Rare event simulation for the ruin problem with investments via importance sampling and duality Jerey Collamore University of Copenhagen Joint work with Anand Vidyashankar (GMU) and Guoqing Diao (GMU).
Background: Classical risk theory Cramér-Lundberg process (Lundberg 1903, Cramér 1930): N t X t = u + ct i=1 X t =capital of insurance company at time t (t 0), X 0 = u; c =premiums income rate; {ξ i } claims losses. ξ i
Risk theory (cont.) Let ˆΨ(u) = P { X t < 0, for some t 0 X0 = u }. It is known that ˆΨ(u) Ce Ru as u.
Ruin with investments (JC '09, JC-Vidyashankar '10) Start with Cramér-Lundberg process, {X t }. Consider the discrete losses at time n: L n := (X n X n 1) (namely, claims losses premiums income). Investment process: Returns R n = (1 + r n )/$ Initial capital Y 0 = u. Capital at time n is (i.i.d.). Y n = R n Y n 1 L n.
Ruin with investments: analytical estimates Cumulative discounted loss: L n = L 1 + D 1 L 2 + + (D 1 D n 1)L n, where D n = 1/R n (discounted returns), and L n = L n /R n. Probability of ruin: Ψ(u) := P {Y n < 0, some n} = { P sup L n > } u. n
Rough comparison of the two ruin problems Figure: Classical ruin. Figure: Ruin with investments. L := sup n L n, Determine P {L > u}.
Random recurrence equations Need to determine tail of L := sup n L n as u. Can show L satises a random recurrence equation: L d = L + D max {0, L}, i.e., a special case of the equation Z d = Φ(Z).
Ruin with investments: analytical estimates (cont.) General recurrence equation: Z d = Φ(Z). (1) Theorem (Goldie '91; Collamore-Vidyashankar '10) Assume (1) with Φ(Z) AZ+error. Then where R = sup {α : Λ A (α) 0}. Here P {Z > u} Cu R as u, ] Λ A (α) = log E [e α log A. Note: In ruin setting, A = D (discounted returns). Decay at polynomial rate, determined by investments.
Importance sampling problem Again, consider risk process where Y n = R n Y n 1 L n, n = 1, 2,... (Y 0 = u), R n = 1/A n are nancial returns at time n. L n are insurance losses at time n. Goal: an exact estimate for Ψ(u) := P {Y n < 0, some n} Cu R. Note: Rare event probability (tends to 0 as u ). Suggests importance sampling. {Y n } is highly dependent. (Straightforward IS infeasible.)
Importance sampling problem (cont.) To simulate Ψ(u) = P {ruin}: Would like to write as large deviations process; roughly, ( ) log L n log L n 1 + log A n (i.e., random walk). This representation is not valid (with {log A n } or any other i.i.d. seq.). Similar to another process satifying ( )? Dual" (reversed) process.
Duality (Siegmund, Asmussen-Sigman) Main idea (duality): Equate ruin probability of risk process" to tail exceedance probability of content process." Start with stationary driving seq. {U n : n Z}. Content process: V n = f (V n 1, U n ), Risk process: Y n = g(y n 1, U n 1). (Roughly, g(, u) = f 1 (, u).)
Duality (cont.) Lemma (Siegmund, 1976; Asmussen and Sigman, 1996) { } { } P lim V n > u = P T (u) <, n where T (u) = inf { n : Y n 0 Y0 = u }.
Return to our problem Our risk process: Y n = R n (Y n 1 L n), Y 0 = u. Dual content process" is: where A d = R 1 and B d = L. Consequently, V n = (A n V n 1 + B n ) +, V 0 = 0, Ψ(u) := P {ruin} = P {V > u}. ruin probab. exceedance probab. of {V n } in steady state
Comments about the dual process In a nancial context, V n = (A n V n 1 + B n ) +, V 0 = 0. (2) V n = σ 2 n = squared volatility of GARCH(1,1) pr. (and then B n > 0). Other processes satisfying (2): Etc. AR(1) pr. with random coecients. Branching pr. with immigration.
Duality (cont.) Figure: Content process. Figure: Risk process. V n = (A n V n 1 + B n ) + Y n = R n (Y n 1 L n)
General approach Simulate the dual process {V n }, not {Y n }. {V n } is a Markov chain with atom" at 0, so the following representation formula" holds: P π {V > u} = E [N u] E [τ]. Regeneration cycle of {V n } (returns to 0). Estimate exceedances above level u:
Approach (cont.) To determine E [N u ], note that {V n > u} is a rare event. Suggests a large deviation change of measure. Following JC-Vidyashankar (2010), introduce a dual" change of measure. Set µ (dx, dy) = e Rx µ(dx, dy), where µ denotes probab. law of (log A, B), and Λ A (α) = 0. (Cramér transform.) Λ A (α) = log E ˆe α log A Now assume that (log A n, B n ) where T (u) := inf {n : V n > u}. { µ if n T (u), µ if n > T (u),
Approach (cont.) The process {V n } under the LD change of measure µ (followed by µ): Heuristics: Since V n = (A n V n 1 + B n ) +, for V n large" we have log V n log V n 1 + log A n. (Suggests ecient, asymptotically.)
Results Importance sampling estimator. Change of measure formula: where E [N u ] = E [N u e RS T u T u = inf {n : V n > u} τ, S n := log A 1 + + log A n. ], IS estimator is: E u = N u e RS T u. (Unbiased estimator for E [N u ].)
Results (cont.) Theorem Assume Λ A (2R) <, Λ B (R) <, etc. Then the above simulation regime has bounded relative error; namely, Relative Error = lim sup u Var(E u ) <. E [E u ] Theorem In this regime, the R-shifted change of measure is the only choice which is ecient, in the sense that no other choice can yield bounded relative error or logarithmic eciency.
Some numerical results RE(u) 0 5 10 15 C 0.0 0.5 1.0 1.5 0 10000 20000 30000 40000 50000 u 0 10000 20000 30000 40000 50000 u Figure: Relative error. Figure: True C(u) = Ψ(u)u R. Example: Normal log-returns, exponential claims.
Results (cont.) Remark: While above results are general, we may actually need to simulate with the k-shifted chain, i.e., {V kn } rather than {V n }, so that inf α E [(A 1 A k ) α ] 1 2. Otherwise, obtain a build-up" prior to rare large deviation" cycle: (Cf. Glasserman-Wang '97). Extension: While not strictly needed, consider state dependence to avoid long cycles" and minimize relative error. (after α shift)
Further extension In practical situations, true distribution unknown. Instead, use empirical distribution. Objective: show theory still holds when working with empirical rather than theoretical distributions. Proofs involve triangular arrays of Markov chains. Simulations show that the methodology works well; theory in progress.
References COLLAMORE, J.F. and VIDYASHANKAR, A.N. (2010). Nonlinear renewal theory for perpetuities and related actuarial applications. Preprint. COLLAMORE, J.F., VIDYASHANKAR, A.N. and DIAO, G. (2010). Rare event simulation for the ruin problem with investments via importance sampling and duality. In preparation. COLLAMORE, J.F., VIDYASHANKAR, A.N. and DIAO, G. (2010). Statistical aspects of importance sampling for the ruin problem with stochastic investments. In preparation. COLLAMORE, J.F. (2009). Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. Ann. Appl. Probab. 19 1404-1458. (Markovian setting.)
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