Acta Applicandae Mathematicae 63: 219–231, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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Donsker’s Delta Function of Lévy Process YUH-JIA LEE and HSIN-HUNG SHIH Department of Mathematics, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C. (Received: 31 March 1999) Abstract. Let X = {X(t) : t ∈ R} be a Lévy process and β a non-decreasing, right continuous, bounded function with β(−∞) = 0 (((1 + u2 )/u2 )dβ(u) is the Lévy measure). In this paper we define the Donsker delta function δ(X(t) − a), t > 0 and a ∈ R, as a generalized Lévy functional under the condition that β(0)−β(0−) > 0. This leads us to define F (X(t)) for any tempered distribution F , and as an application, we derive an Itô formula for F (X(t)) when β has jumps at 0 and 1. Mathematics Subject Classifications (2000): primary: 60H40, 46F25; secondary: 60B05. Key words: white noise, Lévy process, Donsker’s delta function.

1. Introduction Let δ be the Dirac delta function and {B(t)} a Brownian motion in R. It is well known that Donsker’s delta function δ(B(t)−a), t > 0 and a ∈ R, may be realized as a generalized functional in (Gaussian) white noise analysis (see, e.g., [2, 9] and the papers cited there). In definition, the Donsker’s delta function may be defined by the following formula Z +∞ 1 δ(B(t) − a) = eir(B(t )−a) dr. (1.1) 2π −∞ In this paper we are interested in defining Donsker’s delta function for a nonGaussian Lévy process. Let β be a nondecreasing, right continuous, bounded function with β(−∞) = 0 and β(0) − β(0−) > 0. Let X = {X(t) : t ∈ R} be a Lévy process in one dimension with X(0) = 0 with the Lévy measure being given by ((1 + u2 )/u2 ) dβ(u) (see (2.1)). We shall always assume that {X(t)} is right continuous with left limit. In this paper, we adapt the Lévy white noise calculus scheme to define and study Donsker’s delta function of the Lévy process. We shall show that Donsker’s delta function δ(X(t) − a) of X may be realized as a generalized Lévy functional. To start with, in Section 2 we give a brief introduction to the analysis of Lévy functionals and then, in Section 3, we introduce the spaces of test and generalized Lévy functionals. For details, we refer the reader to Lee and Shih [12, 13]. Donsker’s delta function δ(X(t) − a) of X is defined as a generalized Lévy functional in Section 4. When {B(t)} is replaced by {X(t)}, we show that the identity (1.1) also makes sense as a Bochner integral and, as a consequence, we define

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F (X(t)) as a generalized Lévy functional for a tempered distribution F . To demonstrate an application, in Section 5 we derive an Itô formula of F (X(t)) for a simple Lévy process X by which we mean that the corresponding function β has the form p1[0,∞) + q1[1,∞) , p, q > 0. 2. Preliminaries Let X = {X(t) : t ∈ R} be a real-valued Lévy process with X(0) = 0, that is, X is an additive process, continuous in probability and the characteristic function of the increment X(t) − X(s) (t > s) is given by exp{(t − s)fX (r)}, where  Z +∞  1 + u2 iru fX (r) = iµr + eiru − 1 − dβ(u). (2.1) 1 + u2 u2 −∞ Here the function β is a nondecreasing, right continuous, bounded function with β(−∞) = 0; and µ is a real constant. Let S be the Schwartz space of rapidly decreasingRfunctions and S 0 the space of tempered distributions on R. Under the +∞ condition −∞ |u| dβ(u) < +∞, S 0 carries a probability measure 3 on (S 0 , B) (see [12]) such that Z +∞  Z ei(x,η) 3(dx) = exp fX (η(t)) dt , η ∈ S, (2.2) S0

−∞

0

where (·, ·) is the S -S pairing; B the Borel field of S 0 . Throughout R +∞ this paper, we assume that the function β satisfies the moment condition −∞ |u|n dβ(u) < ∞ for any n ∈ N. Then (S 0 , B, 3) serves as the underlying probability space. In addition, we shall assume that (S 0 , B, 3) is complete; otherwise, we replace it by its completion. For each η ∈ S, the mean E[(·, η)] and the variance Var[(·, η)] of the random variable (·, η) are given by Z +∞ Z +∞ E[(·, η)] = κ1 η(t) dt and Var[(·, η)] = κ2 η(t)2 dt, (2.3) −∞

where

Z

κ1 = µ +

−∞

Z

+∞

u dβ(u) −∞

and

κ2 =

+∞

−∞

(1 + u2 ) dβ(u).

