CLARK-OCONE FORMULA BY THE S-TRANSFORM ON THE POISSON WHITE NOISE SPACE

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1 CLARK-OCONE FORMULA BY THE S-TRANSFORM ON THE POISSON WHITE NOISE SPACE YUH-JIA LEE*, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH* Abstract. Given ϕ a square-integrable Poisson white noise functionals we show that the Segal-Bargmann (S- transform t Sϕ(P t(g is absolutely continuous with t Sϕ(Pt(g = S( t E[ tϕ Ft ] (g for almost all t R, where P t(g = 1 (, t] g for g in the Schwartz space S on R, an t means the Poisson white noise erivative. After integration with respect to t an applying the inverse S-transform, this ientity recovers the Clark-Ocone formula for ϕ. 1. Introuction The Clark-Ocone formula allows one to express a given ranom variable ϕ efine on the classical Wiener space as the sum of its expectation value an an Itô integral with respect to Brownian motion. This formula was first erive by J. M. C. Clark (1970 in [2], an later, generalize to weakly ifferentiable functionals by D. Ocone (1984 in [13]. Such a formula is a useful tool in mathematical finance, for example, to compute the replicating portfolio of a claim with given maturity (see [1]. Recently, non-gaussian processes an more general Lévy processes have been extensively stuie an new applications have been foun in financial mathematics, quantum probability, stochastic analysis, etc. (see [1, 10, 12, 14, 16]. It is natural to ask that if the Clark-Ocone formula for Gaussian functionals can be generalize to non-gaussian Lévy white noise functionals. In Hia s white noise theory, the Segal-Bargmann transform (S-transform, for abbrev. plays a key role. There the generalize white noise fuctionals have been efine an stuie through their S-transforms (see [6]. In [9], Lee an Shih have shown that the S-transform allows one to connect the Clark-Ocone formula of Brownian functionals with the funamental theorem of calculus for Segal-Bargmann analytic functions. Applying the integral representation of the S-transform of Lévy white noise functionals in the non-gaussian case, Lee an Shih [7], have evelope a theory of the Lévy 2000 Mathematics Subject Classification. Primary 60H40; Seconary 60H05, 60H07, 60J75. Key wors an phrases. Clark-Ocone formula, Poisson white noise, Poisson measure, white noise analysis. * Research supporte by the NSC of Taiwan. Research supporte by NTU Start-Up Grant M

2 2 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH white noise analysis for case of Lévy processes following Hia s iea an by means of the S-transform, cf. [10] (see also [12]. In this paper we are concerne with the erivation of the Clark-Ocone formula for Poisson white noise functionals. The main result is obtaine in Section 4, in which we apply the results of [10] to the erivation of the Clark-Ocone formula for square-integrable Poisson white noise functionals. The main argument can be briefly escribe as follows. Given g an element of the Schwartz space S on R an P t (g = 1 (, t] g, we prove that the function t Sϕ(P t (g is absolutely continuous, where Sϕ is the Segal-Bargmann (S- transform of the square-integrable Poisson white noise functional ϕ. As a consequence we obtain the following ientity Sϕ(g = E[ϕ] + + t Sϕ(P t(g t, g S, (1.1 by the funamental theorem of calculus. Comparing with the Clark-Ocone formula (Theorem 4.2, we show that for almost all t R we have t Sϕ(P t(g = S ( t E[ t ϕ F t ] (g, g S, where t enotes the Poisson white noise erivative. This also shows that we can recover the Clark-Ocone formula for ϕ simply by taking the inverse S-transform for the equation (1.1. We note that the results of this paper can be extene to compoun Poisson or more general Lévy white noise functionals. Since the arguments are much more involve, they will be iscusse in a forthcoming paper. 2. Elements of Poisson white noise analysis In this section we briefly review some notions an basic results of Poisson white noise analysis. The intereste reaer is referre to [10] for etails an generalization to Lévy white noise analysis. Let X = {X(t; t R} be a stanar Poisson process with X(0 = 0 almost surely, an E[e i r(x(t X(s ] = exp((t s(e i r 1, r R, 0 s t, (2.1 where i = 1. Let S be the Schwartz space with the ual S of tempere istributions on R. It is well-known that there exists a unique probability measure Λ on (S, B(S with the characteristic functional C on S given by ( + C(η E[e i, η ( ] = exp e i η(t 1 t, η S, where, is the S -S pairing an t refers to the Lebesgue measure (see [7]. The probability measure Λ is then calle the Poisson white noise measure an (S, B(S, Λ will serve as the unerlying probability space.

