LARGE DEVIATION ESTIMATES FOR CERTAIN HEAVY-TAILED DEPENDENT - PDF document

LARGE DEVIATION ESTIMATES FOR CERTAIN HEAVY-TAILED DEPENDENT SEQUENCES ARISING IN RISK MANAGEMENT by JEFFREY F. COLLAMORE Laboratory of Actuarial Mathematics University of Copenhagen (Joint work with ANDREA H ¨ OING, Dept. Mathematics, ETH Z¨ urich) 1

The classical Ruin Problem Let S t = capital gain of an insurance co. by time t N ( t ) � capital outflow (claims) = Z i i =1 capital inflow (premiums) = ct N ( t ) � → S t = − Z i + ct − i =1 Find P { S t ever < − u } = P { ruin } (Lundberg, ’03). Theorem (Cram´ er, ’30). If { S t } has positive drift, “light–tailed” claim sizes, then ∼ Ce − Ru as u → ∞ . � � S t < − u, some t P Some extensions : (i) “Heavy–tailed” claims: � ∞ � � ∼ ˜ ¯ S t < − u, some t F Z ( s ) ds. P C u (ii) Finite-time estimates for light tails (Arfwedsen’55): P { ruin before time τu } ∼ D √ ue − uJ ( τ ) . 2

A modified ruin problem Now consider discrete-time process, S n = ξ 1 + · · · + ξ n , where E ξ i > 0, and assume: I. Subexponential claims (“heavy-tails”). � e ǫξ i � = = ∞ , all ǫ > 0 . ⇒ E II. Positive barrier for ruin. Ruin occurs if S n > u , some n ≤ δu . Thus, finite-time ruin est. (cf. Arfwedsen). III. Markov dependence in general state space: ξ i = f ( X i ) , where: • f ( · ) is a random function, • { X i } ⊂ S is a general (e.g. infinite) state M.C. 3

Motivating examples I. Operational risk losses. (E.g., back office errors at a bank.) • “Claims” arrive at a Poisson rate. • Claim sizes are heavy-tailed. • Frequency of claims depends on traded volume in the stock market. For example, if X i = traded volume at time i, then could model { X i } as pos.-drift AR(1) pr. (say). Losses at time i : N ( X i ) � ξ i = f ( X i ) = Z i,j , j =1 where, for each i , { Z i,j } j ≥ 1 is i.i.d., heavy-tailed. Study total loss by time n : S n = f ( X 1 ) + · · · + f ( X n ) . Related work (Rogers-Zane ’05): S n = price increase in high-freq. financial market; N ( X i ) = number of quotes (price changes) during interval i (where N ( · ) is Poisson, Markov-dep.). 4

II. Financial losses (GARCH(1,1) model). Log. returns on a stock: R i = σ i Z i , for Z i ∼ N (0 , 1) , where σ 2 i = a 0 + b 1 σ 2 i − 1 + a 1 R 2 i − 1 . Motivation. Volatility shows: Correlation with absolute log. returns (and previous volatility); Little correlation with actual log. returns ( R i − 1 ). Set: σ 2 A i = ( b 1 + a 1 Z 2 i = X i , i − 1 ) , B i = a 0 . Then above model becomes: X i = A i X i − 1 + B i , ( ∗ ) where { ( A i , B i ) } is i.i.d., E [log A i ] < 0. ( ∗ ) is called a “stochastic recurrence equation.” Note: { X i } is a Markov chain on R . Consider: S n = X 1 + · · · + X n . (cf. Mikosch–Konstantinides ’05). 5

General Problem Now suppose S n = f ( X 1 ) + · · · + f ( X n ) , where: • f ( · ) is a random function. • { X i } is an underlying Markov chain (on R , or S ). Nummelin-Athreya-Ney regeneration method: Assume { X i } satisfies: Minorization . � � (M) h ( x ) ν ( A ) ≤ P k ( x, A ) ≡ P X n + k ∈ A | X n = x . Then: • τ i ≡ T i − T i − 1 “inter-regen. times” exist, i.i.d. • U i ≡ S T i +1 − S T i i.i.d. • Probab. law of S T i is ν ( · ). 6

Results Objective : Determine P { S n > u, some n ≤ δu } , where S n = f ( X 1 ) + · · · + f ( X n ) , and µ ≡ E π [ f ( X )] > 0 (positive drift). Let U d = S T i +1 − S T i . Assumptions: (A1) U is subexponential . U − < − u � � (A2) P = o ( P { U > u } ), u → ∞ . (A3) Markov chain is geometrically recurrent , i.e., e ǫ ( T i +1 − T i ) � � < ∞ , some ǫ > 0 . E Thm. (C-H., ’05). Assume M.C. satisfies (M), and (A1)-(A3) hold. Then P { S n > u, some n ≤ δu } ∼ δu E τ · P { U > (1 − δµ ) u } . 7

Characterizing exceedence over regeneration cycle Case 1 : Operational risk losses . For this case, S n = f ( X 1 ) + · · · + f ( X n ) , where f ( X i ) = � N ( X i ) Z i,j . i =1 Here, N ( x ) ∼ Poisson( λ ( x )). Assumption: (A4) Λ( α ) < ∞ , some α > 0, where 1 � e α ( λ ( X 1 )+ ··· + λ ( X n )) � Λ( α ) = lim n log E . n →∞ (Spectral radius, “G¨ artner-Ellis limit.”) Means: the intensity process { λ ( X i ) } has light tails. Proposition 1 (C-H.,’05). Assume cond’s. of prev. thm., and that (A4) holds. Then P { U > u } ∼ E τ E π [ λ ( X )] ¯ F Z ( u ) as u → ∞ . Equiv.: P { U > u } ∼ E τ P π { f ( X ) > u } . 8

Case 2 : Stochastic recurrence eqn’s . Here, = A i X i − 1 + B i , X i and S n = X 1 + · · · + X n . Suppose: E [log A i ] < 1 ( A i < 1 “on average”). Define: e α log A i � � Λ A ( α ) = log E , (c.g.f. of log A ); and let Λ B ( · ) = c.g.f. of log B . Assumptions: (A5) Λ A ( κ ) = 0 some κ > 0. (A6) Λ A ( α ) , Λ B ( α ) finite for α ∈ N ( κ ). Proposition 2 (C-H.,’05). Assume (A5) and (A6). Then P { U > u } ∼ Cu − κ as u → ∞ . (Build-up of log A i ’s over long interval of length = ρ · log u .) 9

Summarizing: � � � � S n > u, some n ≤ δu f ( X ) > (1 − δµ ) u ; P ∼ Cu P π but C (and its derivation) is different in the two sep- arate cases. Related extension (cf. Mikosch-Konstantinides ’05): In GARCH(1,1) case, but with neg. drift, consider P { S n > u, some n } = P { ruin } . Then a simple application of Prop. 2 yields P { ruin } ∼ Du − ( κ − 1) , u → ∞ . Reference: C OLLAMORE , J. F. and H ¨ OING , A. (2005). Small- time ruin for a financial process modulated by a Harris recurrent Markov chain. Submitted. ( Available from http://www.math.ku.dk/ ∼ collamore/) 10