Itô’s integrated formula for strict local martingales with jumps. Oleksandr Chybiryakov
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Oleksandr Chybiryakov. Itô’s integrated formula for strict local martingales with jumps.. 2005. hal- 00008998
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Oleksandr Chybiryakov∗
Abstract This note presents some properties of positive càdlàg local martingales which are not martingales - strict local martingales - extending the results from [MY05] to local mar- tingales with jumps. Some new examples of strict local martingales are given. The con- struction relies on absolute continuity relationships between Dunkl processes and absolute continuity relationships between semi-stable Markov processes.
Key Words: Strict local martingales, local time, semi-stable Markov processes, Dunkl Markov processes.
Mathematics Subject Classification (2000): 60G44, 60J75, 60J55, 60J25.
1 Main results.
Let (Ω, F, Ft, P) be a filtered probability space. On Ω × R+ we denote by O and P respectively - the optional and predictable sigma fields and by B (R) the Borel sigma field. Consider (St)t≥0 - an R+ valued local martingale with respect to the filtration (Ft)t≥0. For the definitions of local time for discontinuous local martingales we follow ([TL78], p. 17-22 (see also [Mey76] and a [Pro05]). For each a ∈ R there exists a continuous increasing process (Lt , t ≥ 0) , such that Tanaka’s formula holds: t + + (St − a) = (S0 − a) + 1{Su−>a}dSu 0+ − + 1 a + 1 − (S − a) + 1 − (S − a) + L , (1) {Su >a} u {Su ≤a} u 2 t 0 t + + 1 a (S − a) = (S − a) + 1 − dS + L . t 0 {Su >a} u 2 t 0+ ∗Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 175 rue du Chevaleret, F-75013 Paris. [email protected] 1 Furthermore, there exists a B (R)×O measurable version of L, a.s. càdlàg in t, and a B (R)×P measurable version of L, which is a.s. continuous in t. We will only consider such versions. Also note that for any f ≥ 0, Borel, t +∞ c c a f (Xs) d X ,X s = f (a) Lt da. 0 −∞ We shall say that T is a (Ft) stopping time which reduces the local martingale S if (St∧T ) is a uniformly integrable martingale. We shall say that a process X is in class (D) if the family {Xτ , τ - a.s. finite (Ft) stopping time} is uniformly integrable. The following Theorem is a straightforward generalization of Theorem 1 in [MY05]. Theorem 1 Let τ be an (Ft) stopping time such that τ < +∞ a.s. and K ≥ 0. Then there is the following identity + + K 1 K E (Sτ − K) = E (S0 − K) + EJτ + ELτ − cS (τ) , (2) 2 where cS (τ) := E (S0 − Sτ ), K − + Jτ : = 1{Su−>K} (Su − K) + 1{Su−≤K} (Su − K) (3) 0 K 1 K + + EJt∧τ + ELt∧τ = E (St∧τ n − K) − (S0 − K) ≤ ESt∧τ n = ES0. n 2 n Finally, by Beppo-Levi: K 1 K EJt + ELt ≤ ES0 2 and K 1 K EJ∞ + EL∞ ≤ ES0, 2 then for any (Ft) stopping time τ 1 N K1 ≤ 2K + J K + LK a.s., τ {τ<+∞} ∞ 2 ∞ K K which ensures that Nt t≥0 is in class (D ) . Therefore Nt t≥0 is a uniformly integrable mar- tingale and the result follows. Let τ be an (Ft) stopping time which is a.s. finite. With notations from [LN05] suppose 2 + that S ∈ Mloc, and moreover that: S ∞ < ∞ a.s., S is in class (D) , |∆S| ≤ C