3 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 3 For any η S, the mean an variance of the ranom variables, η are given by E[, η ] = + η(t t, an V[, η ] = + η(t 2 t. (2.2 For each ρ L 1 L 2 (R, t, choose a sequence {η n } S so that η n ρ in L 1 L 2 (R, t uner the norm L1 (R,t + L2 (R,t. Then it follows from (2.2 that {, η n } forms a Cauchy sequence in L 2 (S, Λ. Denote by, ρ the L 2 -limit of {, η n }. When ρ = 1 (s, t], the inicator of (s, t], the characteristic function of, ρ is exactly the same as in (2.1. So the Poisson process X on (S, B(S, Λ can be represente by x, 1 [0, t], if t 0 X(t ; x = x, 1 [t, 0], if t < 0, x S. Taking the time erivative formally, we get Ẋ(t; x = x(t, x S. From this viewpoint, the elements of S can be regare as the sample paths of Poisson white noise, an thus members of L 2 (S, Λ are also calle square-integrable Poisson white noise functionals. For η = η 1 +i η 2 L 1 c L 2 c(r, t with η 1, η 2 L 1 L 2 (R, t,, η is efine to be, η 1 + i, η 2, where V c enotes the complexification of a real locally convex space V. Then the above ientity in (2.2 also hol for complex-value ranom variable, η. Fock-type ecomposition. Let B b (R be the class of all boune Borel subsets E of R. For any E B b (R, let N(E ; be the integer-value function on (S, B(S efine by N(E ; x = # {t E : X(t ; x X(t ; x = 1}, x S. Then N(E ; is a ranom variable which has Poisson istribution an the Lebesgue measure Leb on R is the corresponing intensity measure. The system of {N(E ; x Leb(E : E B b (R ; x S } forms an inepenent ranom measure with zero. We enote N(E ; x Leb(E by Ñ(E ; x. Then, for f L2 c(r, the stochastic integral f(t Ñ(t has mean zero an the variance f(t 2 t. We note that as f L 1 c L 2 c(r, f(t Ñ(t; x = f(t X(t; x f(t t [Λ]-almost everywhere x S, where the right-han sie integral with respect to X exists pathwise in the sense of the Lebesgue-Stieltjes integrals. In particular, for b > a, X(b X(a = (b a + 1 (a,b] (t Ñ(t. Notice that, for any E, F B(R, E[Ñ(EÑ(F ] = Leb(E F. Denote by I n (g, g ˆL 2 c(r n, the close subspace of L 2 c(r n consisting of all symmetric