Let ρ ∈ L ∩ L (R, dt) and let {ηn } be a sequence in S so that ηn → ρ in L1 ∩ L2 (R, dt) with respect to the norm | · |L1 (R,dt ) + | · |L2 (R,dt ). Then, by (2.3), {(·, ηn )} forms a Cauchy sequence in L2 (S 0 , 3). Define h·, ρi by the L2 -limit of {(·, ηn )}. Then h·, ρi is defined almost everywhere as a random variable with mean and variance as those given in (2.3) with η being replaced by ρ. When ρ is the indicator 1(s,t ] of (s, t], the characteristic function of h·, ρi is exactly exp{(t −s)fX (1)}. Thus we may represent the Lévy process X on (S 0 , B, 3) by  hx, 1[0,t ] i, if t > 0; X(t; x) = −hx, 1[t,0] i, if t < 0, x ∈ S 0 . 1

2

´ DONSKER’S DELTA FUNCTION OF LEVY PROCESS

221

˙ x) = x(t) for x ∈ S 0 which are independent and form Note that, formally, X(t; a generalized coordinate system. Thus, members of L2 (S 0 , 3) are also referred as Lévy white noise functionals. For any real locally convex space V , Vc will denote the complexification of V . For η = η1 + iη2 ∈ L1c ∩ L2c (R, dt) with η1 , η2 ∈ L1 ∩ L2 (R, dt), we define h·, ηi as h·, η1 i + ih·, η2 i. Then hx, ηi is defined for a.e. x ∈ S 0 such that its mean and variance satisfy the equalities (2.3). 2.1.

THE LÉVY– ITÔ INTEGRALS

For notational convenience, we denote the measure associated with β also by β. Also, we denote R \ {0} (resp. R2 \ {(t, 0) : t ∈ R}) by R∗ (resp. R2∗ ); and let Bb (R2∗ ) be the class of all bounded Borel sets E ⊆ R2∗ , away from the t-axis. For E ∈ Bb (R2∗ ), let N(E; ·) be a random variable on (S 0 , B) defined by N(E; x) = {(t, u) ∈ E : X(t; x) − X(t−; x) = u} . N(E; ·) is Poisson distributed with intensity measure ν, where ν is the measure dt⊗ (1 + u2 /u2 ) dβ(u) on B(R2∗ ) and the family {N(E; ·) − ν(E) : E ∈ Bb (R2∗ )} of random measures are independent with zero mean. For simplicity, we shall denote ‘N(E; ·) − ν(E)’ by ‘N0 (E; ·)’ for E ∈ Bb (R2∗ ). Let B = {B(t) : t ∈ R} be a process defined by Z σ B(t) := X(t) − κ1 t − u1[0∧t,0∨t ] (s) dN0 (s, u), (2.4) R2∗

where σ 2 = β(0) − β(0−), and the integral on the right-hand side of (2.4) is a stochastic integral with respect to N0 . Then B is a one-dimensional Wiener process which is independent of the system {N(E) : E ∈ Bb (R2∗ )} (see [4]). Let λ be a positive measure on B(R2) given by dλ = dt ⊗ (1 + u2 ) dβ(u). Define a L2 (S 0 , 3)-valued function M on {E ∈ B(R2 ) : λ(E) < +∞} by Z +∞ Z M(E) = σ 1E (t, 0) dB(t) + u1E (t, u) dN0 (t, u). (2.5) R2∗

−∞

Then the family of random measures {M(E; ·) : E ∈ B(R2 )} is also indepenbn b dent with zero mean. According to Itô [7], for φn ∈ L2c (R2 , λ)⊗ (⊗ denotes the symmetric tensor), the nth order Lévy–Itô integral In (φn ) of φn is defined as Z Z In (φn ) = ··· φn (s1 , . . . , sn ) dM(s1 ) . . . dM(sn ). R2

R2

Then every ϕ ∈ L2 (S 0 , 3) enjoys the following orthogonal decomposition ϕ = E[ϕ] +

∞ X n=1

⊕In (φn ),

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YUH-JIA LEE AND HSIN-HUNG SHIH

known as the Lévy–Itô decomposition. In notation, we write ϕ ∼ (φn ). Moreover, we have the following isometry: E[|ϕ| ] = |E[ϕ]| + 2

2

∞ X

n!|φn |2L2 (R2 ,λ)⊗bn .