and εS Ee τ < ∞ for some positive constants C and ε. Then from Theorem 1.1 in [LN05] the term cS (τ) := E (S0 − Sτ ) in (2) can be characterized as π 1/2 π 1/2 cS (τ) = lim λ P S τ > λ = lim λ P [S,S]τ > λ . λ→∞ 2 λ→∞ 2 Besides as a consequence of (2) one obtains K 1 K cS (τ) = lim EJτ + ELτ . K→∞ 2 Let τ be an a.s. finite (Ft) stopping time. Define strict + C (K, τ) := lim E (Sτ∧Tn − K) , (4) n→∞ where Tn → ∞ a.s., Tn reduces (St)t≥0 . The following proposition shows that this limit exists and does not depend on the reducing sequence Tn, n ≥ 1. Proposition 2 Let τ be an a.s. finite (Ft) stopping time. Then strict + K 1 K C (K, τ) = E (S0 − K) + EJτ + ELτ . (5) 2 Furthermore, if the process (∆St)t≥0 is in class (D) , then strict + ∗ C (K, τ) = E (Sτ − K) + lim nP (Sτ > n) , n→∞ ∗ where St := sup0≤u≤t Su. 3 Proof. By Tanaka’s formula t + + K 1 K (S − K) − (S − K) = 1 − dS + J + L , t 0 {Su >K} u t 2 t 0+ K where Jt is defined by (3) . Since (St)t≥0 is a local martingale, t K St := 1{Su−>K}dSu 0+ is also a local martingale. We have seen in the proof of Theorem 1 that t K Nt = 1{Su−>K}dSu + S0 − St 0+ is a uniformly integrable martingale, then K K St := Nt + St − S0 is the sum of a uniformly integrable martingale and a local martingale (St)t≥0 . Therefore a K stopping time which reduces (St)t≥0 reduces St t≥0 as well. K Let T be an (Ft) stopping time which reduces (St)t≥0 . Then St∧T and St∧T are uniformly integrable martingales. For any τ - an a.s. finite (Ft) stopping time one gets + + K 1 K E (Sτ∧T − K) = E (S0 − K) + EJτ∧T + ELτ∧T , (6) 2 now taking for T a stopping time Tn, such that Tn → ∞ a.s. and Tn reduces (St)t≥0 , one obtains that strict + C (K, τ) := lim E (Sτ∧Tn − K) n→∞ exists and does not depend on the sequence of stopping times reducing (St)t≥0 . Furthermore, strict + K 1 K C (K, τ) = E (S0 − K) + EJτ + ELτ . 2 In order to move further suppose that the process (∆St)t≥0 is in class (D). Take Tn := inf {u > 0 |Su > n} . Since (St)t≥0 is an adapted càdlàg process, Tn is an (Ft) stopping time and Tn → ∞ a.s. Since |St∧Tn | ≤ n + ∆STn , (St∧Tn )t≥0 is in class (D) and subsequently is a uniformly integrable martingale. In particular Tn is an (Ft) stopping time which reduces (St)t≥0. Now one can get for any τ - an a.s. finite (Ft) stopping time + + + E (Sτ∧Tn − K) = E (Sτ − K) 1{τ≤Tn} + E (STn − K) 1{τ>Tn} . 4 The left hand side converges and equals Cstrict (K, τ) . The first expression on the right hand side + + converges as well (by Beppo-Levi) to E (Sτ − K) . Hence E (STn − K) 1{τ>Tn} converges as well. Besides one has for n > K + (n − K) P (τ > Tn) ≤ E (STn − K) 1{τ>Tn} ≤ (n − K) P (τ > Tn) + E ∆STn 1{τ>Tn} and + + E (STn − K) 1{τ>Tn} − E ∆STn 1{τ>Tn} ≤ (n − K) P (τ > Tn) ≤ E (STn − K) 1{τ>Tn} . 