4 4 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH complex-value functions in L 2 c(r, the multiple stochastic integral of orer n with respect to Ñ g(t 1,..., t n Ñ(t 1 Ñ(t n, which satisfies the isometry I n (g 2 L 2 (S,Λ = n! g(t 1,..., t n 2 t 1 t n. The multiple stochastic integrals with respect to Ñ provie an orthogonal ecomposition theorem for the square-integrable Poisson white noise functionals. Theorem 2.1. (Itô [4] Let ϕ be given in L 2 (S, Λ. Then (i there exist uniquely a series of kernel functions φ n ˆL 2 c(r n, n N {0}, such that ϕ is equal to the orthogonal irect sum I n(φ n. In notation, we write ϕ (φ n. (ii L 2 (S, Λ is isomorphic to the symmetric Fock space F s (L 2 c(r over L 2 c(r by carrying ϕ ( φ n into ( 0! φ 0, 1! φ 1,..., n! φ n,.... The Segal-Bargmann transform. For ϕ L 2 (S, Λ, say ϕ (φ n, the Segal- Bargmann (or the S- transform of ϕ is a complex-value functional on L 2 c(r by Sϕ(g ϕ(x E X (g(x Λ(x where = S φ n (t 1,..., t n g(t 1 g(t n t 1 t n, E X (g = 1 n! I n(g n is the coherent state functional associate with g. The S-transform is a unitary operator mapping from L 2 (S, Λ onto the Bargmann-Segal-Dwyer space F 1 (L 2 c(r over L 2 c(r,. Moreover, we have ϕ 2 L 2 (S,Λ = 1 n! Dn Sϕ(0 2 L (n (2 (L2 c (R, where D is the Fréchet erivative of Sϕ an (n L enotes the Hilbert-Schmit (H (2 operator norm of a n-linear functional on a Hilbert space H. Theorem 2.2. (Lee-Shih [7] Let ϕ L 2 (S, Λ. Then, for any g L 1 c(r L c (R we have ( E X (g = exp g(t t (1 + g(t, t J X (x where J X (x is the set {t R : X(t; x X(t ; x = 1}.

5 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 5 Remark 2.3. Let H be a complex Hilbert space. For r > 0, enote by F r (H the class of analytic functions on H with norm F r (H such that f 2 F r (H = r n calle the Bargmann-Segal-Dwyer space. n! Dn f(0 2 < +, L (n [H] (2 Test an generalize Poisson white noise functionals. Let A = 2 /t 2 + (1 + t 2 enote the ensely efine an self-ajoint operator on L 2 (R, t an {e n : n N 0 } be eigenfunctions of A with corresponing eigenvalues 2n + 2, n N {0}. {e n : n N 0 } forms a complete orthonormal basis (CONS, in abbreviation of L 2 (R, t. For any p R an η L 2 (R, t, efine η p := A p η L 2 (R,t an let S p be the completion of the class {η L 2 (R, t : η p < + } with respect to p -norm. Then S p is a real separable Hilbert space an we have the continuous inclusions: S = lim p>0 S p S p S q L 2 (R, t S q S p S = lim p>0 S p, p > q 0. We next procee to construct the spaces of test an generalize functions as follows. For p R an ϕ L 2 (S, Λ, efine ϕ 2 p = 1 n! Dn Sϕ(0 2 L (n (2 (S p an let L p be the completion of the class { ϕ L 2 (S, Λ : ϕ p < + } with respect to p -norm. Then L p, p R, is a Hilbert space with the inner prouct, p inuce by p -norm. For p, q R with q p, L q L p an the embeing L q L p is of Hilbert-Schmit type, whenever q p > 1/2. Let L = lim L p. p>0 Then L is a nuclear space which will serve as the space of test functions an the ual space L of L the space of generalize functions. The members of L are calle generalize Poisson white noise functionals. In this way, we obtain a Gel fan triple L L 2 (S, Λ L an have the continuous inclusion: L L p L q L 2 (S, Λ L q L p L = lim p>0 L p, p q > 0. In what follows, the ual pairing of L an L will be enote by,. For g L 2 c(r, it is easy to see that E X (g p = e (1/2 g 2 p for any p > 0. Hence E X (g L p if an only if g S p,c for p > 0. We efine the S-transform for F L by SF (g = F, E X (g, g S c. Annihilation an creation operators. Let F L p an ξ S p,c, p R. The Gâteaux erivative (/z z=0 SF ( + z ξ in the irection ξ is an analytic function on S p,c. In fact, by using the Cauchy integral formula, one can show that (/z z=0 SF ( + z ξ L p 1. Define ( ξ F = S 1 z SF ( + z ξ z=0.