(2.6)

c

n=1

2.2.

THE T - TRANSFORM AND S - TRANSFORM

As in the cases of Brownian functionals or Poisson functionals (see [6, 7]), the S-transform and T -transform for a function ϕ ∈ L2 (S 0 , 3) also play an important role in the study of generalized Lévy functionals. Their definitions are given as follows: DEFINITION 2.1 [12]. Let 2 M = {g ∈ L2 (R2 , λ) : g ∗ ∈ L1c ∩ L∞ c (R∗ , ν)}

(g ∗ (t, u) := ug(t, u))

and let ϕ ∈ L2 (S 0 , 3). (i) The T -transform T ϕ(η) of ϕ for η ∈ L1 ∩ L2 (R, dt) is defined by Z T ϕ(η) = ϕ(x)eihx,ηi 3(dx). S0

(ii) The S-transform Sϕ of ϕ from L2c (R2 , λ) into C is defined by Z Sϕ(g) = ϕ(x) Exp(g)(x)3(dx). S0

Here Exp(g) = γg · ϑg , if g ∈ M, where γg and ϑg are given by  Z  Y   ∗ γg (y) = exp − 1 + g ∗ (t, X(t; y) − X(t−; y)) , g (x) dν(x) R2∗

t ∈JX (y)

in which JX (y) = {t ∈ R : X(t; y) − X(t−; y) 6= 0}, and   Z +∞ Z σ 2 +∞ 2 ϑg (y) = exp − g(t, 0) dt + σ g(t, 0) dB(t; y) 2 −∞ −∞ for y ∈ S 0 ; otherwise, Exp(g) is defined as a L2 -limit of {γgm · ϑgm } for an approximating sequence {gm } ⊂ M with gm → g in L2 (R2 , λ). In white noise analysis, S-transform and T -transform are related by the formula: Sϕ(η) = E[eih·,ηi ]T ϕ(−iη),

η ∈ L1 ∩ L2 (R, dt).

For Lévy processes including both Brownian and Poisson parts, a similar relation has been proved in Lee and Shih [12] as follows:

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THEOREM 2.2 [12]. Let ϕ ∈ L2 (S 0 , 3) and η ∈ L1 ∩ L2 (R, λ). Then T ϕ(η) = E[eih·,ηi ] · Sϕ(φη ), where φη : R2 → C is defined by φη (t, u) = (eiuη(t ) − 1)/u if u 6= 0; if u = 0, we define φη (t, u) = iη(t).

3. Test and Generalized Lévy Functionals Let A denote the self-adjoint extension in L2 (R, dt) of Aρ(t) = −ρ 00 (t) + (1 + t 2 )ρ(t), ρ ∈ S. Let {en : n ∈ N0 } the complete orthonormal basis (CONS) of L2 (R, dt) consisting of eigenfunctions of A with corresponding eigenvalues {2n + 2 : n ∈ N0 }. For any p ∈ R and η ∈ L2 (R, dt), define |η|p as |Ap η|L2 (R,dt ). Let Sp be the completion of the class {η ∈ L2 (R, dt) : |η|p < +∞} with respect to | · |p -norm. Then Sp forms a real separable Hilbert space and we have the dense inclusion: S = pr lim Sr ⊂ Sp ⊂ Sq ⊂ L2 (R, dt) ⊂ S−q ⊂ S−p ⊂ S 0 = ind lim S−r , r→∞

