence hypothesis: E[η(χ n+1 ) F n ] = 0, with F n = σ(χ 0,..., χ n ). where ψ(x) = f (x) + η(x) is observable, 2 moment of order 2: σ > 0 such that for all n, E( η(χ n+1 ) 2 F n ) < σ 2 3 f admits a unique minimum x. C. Tauvel proved the convergence to 0 of E(f (S N )) E(f (x )). With an hypothesis of exponential moment on η, see gets the a.e. convergence of f (S N ) f (x ) to / 51

23 Another stochastic algorithm The risk indicator I Recall that we consider d n I (u 1,..., u d ) = E g k (R k p )11 {R k <0}11 p { d k=1 p=1 l=1 Rl p>0}. We are looking for the minimum u R d + v v d = u : under the constrainst I (u ) = inf I (v), v v v d =u Rd +. We shall use a Kiefer-Wolfowitz version of the mirror descent algorithm. 23 / 51

24 The mirror descent algorithm Another stochastic algorithm The set of constraint is U u = {v R d / v i 0, v v d = u}. A possible choice for V is the entropy function V (x) = ln d + d i=1 which is a strongly convex function. x ( i u ln xi ) = ln d + δ(x). u The Legendre-Fenchel transform may be computed easily: W β (ξ) = β ln ( 1 d d [ ] ) u exp ξ i. β i=1 24 / 51

25 Approximate gradient Another stochastic algorithm Y1 1 Y 1 n I (u 1,..., u d ) = E (I(u 1,..., u d, Y)) where Y =. Y1 d Yn d Denote I k ( c + i, Y ) = I(χ 1 i 1,..., χ k 1 i 1, χk i 1 + c i, χ k+1 i 1,..., χd i 1, Y), I k ( c i, Y ) = I(χ 1 i 1,..., χ k 1 i 1, χk i 1 c i, χ k+1 i 1,..., χd i 1, Y), Consider the random vector D ci I whose kth coordinate D k c i I(u 1,..., u d, Y) is I k ( c + i, Y ) I k ( c i, Y ) 2c i. 25 / 51

26 Our algorithm Introduction Another stochastic algorithm Algorithm Initialisation: Update: { ξ 0 = 0 (R m ) χ 0 C for i = 1,..., N do { ξi = ξ i 1 γ i Ψ ci (χ i 1, Y i ) χ i = W βi (ξ i ) Output: S N = N i=1 γ iχ i 1 N i=1 γ i with Ψ ci (χ i 1, Y i ) = D ci I(χ i 1, Y i ) and Y i independent copies of Y. 26 / 51

27 Conditions for the convergence (1) Another stochastic algorithm Condition (on I ) 1 I is a convex function from R d to R, 2 I is C 2 on U u, 3 I has a unique minimum u in U u, 4 σ > 0 such that for all v U u E ( ) I(v 1,..., v d, Y) 2 F i 1 σ 2. F i 1 is generated by (Y 0,..., Y i 1 ). 27 / 51

28 Conditions for the convergence (2) Another stochastic algorithm Condition (on the sequences) Let (β n ) n 0, (γ n ) n 0 and (c n ) n 0 be positive sequences, (β n ) n 0 is non decreasing and: 1 β N / N γ i=1 i 0, N + N 2 γ i=1 i c i / N γ i=1 i 3 4 N γ 2 i i=1 c 2β i i 1 ( ) 2 + γi i=1 c i <. N + 0, / N i=1 γ i N + 0, 28 / 51

29 Result Introduction Another stochastic algorithm Theorem With the above conditions, S N L 1 x. With an hypothesis of moment of order > 2 on I, S N a.s. x. 29 / 51

30 Idea of the proof (1) Another stochastic algorithm Based on the decomposition Ψ cn (χ n 1, Y n ) = I (χ n 1 ) + η cn (χ n 1, Y n ) + r cn (χ n 1 ). The condition of C. Tauvel fails because of the r cn (χ n 1 ) term. η cn (χ n 1, Y n ) is a martingale dierence. η cn (χ n 1, Y n ) = D cn I(χ n 1, Y n ) D cn I (χ n 1 ), r cn (χ n 1 ) = D cn I (χ n 1 ) I (χ n 1 ). and Law of large numbers for martingales Chow Lemma for martingales 30 / 51

31 Idea of the proof (2) Another stochastic algorithm Let ε N = I (S N ) I (x ) 0 Lemma ( N ) γ i ε N β N V (x ) i=1 + N γ i η c i (χ i 1, Y i ), χ i 1 x i=1 N γ i r c i (χ i 1 ), χ i 1 x i=1 N i=1 γ 2 i 2αβ i 1 Ψ c i (χ i 1, Y i ) 2 31 / 51

32 Idea of the proof (3) Another stochastic algorithm N γ i η c i (χ i 1, Y i ), χ i 1 x is a martingale i=1 N γ i r c i (χ i 1 ), χ i 1 x controlled by the fact that r c i (χ i 1 ) i=1 goes to 0 a.e. N i=1 γ 2 i Ψ c i (χ i 1, Y i ) 2 has bounded expectation and 2αβ i 1 controlled a.s. by using the Chow Lemma. 32 / 51

33 Another stochastic algorithm Hypothesis on I 1 Unicity of u : it is sucient that for some k and all u < w k < v k, [ ] E g k (Y k p + v k )1 { d k=1 Y k> u} 1 { v p k <Y k < w p k } < 0. This is satised if for some k, f Sp,Y > 0 on k p ] u, [ ], u[. 2 In the case g k (x) = x, the moment condition for I is equivalent to a moment condition (of the same order) on X n. 33 / 51

34 Introduction Some simulations for some simple models. We considered Normal laws, n = 1 observation of several periods of length 1, so that X k p = Y k p, X p R d are independent and identically distributed random vectors. Some dependencies on the coordinates of X p are allowed. 34 / 51