6 6 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH Then, by the characterization theorem [11], we have ξ F L p 5. It is clear that 2 ξ is continuous from L into itself. Its ajoint operator ξ is then efine from by ξ F, ϕ := F, ξ ϕ for F L an ϕ L. Here, ξ is calle the annihilation operator an ξ is calle the creation operator. For convenience, we enote the Poisson white noise erivative δt an its ajoint δ t respectively by t an t for t R, where δ t is the Dirac measure concentrate on the point t. Proposition 2.4. ([10] Let ϕ ( φ n L p an ξ S p,c for p 0. If p q ln 3/6 ln 2, then ξ ϕ = ξ, φ 1 + n ξ, φ n (, t 2,..., t n Ñ(t 2 Ñ(t n, n=2 where, is the S p,c -S p,c pairing. Moreover, ξ ϕ q 2 ξ p ϕ p. From [10] we have the following proposition. Proposition 2.5. ([10] Let F ( f n L p an ξ S c. Then ξ F = Moreover, for q p ln 3/6 ln 2, I n+1 (ξ ˆ f n in L. ξ F q 2 ξ q F p. 3. Funamental theorem of calculus on Bargmann-Segal-Dwyer spaces First of all, we specify some notations as follows: - For t R an h L 2 c(r we let P t (h = h 1 (,t]. - For g L 1 L (R an x S we set ( γ 1 (g = exp g(t t an γ 2 (g(x = an γ(g = γ 1 (gγ 2 (g. t J X (x (1 + g(t, Theorem 3.1. For any ϕ L 2 (S, Λ an g L 1 L (R, the mapping t R Sϕ(P t (g is absolutely continuous. Moreover, for any < a < b < +, Sϕ(P b (g Sϕ(P a (g = b a t Sϕ(P t(g t. (3.1

7 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 7 Proof. Let g L 1 L (R an ϕ L 2 (S, Λ be fixe. We start by proving the absolute continuity of the mapping t R Sϕ(P t (g. Let {[a i, b i ] : i = 1, 2, 3,...} be any (finite or countably infinite collection of nonoverlapping intervals. Then Sϕ(P bi (g Sϕ(P ai (g i = i ϕ(x(γ(p bi (g(x γ(p ai (g(x Λ(x. (3.2 S We enote this sum by N. To estimate the sum N, we first note that γ 1 (P bi (g γ 1 (P ai (g ϕ(x γ 2 (P bi (g(x Λ(x (3.3 N i S + γ 1 (P ai (g ϕ(x γ 2 (P bi (g(x γ 2 (P ai (g(x Λ(x. (3.4 i S Let N (1 an N (2 stan separately for the sums in the right han sie of (3.3 an (3.4. Since, for any a < b, b ( γ 1 (P b (g γ 1 (P a (g g(t t exp 2 g(t t a an ( γ 2 (P b (g(x exp g(t t E X (P b (g(x, x S, we see, by the Cauchy-Schwarz inequality, that (1 C(ϕ, g g(t t, (3.5 N i[a i, b i] where C(ϕ, g is a constant, epening only on ϕ an g, given by ( C(ϕ, g = ϕ L 2 (S,Λ exp 3 g(t t + 1 g(t 2 t. 2 For the estimation of N (2, we nee the following inequality: Lemma 3.2. Let E, F be two countable subsets of complex numbers such that E F an z F z < +. Then (1 + z (1 + z (1 + z (1 + z. z F z E z F z E By Lemma 3.2, we see that for any a < b, γ 2 (P b (g(x γ 2 (P a (g(x ( b exp b a ln(1 + g(t X(t; x ln(1 + g(t X(t; x exp R R ( a exp ln(1 + g(t X(t; x ( 2 ln(1 + g(t X(t; x. (3.6