r→∞

where p > q > 0. (‘pr lim’ and ‘ind lim’ denote ‘projective limit’ and ‘inductive limit’, respectively.) Next, we construct a Gel’fand triple on L2 (R, β0 ) as follows, where dβ0 = (1 + u2 ) dβ(u). The Lusin theorem (see [15]) assures that {1, u, u2 , . . .} (not necessarily infinite) is total in L2 (R, β0 ). Applying the Gram–Schmidt orthogonalization process to {1, u, u2 , . . .}, one obtains a CONS {ζ0 , ζ1 , . . .} of L2 (R, β0 ). Thus, {en ⊗ ζm : n, m ∈ N0 } forms a CONS of L2 (R2 , dλ). Let Aβ be a linear operator densely defined on L2 (R, β0 ) such that Aβ ζn = (2n + 2)ζn , n = 0, 1, . . .. For each p p > 0, define |ψ|p,β as |Aβ ψ|L2 (R,β0 ) , and let Ep = {ψ ∈ L2 (R, β0 ) : |ψ|p,β < +∞}. Then Ep is a real separable Hilbert space, and let E = pr limp→∞ Ep . Then E is a nuclear space and E ⊂ L2 (R, β0 ) ⊂ E 0 = ind limp→∞ Ep0 forms a Gel’fand triple. Moreover, we have E ⊂ Ep ⊂ Eq ⊂ L2 (R, dt) ⊂ Eq0 ⊂ Ep0 ⊂ E 0 , −p

where p > q > 0 and the norm on Ep0 ’s are given by |ψ|−p,β = |Aβ ψ|L2 (R,β0) . Now, for p ∈ R, let Np be the Hilbert space tensor product Sp ⊗ Ep with norm |·|p;p,β defined by |en ⊗ζm |p;p,β = |en |p ·|ζm |p,β ; and let N = S ⊗E, N 0 = S 0 ⊗E 0 . Then N is a nuclear space induced by the family {(Sp ⊗ Ep , | · |p;p,β ) : p > 0}, and again we have a Gel’fand triple N ⊂ L2 (R2 , λ) ⊂ N 0 such that the inclusions N ⊂ Np ⊂ L2 (R2 , λ) ⊂ N−p ⊂ N 0 , p > 0, are all continuous.

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To construct the spaces of test and generalized functions, we note that for ϕ ∈ L2 (S 0 , 3), Sϕ is an analytic function on L2c (R2 , λ) with the nth Fréchet derivative D n Sϕ(g) being of n-linear Hilbert–Schmidt type on L2c (R2 , λ). Further, kϕk2L2 (S 0 ,3) =

∞ X 1 n |D Sϕ(0)|2Ln [L2 (R2 ,λ)] , (2) n! n=0

where Ln(2)[H ] is the space of all n-linear Hilbert–Schmidt operator on a Hilbert p space H . Let 0p be the second quantization of Ap ⊗Aβ , p ∈ R; and define kϕkp = k0p (ϕ)kL2 (S 0 ,3) . Let Lp be the completion of the class {ϕ ∈ L2 (S 0 , 3) : kϕkp < +∞} with respect to k · kp -norm. Then Lp , p ∈ R, forms a separable Hilbert space and let L = pr limp→∞ Lp . Then L is a nuclear space. Moreover, we have the continuous inclusion: L ⊂ Lp ⊂ Lq ⊂ L2 (S 0 , 3) ⊂ L−q ⊂ L−p ⊂ L0 = ind lim L−r , r→∞

where p > q > 0. L will serves as the space of test functions in our investigation and members of the dual space L0 of L are called generalized Lévy white noise functions. In the sequel, the bilinear pairing of L0 and L will be denoted by hh·, ·ii. b

⊗n PROPOSITION 3.1 ([13]). (i) For p > 0 and F ∈ L−p , there exists Fn ’s ∈ N−p,c such that

hhF, ϕii =

∞ X

n!(Fn , φn )

and

kF k2−p =

n=0

∞ X

n!|Fn |2N ⊗bn ,

n=0

−p,c

b

bn ⊗n ⊗ where ϕ ∼ (φn ) ∈ Lp and (·, ·) is the N−p,c -Np,c pairing. We write it as F ∼ (Fn ). (ii) For g ∈ Np with p ∈ R, let EM (g) ∼ (1/n!g ⊗n ). Then EM (g) ∈ Lp and

kEM (g)kp = e 2 |g|p;p,β . 1

2

We note that, when p > 0, EM (g) = Exp(g) for g ∈ Np,c . According to Proposition 3.1(ii), we define the S-transform SF of F ∈ L−p , p ∈ R, as a complexvalued function on Np,c by SF (g) = hhF, EM (g)ii. Then SF is an analytic function on Np,c satisfying the growth condition: 1

|SF (g)| 6 kF k−p e 2 |g|p;p,β , 2

g ∈ Np,c , p ∈ R.