35 Parameters Introduction The algorithm has been performed with the following sequences (γ n ) n N, (c n ) n N and (β n ) n N : 1 γ n = (n + 1) α with α = , 1 c n = (n + 1) δ with δ = 1 4, β n = 1. Also, we have chosen u = 2 and the initialization of the algorithm (χ 0 ) is done at random uniformly in the simplex U u. 35 / 51

36 No dependence between the coordinates d = 2, n = 1, X 1 i and X 2 i are independent and the vectors X i are also independent. rst X 1 i and X 2 i have the same normal laws, then we consider dierent normal laws. 36 / 51

37 No dependence between the coordinates Same normal laws: X d i N (0.3, 1). For N = independent simulations we obtain : S N = (0.996; 1.004) Estimation of the minimum Zoom first coordinate first coordinate time time 37 / 51

38 No dependence between the coordinates Same normal laws: X d i N (0.3, 1). For N = independent simulations we obtain : S N = (0.996; 1.004) Estimation of the minimum Zoom second coordinate second coordinate time time 38 / 51

39 No dependence between the coordinates Same normal laws: For 50 simulations of length N = 1000, with the same parameters as above. The table below gives for each of the two coordinates, the mean of the estimation (û 1, û 2 ) of the minimum (u 1, u 2 ) and the standard error. rst coord. second coord. mean standard error / 51

40 No dependence between the coordinates Dierent normal laws: 50 simulations of length N = 1000, with two independent normal laws. X 1 p N (0.3, 1) and X 2 p N (0.8, 1). The table below gives for each of the two coordinates, the mean of the estimation (û 1, û 2 ) of the minimum (u 1, u 2 ) and the standard error. rst coord. second coord. mean standard error / 51

41 No dependence between the coordinates Dierent normal laws: 50 simulations of length N = 1000, with two independent normal laws. X 1 p N (0.3, 1) and X 2 p N (0.3, 4). The table below gives for each of the two coordinates, the mean of the estimation (û 1, û 2 ) of the minimum (u 1, u 2 ) and the standard error. rst coord. second coord. mean standard error / 51

42 Some dependence between the coordinates Two simple models of vectorial dependence: a comonotonic example: random vectors in R 3 with L = X 2 p N (0.3, 1) and X 3 p = 2X 2, X 1 p gaussian vectors. 42 / 51

43 Some dependence between the coordinates Comonotonic example. 50 simulations of length N = 1000 of this dimension 3 model. As before, we consider n = 1 (the periods are of length 1). The table below gives for each of the two coordinates, the mean of the estimation (û 1, û 2, û 3 ) of the minimum (u 1, u 2, u 3 ) and the standard error. rst coord. second coord. third coord. mean standard error In this model, X 2 and X 3 fail simultaneously and when they fail, X 3 is 2 two times worse than X / 51

44 Some dependence between the coordinates Gaussian vectors. We have also performed simulations for a Gaussian vector in R 3 with covariance matrix Σ = and expectation m = (0.3, 0.3, 0.3). As above, we have performed 50 simulations of length N = 1000, of the Gaussian vector X. 44 / 51

45 Some dependence between the coordinates The table below gives for each of the three coordinates, the mean of the estimation (û 1, û 2, û 3 ) of the minimum (u 1, u 2, u 3 ) and the standard error. rst coord. second coord. third coord. mean standard error / 51

46 A example with some temporal dependence The rst two coordinates are independant AR(1) with the same law: X i = 0.4X i 1 + ε i with (ε i ) i N gaussian white noise (N (0, 1)). The third coordinate is 2 times the second one. We have simulated 5 times 500 independent periodsof length n = 5. rst coord. second coord. third coord. mean standard error / 51

47 To do... Introduction A deeper study involving other laws (with heavy tail e.g.), more realistic models and temporal dependencies. Asymptotic theory for u. Some dependence on the Y i (using Benveniste, Métivier, Priouret results or methods for example). 47 / 51

48 Thanks for your attention 48 / 51

49 Recall the Chow Lemma Theorem Suppose (a N ) N N is a bounded sequence of positive numbers, N suppose that 1 < p 2. For N N, let A N = 1 + a k and k=0 A = lim N A N. Suppose that (Z N ) N N is a positive sequence of random variables adapted to F N and K is a constant such that E(Z N+1 F N ) K and sup E(Z p F N+1 N) < N then we have the following properties almost surely : on {A < } A k Z k+1 converges k=1 49 / 51

50 Recall the Chow Lemma Theorem Suppose (a N ) N N is a bounded sequence of positive numbers, N suppose that 1 < p 2. For N N, let A N = 1 + a k and k=0 A = lim N A N. Suppose that (Z N ) N N is a positive sequence of random variables adapted to F N and K is a constant such that E(Z N+1 F N ) K and sup E(Z p F N+1 N) < N then we have the following properties almost surely : on {A = } A 1 n 1 A k Z k+1 K. k=1 50 / 51

51 Recall the Chow Lemma Theorem Suppose (a N ) N N is a bounded sequence of positive numbers, N suppose that 1 < p 2. For N N, let A N = 1 + a k and k=0 A = lim N A N. Suppose that (Z N ) N N is a positive sequence of random variables adapted to F N and K is a constant such that E(Z N+1 F N ) K and sup E(Z p F N+1 N) < N then we have the following properties almost surely : Back to idea of proof. 51 / 51

Estimating Bivariate Tail: a copula based approach

Estimating Bivariate Tail: a copula based approach Elena Di Bernardino, Université Lyon 1 - ISFA, Institut de Science Financiere et d'assurances - AST&Risk (ANR Project) Joint work with Véronique Maume-Deschamps