1 1 Since ∆STn {τ>Tn} n≥1 is a uniformly integrable family E ∆STn {τ>Tn} → 0, as n → ∞, and + ∗ lim (STn − K) 1{τ>Tn} = lim (n − K) (τ > Tn) = lim n (Sτ > n) . n→∞ E n→∞ P n→∞ P Finally + + ∗ lim E (Sτ∧Tn − K) = E (Sτ − K) + lim nP (Sτ > n) , n→∞ n→∞ ∗ where St := sup0≤u≤t Su. Remark 3 Note that from Theorem 1 and Proposition 2 for any positive local martingale S, such that (∆St)t≥0 is in class (D) , and for any a.s. finite (Ft) stopping time τ ∗ cS (τ) = lim nP (Sτ > n) . n→∞ Remark 4 Under the conditions of Proposition 2 strict + C (K, τ) = sup E (Sσ∧τ − K) . (7) σ-(Ft) stopping time Proof. From (6) one obtains that for any pair of (Ft) stopping times τ, σ and a sequence of stopping times Rn, such that Rn → ∞ a.s. and Rn reduces (St)t≥0 + + K 1 K E (Sτ∧σ∧Rn − K) = E (S0 − K) + EJτ∧σ∧R + ELτ∧σ∧R . n 2 n Then by Fatou’s Lemma + + K 1 K E (Sτ∧σ − K) ≤ E (S0 − K) + lim inf EJτ∧σ∧R + ELτ∧σ∧R n→∞ n 2 n + K 1 K = E (S0 − K) + EJτ∧σ + ELτ∧σ 2 + K 1 K ≤ E (S0 − K) + EJτ + ELτ . 2 Now (7) follows from (5) and (4) . Remark 5 The original proof of Proposition 2 in [MY05] differs a little from ours. In order K to obtain (6) the fact, that the stopping time which reduces (St)t≥0 reduces as well St t≥0 , is not used. Let us go through this other proof and see that there is no contradiction. 5 K K Proof. Let T be an (Ft) stopping time which reduces (St)t≥0 and Tn , n ≥ 1,Tn → ∞ be K a sequence of stopping times that reduce St t≥0 . Then for any τ - an a.s. finite (Ft) stopping time - one gets + + K 1 K E Sτ∧T ∧T K − K = E (S0 − K) + EJ K + EL K . n τ∧T ∧Tn 2 τ∧T ∧Tn K On the right hand side one can pass to the limit as Tn → ∞ by Beppo-Levi and get a finite limit as soon as we already know from the proof of Theorem 1 that K 1 K EJ∞ + EL∞ ≤ ES0. 2 On the left hand side, S K is a uniformly integrable martingale, thus it converges in τ∧T ∧Tn n≥1 L1 to S . Finally one gets τ∧T + + K 1 K E (Sτ∧T − K) = E (S0 − K) + EJτ∧T + ELτ∧T , (8) 2 which is the same as (6) . For any µ - a finite measure on R+ define +∞ + Fµ (x) := µ (dK)(x − K) 0 +∞ and µ¯ := 0 µ (dK) . As in [MY05] we have the following Proposition and Corollary (the proofs are the same as in the continuous case). Proposition 6 Under the notations and assumptions of Theorem 1 +∞ K 1 K E [Fµ (Sτ )] = Fµ (S0) + E µ (dK) Jτ + Lτ − µc¯ S (τ) . 2 0 Corollary 7 The process +∞ 1 F (S ) − F (S ) − µ (dK) J K + LK − µ¯ (S − S ) , t ≥ 0 µ t µ 0 t 2 t t 0 0 is a martingale. 2 Examples. One can trivially construct strict local martingales from continuous strict local martingales: (c) (d) (c) (d) indeed, Mt := Mt + Mt and Mt is a strict local martingale and Mt is a uniformly integrable martingale, then (Mt) is a strict local martingale. We now obtain strict local martingales with jumps which are generalizations of the strict (3) (3) local martingale 1/Rt , where Rt is a Bessel process of dimension 3. As in the case of 6 (3) 1/Rt such strict local martingales can be obtained from absolute continuity relationships between two Dunkl Markov processes instead of Bessel processes. For simplicity, we consider here only one dimensional Dunkl Markov processes (see [GY05]). The Dunkl Markov process (Xt) with parameter k is a Feller process with extended generator given for f ∈ C2 (R) by 1 1 f (x) − f (−x) L f (x) = f (x) + k f (x) − , k 2 x 2x2 1 (k) where k ≥ 0. Note that |X| is a Bessel process with index ν := k − 2 . Denote by Px the law X of (Xt) started at x ∈ R, and by Ft the natural filtration of X. 1 Proposition 8 Let 0 ≤ k < 2 ≤ k and x > 0. Define T0 := inf {s ≥ 0 |Xs− = 0 or Xs = 0} . (k) Then Px (T0 < +∞) = 1 and there is the following absolute continuity relationship: k −k Nt∧T 2 t∧T 0 2 0 (k ) |Xt∧T0 | k (k ) − k ds (k) Px = exp − Px X , (9) X 2 Ft Ft |x| k 2 0 Xs where Nt denotes the number of jumps of X on [0, t] . Furthermore |x| k −k k Nt (k )2 − k2 t ds M := exp (10) t |X | k 2 X2 t 0 s (k) is a strict local martingale under Px , and (k) (k) Px (T0 > t) = Ex Mt, (11) (k) (k) where Ex is the expectation under Px . (k) 2 Remark 9 Note that the law of T0 under Px is that of x / 2Z 1 , where Z 1 is a ( 2 −k) ( 2 −k) 1 gamma variable of a parameter 2 − k (see p.98 in [Yor01]). Proof. Let X be a Dunkl Markov process. Note that ∆Xs = Xs − Xs− = −2Xs−, when ∆Xs = 0. Hence if Xs = 0, then Xs− = 0 and T0 = inf {s ≥ 0 |Xs− = 0} = inf {s ≥ 0 ||Xs| = 0} . In order to prove (9) we proceed as in the proof of Proposition 4 in [GY05]. First we need to 1 1 extend Theorem 2 in [GY05] for k < 2 . Since |X| is a Bessel process with index k − 2 , for k < 1 ,T < +∞ a.s., and, for k ≥ 1 ,T = +∞ a.s. Denote 2 0 2 0 s du τ := inf s ≥ 0 = t , t X2 0 u 7 then τ t is a continuous strictly increasing time change and τ ∞ = T0. Denote Yu := Xτ u . Since for any f ∈ C2 (R) t f (Xt) − f (X0) − Lkf (Xs) ds 0 is a local martingale, t 2 f (Yu) − f (Y0) − Ys Lkf (Ys) ds 0 is a local martingale. Then as in the proof of Proposition 4 in [GY05] one obtains that Y is in the form (ν) (k/2) Yu = exp βu + iπNu , 1 (ν) (k/2) where ν := k − 2 , βu is a Brownian motion with drift ν, Nu is a Poisson process with (ν) parameter k/2 independent from βu . Denote t du A := , t X2 0 u then τ At = t, for t < T0. Hence Xt = YAt , t < T0. (12) Note also that differentiating the equality Aτ t = t with respect to time one gets d τ = Y 2 dt t t s 2 and At = inf s ≥ 0 0 Yu du = t , t < T0. Note that (9) is equivalent to k −k Nt 2 2 t (k) |x| k (k ) − k ds (k) Px X = exp Px . (13) Ft ∩{t Indeed (9) is equivalent to (k) (k) 1 (k) 1 Ex (F (Xs, s ≤ t)) = Ex F (Xs, s ≤ t) = Ex F (Xs, s ≤ t) 1{t (k) ˆ (k) ˆ Ex Mt1{t to the pair β(ν ),N (k /2) under P (k ). Both pairs consist of a Brownian motion with drift and 1 1 a Poisson process which are mutually independent, and ν := k − 2 , ν := k − 2 . Now in the same way as in the proof of Proposition 4 in [GY05], for any bounded measurable F, (k) (ν) (k) (k) (ν) (k) Ex F βs ,Ns , s ≤ t = Ex DtF βs ,Ns , s ≤ t , 8 where (k /2) N 1 k t 1 D : = exp (ν − ν ) β(ν ) − ν2 − (ν )2 t exp − (k − k ) t t t 2 k 2 (k /2) N 1 k t = exp (k − k ) β(ν ) − k2 − (k )2 t t 2 k (ν ) (k /2) (ν ) (k /2) 1 and DAt = Mt, t < T0. Denote Gt := σ βs ,Ns , s ≤ t , then F βAs ,NAs {As≤u} is GAs∧u measurable and (k) (ν) (k/2) 1 (k ) (k ) (ν ) (k /2) 1 Ex F βAs ,NAs {As≤u} = Ex E (Dt |GAs∧u ) F βAs ,NAs {As≤u} (k ) (ν ) (k /2) 1 = Ex DAs F βAs ,NAs {As≤u} . As u → +∞ one gets (k) (ν) (k/2) 1 (k ) (ν ) (k /2) 1 Ex F βAs ,NAs {As<+∞} = Ex DAs F βAs ,NAs {As<+∞} . (14) Noting that {As < +∞} = {s < T0}, (14) leads to (13) . From (13) one easily obtains (11) . (k) Suppose that (Mt) is a martingale then from (11) for any t ≥ 0 Px (T0 > t) = 1 and (k) 1 Px (T0 = +∞) = 