Bernstein-gamma functions and exponential functionals of Lévy processes

Transcription

1 Bernstein-gamma functions and exponential functionals of Lévy processes M. Savov 1 joint work with P. Patie 2 FCPNLO 216, Bilbao November Marie Sklodowska Curie Individual Fellowship at IMI, BAS, project MOCT 2 Cornell University (spends a year in the nearby town of Pau, France)

2 Bernstein-gamma functions φ is a Bernstein function that is φ B iff φ(z) = m + δz + where m, δ ; (1 y) µ(dy) <. ( 1 e zy ) µ(dy), z {ζ C : Re(ζ) }, 3 The Mellin transform of a positive random variable Y is given by M Y (z) = E [ Y z 1].

3 Bernstein-gamma functions φ is a Bernstein function that is φ B iff φ(z) = m + δz + where m, δ ; (1 y) µ(dy) <. ( 1 e zy ) µ(dy), z {ζ C : Re(ζ) }, The unique solution in the space of Mellin transforms of positive random variables 3 to the recurrent equation f (z + 1) = φ(z)f (z) on z C (, ) = {ζ C : Re(ζ) > } (.1) we denote by W φ and call a Bernstein-gamma function. 3 The Mellin transform of a positive random variable Y is given by M Y (z) = E [ Y z 1].

4 Motivation - Bernstein-gamma functions 1 W φ appear crucially in the spectral studies of the generalized Laguerre semigroups and the positive self-similar Markov processes as instances of non-selfadjoint Markov semigroups. 2 W φ are related to the phenomenon of self-similarity the same way the Gamma function appears in the study of some diffusions 3 Amongst W φ are some well-known special functions, e.g. the Barnes-Gamma function, the q-gamma function 4 W φ participates in the computation of the Mellin transform of the exponential functionals of Lévy processes

5 Motivation - Bernstein-gamma functions 1 W φ appear crucially in the spectral studies of the generalized Laguerre semigroups and the positive self-similar Markov processes as instances of non-selfadjoint Markov semigroups. 2 W φ are related to the phenomenon of self-similarity the same way the Gamma function appears in the study of some diffusions 3 Amongst W φ are some well-known special functions, e.g. the Barnes-Gamma function, the q-gamma function 4 W φ participates in the computation of the Mellin transform of the exponential functionals of Lévy processes

6 Motivation - Bernstein-gamma functions 1 W φ appear crucially in the spectral studies of the generalized Laguerre semigroups and the positive self-similar Markov processes as instances of non-selfadjoint Markov semigroups. 2 W φ are related to the phenomenon of self-similarity the same way the Gamma function appears in the study of some diffusions 3 Amongst W φ are some well-known special functions, e.g. the Barnes-Gamma function, the q-gamma function 4 W φ participates in the computation of the Mellin transform of the exponential functionals of Lévy processes

7 Motivation - Bernstein-gamma functions 1 W φ appear crucially in the spectral studies of the generalized Laguerre semigroups and the positive self-similar Markov processes as instances of non-selfadjoint Markov semigroups. 2 W φ are related to the phenomenon of self-similarity the same way the Gamma function appears in the study of some diffusions 3 Amongst W φ are some well-known special functions, e.g. the Barnes-Gamma function, the q-gamma function 4 W φ participates in the computation of the Mellin transform of the exponential functionals of Lévy processes

8 Quantities of φ B that describe the analytic structure of W φ We use A (a,b) (resp. M (a,b) ) to denote the holomorphic (resp. meromorphic) functions on the complex strip C (a,b) = {z C : Re(z) (a, b)}.

9 Quantities of φ B that describe the analytic structure of W φ We use A (a,b) (resp. M (a,b) ) to denote the holomorphic (resp. meromorphic) functions on the complex strip C (a,b) = {z C : Re(z) (a, b)}. If φ B then φ A (, ) and set a φ = inf u< { φ A(u, ) } [, ], u φ = sup {φ(u) = } [, ], u d φ = sup {φ(u) = or φ(u) = } [a φ, ]. u iy iy ()> ()> d =u x d =a x -

10 Main representation of the solution to W φ (z + 1) = φ(z)w φ (z) Theorem For any φ B Morever W φ (z) = 1 φ(z) e γ φz φ(k) φ (k + z) e φ (k) φ(k) z A (dφ, ) M (aφ, ). k=1 W φ solves f(z + 1) = φ(z)f(z), f(1) = 1, on C (dφ, ), [ ] W φ (z + 1) = E Yφ z for some positive random variable Y φ.