8 8 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH It follows from (3.6 that ( (2 e g(t t ϕ(x ln(1 + g(t X(t; x N S i[a i,b i] ( exp 2 ln(1 + g(t X(t; x Λ(x e g(t t ϕ L 2 (S,Λ ( ln(1 + g(t X(t; x S i[a i,b i] ( exp 4 D(ϕ, g ( exp 5 { S 2 ] } 1/2 ln(1 + g(t X(t; x Λ(x ( ln(1 + g(t X(t; x i[a i,b i] 1/2 ln(1 + g(t X(t; x Λ(x}, (3.7 where D(ϕ, g = e g(t t ϕ L 2 (S,Λ is a constant epening only only on ϕ an g. Note that for any two non-negative Borel measurable functions p(t, q(t, t R, we have ( ( p(t X(t; x exp q(t X(t; x Λ(x S ( ( = p(t e q(t t exp (e q(t 1 t. By applying this formula to (3.7, we see that ( (2 D(ϕ, g N exp (1 + g(t 5 ln(1 + g(t t i[a i,b i] ( 1 2 ( (1 + g(t 5 1 t, 1/2 which tens to zero whenever (b i a i tens to zero. Combining this boun with (3.5 yiels the absolute continuity property an ( Clark-Ocone formula for Poisson white noise functionals In [2], Clark prove that a sufficiently well-behave Fréchet-ifferentiable Brownian functional can be represente as a stochastic integral in which the integran has the form of conitional expectations of the ifferential. We note that this result can be extene to all square integrable Poisson white noise functionals.

9 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 9 Lemma 4.1. [8] Let F ( f n L 2 (S, Λ. The conitional expectation E[ F F t ] of F relative to F t, t R, is given by E[ F F t ] = I n ( 1 (,t] n f n, where E n means the prouct E E for any E B(R. Let f n ˆL 2 c(r n an ψ ( ψ n L. We observe that for almost all t R, t I n (f n L 2 (S, Λ, an thus, for p > 1, by Lemma 4.1, E[ t I n (f n F t ], ψ 2 E[ t I n (f n F t ] 2 p ψ 2 p where f n (t is the function on R n 1 to C efine by = (n 1! n 2 1 (,t] n 1 f n (t 2 p ψ 2 p (by Proposition 2.4 an Lemma (p 1(n 1 n! f n (t 2 0 ψ 2 p, (4.1 f n (t(t 1,..., t n 1 = f n (t 1,..., t n 1, t. So, for a fixe F ( f n L 2 (S, Λ, it follows from (4.1 that lim sup m,m m n=m E[ t I n (f n F t ] 2 p t lim m n=m n! f n 2 0 = 0. In other wors, for almost all t R, { m E[ ti n (f n F t ] } is a Cauchy m sequence in L p, p > 1. Consequently we can efine { m E[ t F F t ] := the L p -limit of } E[ t I n (f n F t ] as m. Theorem 4.2. (Clark-Ocone formula for Poisson white noise functionals Let F ( f n be in L 2 (S, Λ. Then F = E[F ] + t E[ t F F t ] t in L p, p > 5/4, where the above integral is regare as a Bochner integral. Proof. Letting n (t = {(t 1,..., t n 1 R n 1 ; t j < t, j = 1, 2,..., n 1},

10 10 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH we have F = E[F ] + = E[F ] + = E[F ] + I n ( f n n ( 1 n(t(t 1,..., t n 1 f n (t 1,..., t n 1, t Ñ(t 1 Ñ(t n 1 E[ t I n ( f n F t ] Ñ(t, Ñ(t, cf. also Proposition 12 of [15] an Proposition of [16]. Then, by applying [10, Theorem 7.2], F = E[F ] + Moreover, since for p > 5/4, 2 t E[ t I n ( f n F t ] p t t E[ t I n ( f n F t ] t in L. (4.2 δ t p+1/4 E[ t I n ( f n F t ] p+1/4 t (by Proposition (p 5/4(n 1 (n! 1/2 δ t p+1/4 f n (t 0 t (by 4.1 ( 1/2 ( 1/2 2 δ t 2 1 t 2 2(p 5/4(n 1 F 0 < +, we see that the ientity (4.2 becomes that F = E[F ] + = E[F ] + t E[ t I n ( f n F t ] t t E[ t F F t ] t in L p, where the above integrals are Bochner integrals.