1, which is impossible because k < 2 . Hence (Mt) is a strict local mar- tingale. Other examples of strict local martingales with jumps can be obtained from absolute conti- nuity relationships between two non-negative semi-stable Markov processes. We shortly recall the definition of a semi-stable Markov process (see [Lam72]): A semi-stable Markov process (with index of stability α = 1) on R+ := [0, +∞) is a Markov process (Xt) with the following scaling property: for any c > 0 − 1 (d) (xc 1) X(x) = X , c ct t t≥0 t≥0 (x) where Xt denotes a semi-stable Markov process started at x > 0. Denote T0 := inf {s ≥ 0 |Xs− = 0 or Xs = 0} , (15) then Lamperti in [Lam72] showed that: either T0 = +∞ a.s., or T0 < +∞ a.s. and XT0− = 0 a.s., or T0 < +∞ a.s. and XT0− > 0 a.s. Furthermore this does not depend on the starting point x > 0. Note that for a semi-stable Markov process the following Lamperti relation is true. We suppose that there is no killing inside (0, ∞) . Proposition 10 Let (ξt) be a one-dimensional Lévy process, starting at 0. Define t (x) At := x exp (ξs) ds, 0 9 for any x > 0. Then the process (Xu) , defined implicitly by x exp ξt = X (x) , t < T0, (16) At is a semi-stable Markov process, starting at x, and (x) A∞ = T0, (17) where T0 is defined by (15) . The converse is also true. Denote (x) (x) τ t := inf s ≥ 0 As = t . ξ X Let Ft be the natural filtration of (ξt) and Ft be the natural filtration of (Xt) . As in [CPY94], using Proposition 10, one obtains the following absolute continuity relationship between two semi-stable Markov processes. Proposition 11 Suppose that (Xt) is a semi-stable Markov process associated with Lévy process bξt tρ(b) (ξt) via Lamperti relation (16) and EP e = e < ∞. Define Q by b t Xt ds Q| X = exp −ρ (b) P| X , Ft ∩{t Ψ(˜ u) = Ψ (u − ib) − Ψ(−ib) , ˜ where Ψ, Ψ are the characteristic exponents of (ξt) under P and Q respectively. Proof. Let us consider the change of measure given by the Esscher transform: Q| ξ = exp (bξ − ρ (b) t) P | ξ . Ft t Ft Since Eebξt = etρ(b) < ∞ Mt := exp (bξt − ρ (b) t) (x) is a martingale. Furthermore (ξt) is still a Lévy process under Q. Note that τ t < +∞ = {t < T0} and for any t < T0 (x) A (x) = t. (18) τ t ξ Denote Gt := F (x) , then for any A ∈ Gt τ t (x) (x) Q A ∩ τ t ≤ u = EP 1 (x) exp bξ (x) − ρ (b) τ t . (19) A∩ τ t ≤u τ t 10 b Note that (Xt/x) = exp bξ (x) on {t < T0} . Differentiating (18) one gets that τ t d (x) 1 τ t = 2 . dt Xt Letting u tend to infinity, from (19) one gets b t (x) Xt ds Q A ∩ τ < +∞ = 1 (x) exp −ρ (b) . t EP A∩ τ <+∞ 2 t x 0 Xs (x) But from (17) τ t < +∞ = {t < T0} . Hence b t Xt ds Q (A ∩ {t < T0}) = EP 1A∩{t Let us find the range of the parameter b such that X b t ds M := t exp −ρ (b) t x X2 0 s is a strict local martingale. Note that it is sufficient to find b such that Q (T0 < +∞) = 1 and P (T0 = +∞) = 1. Indeed, given such a parameter Q (t < T0) = EP (Mt) and as in the proof of Proposition 8 (Mt) is a strict local P-martingale. Now let ξ be a Lévy process under P with the characteristic exponent Ψ, given by Lévy- Khintchine formula 1 Ψ(λ) = iaλ + σ2λ2 + 1 − eiλx + iλx1 Π(dx) , 2 {|x|<1} R 