(3.1)

Moreover, we have the following characterization theorem. THEOREM 3.2 ([13]). Suppose that a complex-valued function G is analytic on Np,c and satisfies the following growth condition: 1

∃p ∈ R, ∃c > 0 3 |G(ξ )| 6 c · e 2 |ξ |−p;−p,β , 2

ξ ∈ Nc .

Then there exists a unique F ∈ Lp−1 such that SF = G.

´ DONSKER’S DELTA FUNCTION OF LEVY PROCESS

225

4. Donsker’s Delta Function of Lévy Processes In this section, we always assume that σ 2 = β(0) − β(0−) > 0. Let f ∈ S. Then the inversion formula of Fourier transform implies that Z +∞ 1 f (X(t) − a) = √ fˆ(r) eir(X(t )−a) dr, 2π −∞ √ R +∞ where fˆ(r) = 1/ 2π −∞ f (s) e−irs ds. Clearly, f (X(t) − a) ∈ L1 (S 0 , 3). Let t > 0 be fixed and regard f (X(t)) as a generalized function in L0 . Then, for ϕ ∈ L, Z hhf (X(t) − a), ϕii = f (X(t; x) − a)ϕ(x)3(dx) S0 Z +∞ Z 1 = √ fˆ(r)ϕ(x) eir(X(t ;x)−a)3(dx) dr 2π −∞ S 0 Z +∞ 1 = √ fˆ(r) e−ira T ϕ(r1[0,t ] ) dr 2π −∞ Z +∞ = fˆ(r) e−ira Gt,ϕ (r) dr, √

−∞

where Gt,ϕ (r) = 1/ 2π T ϕ(r1[0,t ] ). Choose a sequence {δn } ⊂ S such that δn → δ in S 0 . Then δbn → δˆ in S 0 . We shall show that Donsker’s delta function δ(X(t)−a), t > 0 and a ∈ R may be defined by Z +∞ hhδ(X(t) − a), ϕii := lim δbn (r) e−ira Gt,ϕ (r) dr. (4.1) n→∞

−∞

We have to show that (i) the limit in (4.1) does exists, (ii) such a limit is independent of the choice of the sequence {δn }; and (iii) δ(X(t) − a) ∈ L0 . The proof will be accomplished in Theorem 4.3. By Theorem 2.2 we have Gt,ϕ (r) =

E[eirX(t )] √ × Sϕ(φt (r)), 2π

where φt : R → L2 (R2 , λ) is given by eiru1[0,t] (s) − 1 , u if u 6= 0; otherwise, φt (r) = ir1[0,t ] (s). φt (r)(s, u) =

LEMMA 4.1. Let ϕ ∈ L and t > 0 be fixed. Then Sϕ(φt (·)) is infinitely differentiable on R. Moreover, for any n ∈ N, there exists a constant c(n) > 0, depending only on n, such that for any p > 0 and r ∈ R, n d 2−(4p+1) c(n)(r 2 +1)t . (4.2) dr n Sϕ(φt (r)) 6 c(n)kϕkp e

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YUH-JIA LEE AND HSIN-HUNG SHIH

Proof. Let p > 0 be fixed. It is easy to see that φt is an infinitely differentiable N−p,c -valued function on R. Indeed, for any r ∈ R and m ∈ N, m d 1 dm φ (r) 6 φ (r) t t dr m 2 2 4p dr m −p;−p,β L (R ,λ) √ Z +∞ 1/2 t 6 p u2m−2 (1 + u2 ) dβ(u) (4.3) 4 −∞ and