11 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 11 Combining Theorem 4.2 with Theorem 3.1 yiels that, for any ϕ L 2 (S, Λ, b a t Sϕ(P t(g t ( ( = S t E[ t F F t ] t (P b (g S = = = b b a S ( t E[ t F F t ] (P b (g t t E[ t F F t ] t (P a (g S ( t E[ t F F t ] (P a (g t S ( t E[ t F F t ] a (g t S ( t E[ t F F t ] (g t S ( t E[ t F F t ] (g t, a < b, g L 1 L (R. Therefore, by applying Lebesgue s ifferentiation theorem, we arrive at the following result, which can be rea as the S-transform version of the Clark-Ocone formula. Corollary 4.3. Let ϕ L 2 (S, Λ. Then, for any g L 1 L (R, t Sϕ(P t(g = S ( t E[ t ϕ F t ] (g, [Leb] a.e. t R. References [1] K. Aase, B. Øksenal, N. Privault, J. Ubøe, White noise generalizations of the Clark- Haussmann-Ocone theorem with application to mathematical finance, Finance Stochast. 4 (2000, [2] J. M. C. Clark, The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Stat. 41 (1970, [3] M. e Faria, M. J. Oliveira, an L. Streit, A generalize Clark-Ocone formula, Ranom Oper. an Stoch. Equ. 8 No.2 ( [4] K. Itô, Spectral type of shift transformations of ifferential process with stationary increments, Trans. Amer. Math. Soc. 81 (1956, [5] I. Karatzas, D. Ocone, J. Li, An extension of Clark s formula, Stochastics an Stochastics Reports 37 (1991, [6] H.-H. Kuo, White Noise Distribution Theory, CRC Press, [7] Y.-J. Lee, H.-H. Shih, The Segal-Bargmann Transform for Lévy Functionals, J. Funct. Anal. 168 ( [8] Y.-J. Lee, H.-H. Shih, Itô formula for generalize Lévy functionals, In: Quantum Information II, Es. T. Hia an K. Saitô, Worl Scientific (2000, [9] Y.-J. Lee, H.-H. Shih, The Clark formula of generalize Wiener functionals, In: Quantum Information IV, Es. T. Hia an K. Saitô, Worl Scientific, 2002, [10] Y.-J. Lee, H.-H. Shih, Analysis of generalize Lévy white noise functionals, J. Funct. Anal. 211 (2004, [11] Y.-J. Lee, H.-H. Shih, A characterization of generalize Lévy functionals, In: Quantum Information an Complexity, Worl Scientific, 2005, [12] G. D. Nunno, B. Øksenal, F. Proske, White noise analysis for Lévy processes, J. Funct. Anal. 206 (2004, [13] D. Ocone, Malliavin s calculus an stochastic integral representations of functionals of iffusion process, Stochastics. 12 (1984, [14] K. R. Parthasarathy, An Introuction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992.

12 12 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH [15] N. Privault, An extension of stochastic calculus to certain non-markovian processes, Prépublication 49, Université Evry, [16] N. Privault, Stochastic Analysis in Discrete an Continuous Settings, Lect. Notes in Math., vol. 1982, Springer-Verlag, Berlin, Yuh-Jia Lee: Department of Applie Mathematics, National University of Kaohsiung, Kaohsiung, Taiwan 811 aress: Nicolas Privault: School of Physical an Mathematical Sciences, Nanyang Technological University, SPMS-MAS, 21 Nanyang Link, Singapore aress: Hsin-Hung Shih: Department of Applie Mathematics, National University of Kaohsiung, Kaohsiung, Taiwan 811 aress: