Ernesto Mordecki. Talk presented at the. Finnish Mathematical Society

Transcription

1 EXACT RUIN PROBABILITIES FOR A CLASS Of LÉVY PROCESSES Ernesto Mordecki mordecki Montevideo, Uruguay visiting Åbo Akademi, Turku Talk presented at the Finnish Mathematical Society 11 April 2005, Helsinki, Finland

2 Contents: Introduction 1. The problem: Ruin and Maxima 2. Lévy processes (LP) 3. Examples Brownian motion with drift Compound Poisson Processes Jump-diffusion process (I) Processes with positive jumps with rational transform (II)

3 Exact Ruin Probabilities 1. Jump-diffusion (I) Proof: Wiener-Hopf factorization 2. Rational transform positive jumps (II) Heuristics: Wiener-Hopf factorization Proof: Baxter-Donsker (complex analysis) 3. Open questions 4. References

4 The problem: Ruin and Maxima Mathematical model: X t : Ω R (t 0), is a stochastic process defined (Ω, F, P x ). Notation: X 0 = x, P 0 = P. Problem: Compute the Ruin Probability R(x) = P x ( t 0: X t 0) = P x (inf X t 0)

5 Equivalent Problem: Define the maximum M := sup{y t : t 0} of the symmetric process Y, and find 1 F M (x) = P(M > x) = R(x). Ruin for X = maximum for Y Generalized Problem: Take τ(q) exp(q) indep. of X τ(0) = + and consider now M q := sup{y t : 0 t < τ(q)} We want to find F Mq (x) = P(M q x).

6 Lévy Processes: Definition It starts at X 0 = 0, Independent Increments: If 0 t 1 t n, then X t1, X t2 X t1,..., X tn X tn 1 are independent random variables Stationary Increments: X t+h X t same distribution as X h Also called PIIS: Processes with independent and stationary increments

7 Lévy Processes: Characterization Lévy-Kinchine Formula: E e zx t = e tψ(z), where ψ(z) = az+ 1 2 σ2 z 2 + R (ezy 1 zy1 { y <1} )Π(dy) a R is the drift σ 0 is the variance of the Gaussian part Π is a positive measure on R \ {0}, with (1 y 2 )Π(dy) < +, governing the jumps of X: is the jump measure

8 Example: Brownian motion with drift Let B = (B t ) be a Brownian motion. Define X t = at + σb t We have E e zx t = e tψ(z), where ψ(z) = az σ2 z 2 Conclusion: In Levy-Khinchine formula: Π = 0 indicates absence of jumps

9 Example: Compound Poisson Process Taking T = (T k ) i.i.d.r.v. with distribution exp(λ), define N t = inf{k : T 1 + T T k t}. N = (N t ) is a Poisson process. Consider X t = N t Y k k=1 where Y = (Y k ) are i.i.d.r.v. with distribution F (dy) (independent of T ). Simple computations give: In this case ψ(z) = λ R (ezy 1)F (dy), a = σ = 0 Π(dy) = λf (dy).

10 Example: Jump-diffusion processes X t = at + σb t + N t Y k k=1 with B, N, Y independent, gives ψ(z) = az σ2 z 2 + λ If Y 1 has density f(x) = R (ezy 1)F (dy), pαe αy when y > 0 (1 p)βe βy when y < 0 for some positive α, β and p (0, 1), then ψ(z) =az σ2 z 2 + λp z α z z λ(1 p) β + z

11 Denote M q = sup{t 0} I q = inf{t 0} Theorem 1. If X is a jump-diffusion process, the densities of M q and I q are: f Mq (x) = A 1 exp( p 1 x) + A 2 exp( p 2 x), x > 0 f Iq (x) = B 1 exp( r 1 x) + B 2 exp( r 2 x), x < 0 where equation q ψ(z) = 0 has p 1 and p 2 positive roots r 1 and r 2 negative roots and coefficients are: A 1 = 1 p 1/α A 2 = 1 p 2/α 1 p 1 /p 2 1 p 2 /p 1 B 1 = 1 + r 1/β B 2 = 1 + r 2/β 1 r 1 /r 2 1 r 2 /r 1

12 Proof: Consider the characteristic function of M q and I q : φ + q (z) = E exp(zm q) φ q (z) = E exp(zi q ) and use Rogozin s (1966) WH factorization: q q ψ(z) = φ+ q (z)φ q (z) Uniqueness in WH and complex analysis arguments give the result (roots of q ψ(z) are poles, use residue s calculus)

13 LP with rational transform negative jumps Let X be a LP with jump measure Π(dx) = π + (dx) if x > 0, π (dx) = λp(x)dx if x < 0. (1) where π + arbitrary on (0, + ) (arbitrary positive jumps), and p(x) = n m k k=1 j=1 c kj ( αk ) j ( x) j 1 (j 1)! eα kx, x < 0, is the more general form of a r.v. with rational Laplace transform. P = m m n is the Pole count.

14 Characteristic exponent ψ ˆp(z) = So 0 ezy p(y)dy = n m k ( c kj k=1 j=1 αk α k z ) j. ψ(z) = az σ2 z (ezy 1 zh(y))π + (dy) + λ(ˆp(z) 1)

15 Theorem 2 (A. Lewis - EM). (a) On the half space Rz > 0 the equation q ψ(z) = 0 has roots with multiplicities β 1,..., β N n 1,..., n N and such that n n N = P + 1 (σ > 0) of the infi- (b) The characteristic function φ q mum I q is φ q (z) = n k=1 ( αk z ) mk N α k j=1 ( ) nj βj, β j z

16 Proof: (1) Study the roots of q ψ(z) = 0 on Rz > 0 with Rouche s Theorem (we know the poles). (2) Consider Baxter-Donsker (1957) Formula: φ q (iu) = E e ui q = exp { 1 2π ( u ξ(ξ iu) log q q ψ(ξ) ) dξ } (3) Compute the integral over a convenient contour in Rz > 0, computing the residues at the poles, and take limit.

17 Open Questions: Solve a two barrier problem for this class of Processes: Probability of hitting level a > 0 before than hitting level b < 0. (now π + should also be of rational type) Compute the Green Kernel for this class of Processes

18 References Baxter, G. and Donsker M.D. (1957), On the distribution of the supremum functional for processes with stationary independent increments, Transactions of the American Mathematical Society 85, Bertoin, J. (1996), Lévy Processes (Cambridge University Press, Cambridge). Lewis, A., Mordecki, E. Wiener-Hopf factorization for Lévy processes with negative jumps with rational transforms. Preprint (2005). Rogozin, B.A. (1966), On distributions of functionals related to boundary problems for processes with independent increments, Theory of Probability and its Applications XI, Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.