√ 1 |r| κ2 t |φ (r)| 6 , (4.4) t L2 (R2 ,λ) 4p 4p where κ2 is defined in (2.3). Observe that the nth derivative of Sϕ(φt (r)) with respect to r can be expressed as a linear combination of the terms of the form |φt (r)|−p;−p,β 6

D k Sϕ(φt (r))(φt(l1 ) (r), . . . , φt(lk ) (r)), where 1 6 k, l1 , . . . , lk 6 n and D denotes the Fréchet derivative of Sϕ(·). By the Cauchy integral formula, we have for g, g1 , . . . , gn ∈ N−p,c and m ∈ N, |D m Sϕ(g)(g1 , . . . , gm )|    m X (m + 1)2 2 2 6 kϕkp exp |gi |−p;−p,β . |g|−p;−p,β + 2 i=1

(4.5)

Apply the estimations (4.3) and (4.4) to (4.5), we obtain the inequality (4.2).

2

PROPOSITION 4.2. For an arbitrary ϕ ∈ L and t > 0, Gt,ϕ ∈ Sc . Moreover, for any q > 0 and 0 < a < b < +∞, there exists a positive real number p and a constant c > 0, depending only on q, a, b, such that sups∈[a,b] |Gs,ϕ |q 6 ckϕkp . Proof. We note that Gt,ϕ (r) = f1 (r)f2 (r)f3 (r)Sϕ(φt (r)), where eirt κ1 f1 (r) = √ , 2π and

 Z f3 (r) = exp t

1

f2 (r) = e− 2 σ

iru

(e |u|>0

for r ∈ R. Then dn Gt,ϕ (r) dr n X = α1 +···+α4 =n α1 ,...,α4 ∈N∪{0}

2r 2t

,

1 + u2 − 1 − iru) dβ(u) u2



 α4  n! d (α3 ) (α1 ) (α2 ) Sϕ(φt (r)) . f (r)f2 (r)f3 (r) α1 ! · · · α4 ! 1 dr α4

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´ DONSKER’S DELTA FUNCTION OF LEVY PROCESS 1

2 2

It is easy to see that f3(α3 ) is a bounded function and limr→∞ f2(α2 ) (r)e 4 σ r t = 0. Then, by applying (3.1) and Lemma 4.1, there exists a constant d(n) > 0, depending only on n, a, b, such that n 2 d (2−(4p+1) d(n)− σ4 )r 2 t , ∀p > 0. (4.6) dr n Gt,ϕ (r) 6 d(n)kϕkp e Choose a positive real number p such that 2−(4p+1) d(n) − (σ 2 /4) < 0. Then we have, ∀n, m ∈ N, lim (1 + r 2 )m

r→∞

dn Gt,ϕ (r) = 0. dr n

This proves that Gt,ϕ ∈ Sc . Clearly, for any q > 0, there exist nq ∈ N and cq > 0 so that for every p > 0 sup |Gs,ϕ |q s∈[a,b]

6 cq

α d sup sup sup(1 + r ) α Gs,ϕ (r) dr 2 nq

s∈[a,b] |α|6nq r∈R

2

−(4p+1) d− ˜ σ 4 6 kϕkp cq d˜ sup sup(1 + r 2 )nq e(2

)r 2 s

,

s∈[a,b] r∈R

where d˜ = max|α|6nq d(α). Pick p so large that 2−(4p+1) d˜ − (σ 2 /4) < 0 and let 2

−(4p+1) d− ˜ σ 4 c = cq d˜ sup(1 + r 2 )nq e(2

)r 2 a

.

r∈R

Then sups∈[a,b] |Gs,ϕ |q 6 ckϕkp for any ϕ ∈ L. This completes the proof.