2 where a ∈ R, σ2 ≥ 0 and Π is a positive measure on R\{0} such that 1 ∧ |x| Π(dx) < ∞. Let us suppose that Π has compact support. Then ebξt = etρ(b) < ∞ for any t and b. Let EP the semi-stable Markov process X be associated to ξ via Lamperti relation. Conditions for 2 P (T0 = +∞) = 1 or P (T0 < +∞) = 1 bearing on (a, σ , Π) can be deduced from Theorem 1 in [BY05]. Note that T0 < +∞ if and only if ξt → −∞. Since Π has compact support, from the Central Limit Theorem for a Lévy process, ξt → −∞ if and only if EP ξ1 < 0 i.e. −a + xΠ(dx) < 0. |x|>1 Let Q be given by Proposition 11. Denote by Ψ˜ the characteristic exponent of ξ under Q, then 1 Ψ(˜ λ) = iλ a − bσ2 + x 1 − ebx Π(dx) + σ2λ2 + 1 − eiλx + iλx1 Π(˜ dx) , 2 {|x|<1} |x|<1 R 11 bx where Π(˜ dx) = e Π(dx) . Hence, in order to have Q (T0 < +∞) = 1 and P (T0 = +∞) = 1 one can choose b such that −a + xΠ(dx) ≥ 0 (20) |x|>1 and −a + bσ2 − x 1 − ebx Π(dx) + xebxΠ(dx) < 0. (21) |x|<1 |x|>1 It is easy to see that (20) and (21) imply that b < 0. For example, for any given a and Π, such that (20) is true, one can always choose b < 0 such that bσ2 − e−b |x| Π(dx) < a − xΠ(dx) , (22) x<−1 x>1 which implies (21) . Note that condition (22) is more restrictive than (21) . References [BY05] J. Bertoin and M. Yor, Exponential functionals of Lévy processes, To appear in Prob- ability Surveys (2005). [CPY94] P. Carmona, F. Petit, and M. Yor, Sur les fonctionnelles exponentielles de certains processus de Lévy, Stochastics Stochastics Rep. 47 (1994), no. 1-2, 71—101. [GY05] L. Gallardo and M. Yor, Some remarkable properties of the Dunkl martingales, To appear in Séminaire de Probabilités XXXIX. Dedicated to Paul-André Meyer, Lecture Notes in Math., Springer, Berlin, 2005. [Lam72] J. Lamperti, Semi-stable Markov processes. I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205—225. [LN05] R. Liptser and A. Novikov, On tail distributions of supremum and quadratic variation of local martingales, Working paper (2005). [Mey76] P. A. Meyer, Un cours sur les intégrales stochastiques, Séminaire de Probabilités, X (Seconde partie: Théorie des intégrales stochastiques, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), Springer, Berlin, 1976, pp. 245—400. Lecture Notes in Math., Vol. 511. [MY05] D.B. Madan and M. Yor, Itô’s integrated formula for strict local martingales, To appear in Séminaire de Probabilités XXXIX. Dedicated to Paul-André Meyer, Lecture Notes in Math., Springer, Berlin, 2005. [Pro05] P. E. Protter, Stochastic integration and differential equations, second ed., Version 2.1, Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 2005, Stochastic Modelling and Applied Probability. 12 [TL78] Temps locaux, Astérisque, vol. 52, Société Mathématique de France, Paris, 1978, Exposés du Séminaire J. Azéma-M. Yor, Held at the Université Pierre et Marie Curie, Paris, 1976—1977, With an English summary. [Yor01] M. Yor, Exponential functionals of Brownian motion and related processes, Springer Finance, Springer-Verlag, Berlin, 2001, With an introductory chapter by Hélyette Geman, Chapters 1, 3, 4, 8 translated from the French by Stephen S. Wilson. 13