11 Main representation of the solution to W φ (z + 1) = φ(z)w φ (z) Theorem For any φ B Morever W φ (z) = 1 φ(z) e γ φz φ(k) φ (k + z) e φ (k) φ(k) z A (dφ, ) M (aφ, ). k=1 W φ solves f(z + 1) = φ(z)f(z), f(1) = 1, on C (dφ, ), [ ] W φ (z + 1) = E Yφ z for some positive random variable Y φ. When φ(z) = z, d φ =, a φ =, W φ (z) = Γ(z).

12 Stirling asymptotic for W φ Theorem If a >, b R, z = a + ib. Then φ(1) W φ (z) = e G φ(a) A φ (z) e E φ(z) R φ (a) φ(a)φ(1 + a) φ(z) }{{}, error term

13 Stirling asymptotic for W φ Theorem If a >, b R, z = a + ib. Then φ(1) W φ (z) = e G φ(a) A φ (z) e E φ(z) R φ (a) φ(a)φ(1 + a) φ(z) }{{}, error term where A φ (z) = G φ (z) = and 1 b A φ (a + ib) [, π 2 ]. b 1+a 1 arg (φ (a + iu)) du, ln φ(u)du = G φ (a)

14 Lévy process A Lévy process ξ = (ξ s ) sgeq is a stochastic process with the properties: 1 for every t, (ξ t+s ξ t ) s w = (ξs ) s, that is ξ has stationary increments 2 for every t, (ξ t+s ξ t ) s is independent of (ξ s ) t s, that is ξ has independent increments 3 ξ is a.s. with right-continuous paths.

15 Lévy process A Lévy process ξ = (ξ s ) sgeq is a stochastic process with the properties: 1 for every t, (ξ t+s ξ t ) s w = (ξs ) s, that is ξ has stationary increments 2 for every t, (ξ t+s ξ t ) s is independent of (ξ s ) t s, that is ξ has independent increments 3 ξ is a.s. with right-continuous paths. In some loose sence the Lévy process is a Brownian motion with jumps.

16 Lévy process A Lévy process ξ = (ξ s ) sgeq is a stochastic process with the properties: 1 for every t, (ξ t+s ξ t ) s w = (ξs ) s, that is ξ has stationary increments 2 for every t, (ξ t+s ξ t ) s is independent of (ξ s ) t s, that is ξ has independent increments 3 ξ is a.s. with right-continuous paths. In some loose sence the Lévy process is a Brownian motion with jumps. We say that ξ is a killed Lévy process if there exists independent e q Exp(q) such that ξ s =, s e q.

17 Lévy-Khintchine exponent of Lévy processes A killed Lévy process is determined by its Lévy-Khintchine exponent via log E [ e zξ1] = Ψ(z) = σ2 ( 2 z2 + bz + e zr ) 1 zr1 r <1 Π(dr) q, where b R is linear drift, σ 2 is the variance of the Brownian component, Π is a sigma-finite measure describing the structure of the jumps satisfying min { x 2, 1 } Π(dx) <, q is the killing rate.

18 Lévy processes and exponential functionals of Lévy processes Denote by N = } {Ψ : Ψ(z) = σ2 ( 2 z2 + bz + e zr ) 1 zr1 r <1 Π(dr) q the set of all Lévy-Khintchine exponents of possibly killed Lévy processes.

19 Lévy processes and exponential functionals of Lévy processes Denote by N = } {Ψ : Ψ(z) = σ2 ( 2 z2 + bz + e zr ) 1 zr1 r <1 Π(dr) q the set of all Lévy-Khintchine exponents of possibly killed Lévy processes. The random variables I Ψ = eq e ξs ds, e q Exp(q); e = are called exponential functionals of Lévy processes and { } I Ψ < Ψ N = Ψ N : q > or lim ξ s = s N.

20 Some applications of I Ψ 1 Appear in financial and insurance mathematics; branching with immigration; fragmentation; self-similar growth fragmentation; etc. 2 We also use it in our work on the spectral theory of positive self-similar semigroups

21 Some applications of I Ψ 1 Appear in financial and insurance mathematics; branching with immigration; fragmentation; self-similar growth fragmentation; etc. 2 We also use it in our work on the spectral theory of positive self-similar semigroups

22 Background 1 I Ψ introduced and studied by Urbanik when ξ is a subordinator 2 Further important studies have been made by Carmona et al. 4 and Maulik et al. 5 3 Most recent contributions belong to Kuznetsov et al. 4 P. Carmona, F. Petit and M. Yor (1997) Exponential functionals and principal values related to Brownian motion. Bibl. Rev.Mat. Iber. 5 M.Maulik and B. Zwart (26) Tail Asymptotics for exponential functionals of Lévy processes, Stoch. Process. Appl, 11:

23 Background 1 I Ψ introduced and studied by Urbanik when ξ is a subordinator 2 Further important studies have been made by Carmona et al. 4 and Maulik et al. 5 3 Most recent contributions belong to Kuznetsov et al. 4 P. Carmona, F. Petit and M. Yor (1997) Exponential functionals and principal values related to Brownian motion. Bibl. Rev.Mat. Iber. 5 M.Maulik and B. Zwart (26) Tail Asymptotics for exponential functionals of Lévy processes, Stoch. Process. Appl, 11:

24 Background 1 I Ψ introduced and studied by Urbanik when ξ is a subordinator 2 Further important studies have been made by Carmona et al. 4 and Maulik et al. 5 3 Most recent contributions belong to Kuznetsov et al. 4 P. Carmona, F. Petit and M. Yor (1997) Exponential functionals and principal values related to Brownian motion. Bibl. Rev.Mat. Iber. 5 M.Maulik and B. Zwart (26) Tail Asymptotics for exponential functionals of Lévy processes, Stoch. Process. Appl, 11:

25 A method to study the exponential functionals 1 Prove that M IΨ (z + 1) = E [I z Ψ ] solves f (z + 1) = in some meaningful sense. z f(z) on {z ir \ {} : Ψ( z) } (.2) Ψ( z) 2 For any Ψ N to solve and characterize the solutions of (.2) in terms of the global quantities of Ψ

26 A method to study the exponential functionals 1 Prove that M IΨ (z + 1) = E [I z Ψ ] solves f (z + 1) = in some meaningful sense. z f(z) on {z ir \ {} : Ψ( z) } (.2) Ψ( z) 2 For any Ψ N to solve and characterize the solutions of (.2) in terms of the global quantities of Ψ

27 Strategy to solve f (z + 1) = z Ψ( z) f(z) The Wiener-Hopf factorization gives that where φ ± are Bernstein functions. Ψ( z) = φ + (z)φ ( z), at least for z ir

28 Strategy to solve f (z + 1) = z Ψ( z) f(z) The Wiener-Hopf factorization gives that where φ ± are Bernstein functions. Ψ( z) = φ + (z)φ ( z), at least for z ir Then f (z + 1) = z Ψ( z) f(z) = z 1 f(z), (.3) φ + (z) φ ( z) on {z ir \ {} : Ψ( z) }.

29 Strategy to solve f (z + 1) = z 1 f(z) φ +(z) φ ( z) If f 1 (z + 1) = z f 1 (z) φ + (z) 1 f 2 (z + 1) = φ ( z) f 2(z) is solved on a common complex domain then f 1 (z)f 2 (z) is a solution to f (z + 1) = f(z) on this domain. z Ψ( z)

30 Strategy to solve f (z + 1) = z 1 f(z) φ +(z) φ ( z) If f 1 (z + 1) = z f 1 (z) φ + (z) 1 f 2 (z + 1) = φ ( z) f 2(z) is solved on a common complex domain then f 1 (z)f 2 (z) is a solution to f (z + 1) = f(z) on this domain. z Ψ( z) f 1, f 2 are expressed in terms of the Bernstein-gamma functions W φ± that solve f ± (z + 1) = φ ± (z)f ± (z).