2

As an application of Theorem 4.2, we represent δ(X(t) − a) in terms of a Bochner integral and, as a consequence, we define F (X(t) − a) for any tempered distribution F . THEOREM 4.3. Let t > 0 and a ∈ R be fixed. Then Z +∞ 1 δ(X(t) − a) = eir(X(t )−a) dr, in N−p,c , 2π −∞

(4.7)

for any p > 0 which is sufficiently large so that σ 2 − 2−4p κ2 > 0, where the integral in (4.7) exists in the sense of Bochner integral. Moreover, for any ϕ ∈ L, Z +∞ 1 hhδ(X(t) − a), ϕii = e−ira Gt,ϕ (r) dr. (4.8) 2π −∞

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YUH-JIA LEE AND HSIN-HUNG SHIH

Proof. Let ρ(r) ≡ e−ira Gt,ϕ (r) for r ∈ R. By Theorem 4.2, Gt,ϕ ∈ Sc so that ρ ∈ Sc . Then, for any sequence {δn } ⊆ S 0 such that δn → δ in S 0 , we have δbn → δˆ in S 0 and Z +∞ lim δbn (r) e−ira Gt,ϕ (r) dr n→∞ −∞   ˆ ρ), = lim (δbn , ρ) = lim δbn , ρ = (δ, n→∞

n→∞

√ where (·, ·) is the Sc0 -Sc pairing. Since δˆ = 1/ 2π in S 0 , we obtain the formula (4.8). Next, we note that for any r ∈ R, hheir(X(t )−a), ϕii = e−ira Gt,ϕ (r) = E[eir(X(t )−a)] × Sϕ(φt (r)), by Theorem 2.2. Then from (3.1) and (4.4), it follows that 1

|hheir(X(t )−a), ϕii| 6 kϕkp e− 2 (σ

2 −2−4p κ )r 2 t 2

,

where σ 2 − 2−4p κ2 > 0 by the choice of p. It follows that keir(X(t )−a)k−p 6 e− 2 (σ 1

2 −2−4p κ )r 2 t 2

.

Observe that the term on the right-hand side of the above inequality is Lebesgue integrable on R as a function of r and the formula (4.7) follows. Clearly the formula (4.8) follows from (4.7). 2 For s ∈ R and ϕ ∈ S, let ga,ϕ (s) = hhδ(X(t) − a − s), ϕii.

√ d Then it follows from (4.8) that ga,ϕ (s) = 1/ 2π G t,ϕ (s + a) and then, by Proposition 4.2, ga,ϕ ∈ S. This leads to the following definition: DEFINITION 4.4. Let t > 0 and a ∈ R. For F ∈ S 0 , F (X(t) − a) is defined as a generalized Lévy functional by hhF (X(t) − a), ϕii = (F[s] , hhδ(X(t) − a − s), ϕii), where F[s] means that F acts on the test functions in the variable s.

5. Generalized Itô Formula for a Simple Lévy Process In this section, we assume that {X(t)} is a simple Lévy process by which we mean that the measure β0 is of the form σ 2 δ0 + b1 δ1 , where σ 2 , b1 > 0 and δi is the Dirac measure concentrated on the point i, i = 0, 1. In this case, a function g ∈ L2 (R, β0 ) if and only if g = α0 1{0} + α1 1{1} β0 a.e. for some α0 , α1 ∈ C. In this section, we shall derive an Itô formula for the process {F (X(t))} with F ∈ S 0 .

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Let {ζ0 , ζ1 } is a CONS of L2 (R, β0 ). For any (t, u) ∈ R2 , let δ(t,u) be the evaluating map on L2 (R2 , λ), i.e., X (δ(t,u) , g) = (g, en ⊗ ζm )en (t) ⊗ ζm (u), n,m

where (·, ·) is the Nc0 -Nc pairing. Then δ(t,u) ∈ N−1,c for all (t, u) ∈ R2 . Moreover, we have LEMMA 5.1. For any u ∈ R and g ∈ Nc , there exists a positive real number cu such that, for all αi , ti ∈ R, i = 1, 2, |α1 δ(t1 ,u) − α2 δ(t2 ,u) |−1;−1,β 6 cu |α1 δt1 − α2 δt2 |−1 . Let φt : R → L2 (R2 , λ) be defined as in Section 4. Then, for any g ∈ Nc and r ∈ R,   φt + (r) − φt (r) 2 ir δ − 2b (e − 1)δ , g − irσ (t,0) 1 (t,1)  Z |r|(σ 2 + 2b1 )) t + 6 (|δ(s,0) − δ(t,0), g)| + |(δ(s,1) − δ(t,1), g)|) ds  t Z c0 |g|1 |r| t + 6 |δs − δt |−1 ds, (5.1)  t by Lemma 5.1, where (·, ·) is the Nc0 -Nc pairing and c0 is a constant. From (5.1) it follows that the function t 7→ φt (r) is differentiable with respect to t in | · |−1;−1,β -norm; moreover, d φt (r) = irσ 2 δ(t,0) + 2b1 (eir − 1)δ(t,1). dt

(5.2)

DEFINITION 5.2. For (t, u) ∈ R2 , a linear operator ∂(t,u) from L to L0 is defined by ∂(t,u)ϕ := S −1 (DSϕ(·)δ(t,u)),

ϕ ∈ L.

Note that ∂(t,u) is continuous from L into itself. Moreover, we note that for t1 , t2 ∈ R, q > 0, and ϕ ∈ L, k∂(t1 ,u) ϕ − ∂(t2,u) ϕkq 6 |δ(t1 ,u) − δ(t2 ,u) |−q;−q,β kϕkq+1 . For more details about ∂(t,u), we refer the reader to [13]. It immediately follows from (5.2) that we have the following lemma:

(5.3)

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LEMMA 5.3. Let ϕ ∈ L and Gt,ϕ (r), r ∈ R, t > 0, be defined as in (4.1). Then   d σ 2r 2 ir Gt,ϕ (r) = irκ1 − + 2b1 (e − 1 − ir) Gt,ϕ (r) + dt 2 + irσ 2 Gt,∂(t,0) ϕ (r) + 2b1 (eir − 1)Gt,∂(t,1) ϕ (r), where κ1 is defined in (2.3). Let ϕ ∈ L and u ∈ R be fixed. Consider two Sc -valued mappings ρ1 , ρ2 on R by ρ1 (t) = Gt,ϕ and ρ2 (t) = Gt,∂(t,u)ϕ . Then, by applying (5.3), we have LEMMA 5.4 [14]. ρ1 and ρ2 are both Sc -valued continuous mappings on R. Let F ∈ S 0 and ϕ ∈ L. For 0 < a < b < +∞ and t, t +  ∈ (a, b),   Z +∞ F (X(t + )) − F (X(t)) b(r) · Gt +,ϕ (r) − Gt,ϕ (r) dr, (5.4) F ,ϕ =   −∞ where (·, ·) is the L0 -L pairing and the integral is understood to be the Sc0 -Sc pairing. By Lemma 5.3, the term on the right-hand side of (5.4) becomes Z +∞ Z 1 t + d b F (r) Gs,ϕ (r) ds dr  t ds −∞  Z t + Z +∞  σ 2 c00 1 0 b = κ1 F (r) + F (r) Gs,ϕ (r) ds dr+ 2  t −∞ Z +∞ Z 1 t + 0 (r)) b + 2b1 (τd F (r) − 1 − F Gs,ϕ (r) ds dr+ 1  t −∞ Z +∞ Z t + 1 2 0 +σ Fb (r) Gs,∂(s,0) ϕ (r) ds dr+  t −∞ Z +∞ Z 1 t + d + 2b1 (τ1 F (r) − 1) Gs,∂(s,1) ϕ (r) ds dr,  t −∞ where τ1 F (·) = F (·+1), and F 0 , F 00 are the distributional derivatives of F . Letting  → 0 and applying Lemma 5.4, we obtain the following theorem: THEOREM 5.5 (Itô formula). Let F ∈ S 0 and X be a simple Lévy process, i.e., β0 = σ 2 δ0 + b1 δ1 , σ 2 , b1 > 0. Then d σ2 † F 0 (X(t)) + F (X(t)) = κ1 F 0 (X(t)) + F 00 (X(t)) + σ 2 ∂(t,0) dt 2 + 2b1 ((τ1 F )(X(t)) − F (X(t)) − F 0 (X(t))) + † + 2b1 ∂(t,1) ((τ1 F )(X(t)) − F (X(t))), † where ∂(t,u) is the adjoint operator of ∂(t,u) for u = 0, 1.

(5.5)

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Remark 5.6. The Itô formula can also be derived. For an arbitrary β satisfying σ 2 > 0. Since the computation is more involved, we shall prove it in our forthcoming paper [14].

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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