31 Solution to f (z + 1) = z f(z) and representation of Ψ( z) MI (z) = E [ ] Ψ I z 1 Ψ Theorem Let Ψ N. Then M Ψ (z) = Γ(z) W φ+ (z) W φ (1 z) A ( a φ+ 1 {d φ+ =}, 1 d φ ) M (aφ+, 1 a φ ) solves f (z + 1) = z Ψ( z) f(z).

32 Solution to f (z + 1) = z f(z) and representation of Ψ( z) MI (z) = E [ ] Ψ I z 1 Ψ Theorem Let Ψ N. Then M Ψ (z) = Γ(z) W φ+ (z) W φ (1 z) A ( a φ+ 1 {d φ+ =}, 1 d φ ) M (aφ+, 1 a φ ) solves f (z + 1) = z Ψ( z) f(z). Also, if Ψ N, that is I Ψ <, then E [ I z 1 ] Ψ = MIΨ (z) = φ ()M Ψ (z) = Γ(z) W φ+ (z) φ ()W φ (1 z).

33 Solution to f (z + 1) = z f(z) and representation of Ψ( z) MI (z) = E [ ] Ψ I z 1 Ψ Theorem Let Ψ N. Then M Ψ (z) = Γ(z) W φ+ (z) W φ (1 z) A ( a φ+ 1 {d φ+ =}, 1 d φ ) M (aφ+, 1 a φ ) solves f (z + 1) = z Ψ( z) f(z). Also, if Ψ N, that is I Ψ <, then E [ I z 1 ] Ψ = MIΨ (z) = φ ()M Ψ (z) = Γ(z) W φ+ (z) φ ()W φ (1 z). To obtain information for the law of I Ψ via Mellin inversion we need to understand the decay of M IΨ (z) = φ ()M Ψ (z) along lines a + ir.

34 Decay of M Ψ (z) = Γ(z) W φ+ (z) W φ (1 z) along a + ir Theorem Let Ψ N. Then exists N Ψ (, ] such that for any a (, 1 d φ ) lim b b η M Ψ (a + ib) = η (, N Ψ )

35 Decay of M Ψ (z) = Γ(z) W φ+ (z) W φ (1 z) along a + ir Theorem Let Ψ N. Then exists N Ψ (, ] such that for any a (, 1 d φ ) lim b b η M Ψ (a + ib) = η (, N Ψ ) and N t N Ψ < Ψ corresponds to ξ t = δ + t + X j, δ + >. j=1

36 Decay of M Ψ (z) = Γ(z) W φ+ (z) W φ (1 z) along a + ir Theorem Let Ψ N. Then exists N Ψ (, ] such that for any a (, 1 d φ ) lim b b η M Ψ (a + ib) = η (, N Ψ ) and N t N Ψ < Ψ corresponds to ξ t = δ + t + X j, δ + >. j=1 Therefore, by Mellin inversion p Ψ (x) = 1 2π where p Ψ is the density of I Ψ. x a ib M IΨ (a + ib)db C N ( Ψ 1 R + ),

37 Smoothness has been investigated by a number of authors in different contexts, including the works of Carmona et al. 6 and Bertoin et. al P. Carmona, F. Petit and M. Yor (1997) Exponential functionals and principal values related to Brownian motion. Bibl. Rev.Mat. Iber. 7 J. Bertoin, A. Lindner and R. Maller (28) On continuity properties of the law of integrals of Lévy processes. Séminaire de probabilitès,

38 The behaviour of p (n) Ψ at infinity Theorem If θ Ψ < : Ψ (θ Ψ ) = and ( Ψ θ Ψ) + < (Cramer s condition) then lim x x θ Ψ+n+1 p (n) Ψ (x) = C > (.4) provided N Ψ > n A. Kuznetsov (211) On extrema of stable processes. Ann. Probab. 39(3): R. Doney and M. Savov (21) The asymptotic behaviour of densities related to the supremum of a stable process. Ann. Probab. 38(1):

39 The behaviour of p (n) Ψ at infinity Theorem If θ Ψ < : Ψ (θ Ψ ) = and Ψ ( θ + Ψ) < (Cramer s condition) then provided N Ψ > n + 1. lim x x θ Ψ+n+1 p (n) Ψ (x) = C > (.4) Application: If ξ is a stable Lévy process of index α and S 1 = sup s 1 ξ s then it is known that S 1 d = I 1/α Ψ for some Ψ N. Then the statements of Kuznetsov 8 and Doney et al. 9 for the asymptotic behaviour at zero of the density of S 1 are immediate corollaries of the theorem above. 8 A. Kuznetsov (211) On extrema of stable processes. Ann. Probab. 39(3): R. Doney and M. Savov (21) The asymptotic behaviour of densities related to the supremum of a stable process. Ann. Probab. 38(1):

40 The behaviour of p (n) Ψ at infinity Theorem If θ Ψ < : Ψ (θ Ψ ) = and Ψ ( θ + Ψ) < (Cramer s condition) then provided N Ψ > n + 1. lim x x θ Ψ+n+1 p (n) Ψ (x) = C > (.4) Application: If ξ is a stable Lévy process of index α and S 1 = sup s 1 ξ s then it is known that S 1 d = I 1/α Ψ for some Ψ N. Then the statements of Kuznetsov 8 and Doney et al. 9 for the asymptotic behaviour at zero of the density of S 1 are immediate corollaries of the theorem above. Another result of ours recovers the asymptotic behaviour at infinity of the density of S 1. 8 A. Kuznetsov (211) On extrema of stable processes. Ann. Probab. 39(3): R. Doney and M. Savov (21) The asymptotic behaviour of densities related to the supremum of a stable process. Ann. Probab. 38(1):

41 Pricing of Asian options Let the dynamics of the price of an asset, say S t, be driven by a Lévy process, say ξ, that is S t = S e ξt.

42 Pricing of Asian options Let the dynamics of the price of an asset, say S t, be driven by a Lévy process, say ξ, that is S t = S e ξt. The value of an Asian option is given by T + C (S, T, K, r) = e rt S E e ξs ds K T }{{} average price where a + = max {a, } and K is the strike price.

43 Pricing of Asian options Let the dynamics of the price of an asset, say S t, be driven by a Lévy process, say ξ, that is S t = S e ξt. The value of an Asian option is given by T + C (S, T, K, r) = e rt S E e ξs ds K T }{{} average price where a + = max {a, } and K is the strike price. Putting I Ψ (T) = T e ξs ds, it suffices to evaluate [ f(t, K) = E (I Ψ (T) K) +].

44 Pricing of Asian options Putting Ψ q (z) = Ψ(z) q N then the Laplace transform of f(t, K) in T is given by [ e qt f(t, K)dT = e qt E (I Ψ (T) K) +] dt = 1 q E [ ( eq ) ] + e ξs ds K = 1q [ E (IΨq K ) ] + 1 D.Hackmann and A. Kuznetsov (214). Asian options and meromorphic Lévy processes. Finance Stoch. 18(4):

45 Pricing of Asian options Putting Ψ q (z) = Ψ(z) q N then the Laplace transform of f(t, K) in T is given by [ e qt f(t, K)dT = e qt E (I Ψ (T) K) +] dt = 1 q E [ ( eq ) ] + e ξs ds K = 1q [ E (IΨq K ) ] + Then Mellin transform in K yields that 1 [ f (q, z) = K z 1 (IΨq E K ) ] + dk = 1 M IΨq (z + 2), Re(z) ( 2, 1) q q z(z + 1) 1 D.Hackmann and A. Kuznetsov (214). Asian options and meromorphic Lévy processes. Finance Stoch. 18(4):

46 Pricing of Asian options Putting Ψ q (z) = Ψ(z) q N then the Laplace transform of f(t, K) in T is given by [ e qt f(t, K)dT = e qt E (I Ψ (T) K) +] dt = 1 q E [ ( eq ) ] + e ξs ds K = 1q [ E (IΨq K ) ] + Then Mellin transform in K yields that 1 [ f (q, z) = K z 1 (IΨq E K ) ] + dk = 1 M IΨq (z + 2), Re(z) ( 2, 1) q q z(z + 1) Attempt to obtain f(t, K) via numerical or analytical inversion in z and q. In special case Hackmann et al. 1 propose an algorithm based on I Ψq d = Beta (α(k, q), β (k, q)). k= 1 D.Hackmann and A. Kuznetsov (214). Asian options and meromorphic Lévy processes. Finance Stoch. 18(4):

47 Pricing of Asian options Theorem For any Ψ N we have that d I Ψ = Y k, where Y k are random variables whose law is computable in terms of the Wiener-Hopf factors φ ±. k=

48 Pricing of Asian options Theorem For any Ψ N we have that d I Ψ = Y k, where Y k are random variables whose law is computable in terms of the Wiener-Hopf factors φ ±. Can we tackle k= e ζt a+i f(t, K) = C ζ ζ z=a i K M z I Ψζ (z + 2) dzdζ? z(z + 1)

49 If I Ψ = e ξs ds =, i.e. Ψ N \ N, then we consider the asymptotic behaviour of the laws of I Ψ (t) = t e ξs ds. Theorem

50 If I Ψ = e ξs ds =, i.e. Ψ N \ N, then we consider the asymptotic behaviour of the laws of I Ψ (t) = t e ξs ds. Theorem Let Ψ / N, lim sup t ξ t = lim sup t ξ t = and lim P (ξ t < ) = ρ [, 1). t

51 If I Ψ = e ξs ds =, i.e. Ψ N \ N, then we consider the asymptotic behaviour of the laws of I Ψ (t) = t e ξs ds. Theorem Let Ψ / N, lim sup t ξ t = lim sup t ξ t = and lim P (ξ t < ) = ρ [, 1). t Then, for any a (, 1 a φ+ ), f C b (R + ) E [ I a Ψ lim (t)f (I Ψ(t)) ] ( t κ 1 ) = t f(x)ϑ a (dx), where κ RV(ρ) at zero and ϑ a is a finite positive measure.

52 If I Ψ = e ξs ds =, i.e. Ψ N \ N, then we consider the asymptotic behaviour of the laws of I Ψ (t) = t e ξs ds. Theorem Let Ψ / N, lim sup t ξ t = lim sup t ξ t = and lim P (ξ t < ) = ρ [, 1). t Then, for any a (, 1 a φ+ ), f C b (R + ) E [ I a Ψ lim (t)f (I Ψ(t)) ] ( t κ 1 ) = t f(x)ϑ a (dx), where κ RV(ρ) at zero and ϑ a is a finite positive measure. If E [ξ 1 ] =, E [ ξ 2 1] < then κ (r) Cr 1 2.

53 Applications: A diffusion X in a Lévy random potential has generator 1 2 eξx ( ) P sup X s > t s> = [ E a a + I Ψ (t) If E [ [ ξ1] 2 = 1 and E [ξ1 ] = then E ] h(a)da = a a+i Ψ (t) ] c(a)t 1/2. x e ξx x and E [f a (I Ψ (t))] h(a)da. For continuous state branching process Z in random environment in a Lévy random environment ξ P z (Z t < ) = P z (Z t < ξ = ω) P (ξ ω) ] = E [e z( t e ξs ds) 1 = E [f z (I Ψ (t))].

54 Applications: A diffusion X in a Lévy random potential has generator 1 2 eξx ( ) P sup X s > t s> = [ E a a + I Ψ (t) If E [ [ ξ1] 2 = 1 and E [ξ1 ] = then E ] h(a)da = a a+i Ψ (t) ] c(a)t 1/2. x e ξx x and E [f a (I Ψ (t))] h(a)da. For continuous state branching process Z in random environment in a Lévy random environment ξ P z (Z t < ) = P z (Z t < ξ = ω) P (ξ ω) ] = E [e z( t e ξs ds) 1 = E [f z (I Ψ (t))].

55 Thank you!