Product formulas for multiple stochastic integrals associated with Lévy processes

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Product formulas for multiple stochastic integrals associated with Lévy processes © The Author(s) 2024 Published version Di Tella, Paolo; Geiss, Christel; Steinicke, Alexander Di Tella, P., Geiss, C., & Steinicke, A. (2024). Product formulas for multiple stochastic integrals associated with Lévy processes. Collectanea mathematica, Early online. https://doi.org/10.1007/s13348-024-00456-6 2024 Collectanea Mathematica https://doi.org/10.1007/s13348-024-00456-6 Product formulas for multiple stochastic integrals associated with Lévy processes Paolo Di Tella1·Christel Geiss2·Alexander Steinicke3 To the memory of Hans-Jürgen Engelbert ( ∗1944–2021†) Received: 29 October 2023 / Accepted: 23 September 2024 © The Author(s) 2024 Abstract In the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a Lévy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider Lévy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula. Keywords Product formulas for multiple stochastic integrals ·Moment formulas ·Lévy processes ·Central limit theorem Mathematics Subject Classification 60H05 ·60G51 ·60G44 ·60F05 1 Introduction In the present work we consider multiple Brownian–Poisson stochastic integrals, that is, multiple stochastic integrals generated by a random measure that exhibits both a Gaussian and BAlexander Steinicke alexander[email protected] Paolo Di Tella [email protected] Christel Geiss [email protected] 1Institute for Mathematical Stochastics, TU Dresden, Dresden, Germany 2Department of Mathematics and Statistics, University of Jyvaskyla, Jyvaskyla, Finland 3Department of Mathematics and Information Technology, Montanuniversitaet Leoben, Leoben, Austria 123 P. Di Tella et al. a Poisson part. This kind of random measures are associated with Lévy processes. We show in Theorem 4.8 a compact and accessible product formula for the product of N≥2 multiple Brownian–Poisson stochastic integrals, extending previous results obtained for N=2(see for instance [9,17,21,22,25,29,33]). In such formulas, products of multiple stochastic integrals are expressed as a sum of multiple stochastic integrals. To state our product formula, a crucial step is the definition of the star operator l lo(see Definition 4.5). It generalizes the identification and contraction rules introduced e.g. in [8] for N=2, to an arbitrary number N≥2 of multiple integrals. Multiple stochastic integrals play an important role in stochastic analysis because of their use for the chaos expansion and for the definition of the Malliavin derivative. Moreover, their products find applications in certain limit theorems and numerical simulations [29]. The classic example of such a product formula can be found e.g. in Nualart [25, Proposition 1.1.3], where for symmetric functions f∈L2([0,T]n), g∈L2([0,T]m), the product of two multiple integrals w.r.t. the Gaussian measure generated by the Brownian motion is given by In(f)Im(g)= n∧m  r=0 r!n rm rIn+m−2r(f⊗rg). (1) Here, f⊗rgdenotes an operator which ’contracts’ rvariables of fand of g(i.e., they are identified and integrated out). For the extension of (1)toanarbitraryN≥2, we refer to Major [23, Theorem 10.2]. It is well known that for multiple Brownian integrals the product formula always holds for any N≥2, with no need of further integrability conditions on the kernels. Contrarily, for multiple Poisson integrals, i.e., for multiple stochastic integrals generated by a Poisson random measure, this is not true in general and, as a minimal requirement for the product formula, one has to ensure that the product of multiple Poisson stochastic integrals is again square integrable. Even in the elementary case of a compensated Poisson process, the latter condition may be violated: Let Nbe a standard Poisson process and set Lt=Nt−t. For any f∈L2([0,T]), by Itô’s formula, we have ⎛ ⎝ T 0 f(t)dLt⎞ ⎠ 2 =2 T 0 f(t)⎛ ⎝ t− 0 f(s)dLs⎞ ⎠dLt+ T 0 f(t)2dLt+ T 0 f(t)2dt, which is square integrable if and only if f∈L4([0,T]). Interpreting the left-hand side as the product of two multiple stochastic integrals of order one, we see that this is not square integrable, and hence does not have a chaos decomposition, unless f∈L4([0,T]). The first product formula was established by Itô [14] in 1951 and corresponds to (1) for an arbitrary nbut for m=1. Kabanov [17] extended this formula to multiple Poisson stochastic integrals (keeping m=1). Surgailis [33] proved a product formula for N≥2 multiple Poisson integrals in 1984 and asked whether the square integrability of their product is a sufficient condition for the product formula to hold. Recently, in [8, Theorem 2.2], Döbler and Peccati gave a positive answer for the case of two multiple Poisson integrals. However, the case of an arbitrary number of factors seems to be still open and, in the present paper, we formulated this problem in Conjecture 4.9.In[32], Russo and Vallois obtained a representation formula for the product of two multiple integrals of arbitrary order associated with a normal martingale.1A quantum version of this formula has been obtained by Privault, 1A martingale Xis called normal if it is square integrable and (X2 t−t)t≥0is a martingale too. Normal martingales need not have independent increments. 123 Product formulas for multiple stochastic... Solé and Vives in [31]. Lee and Shih were the first authors who in [21] unified the Brownian and the Poisson case considering product formulas for Brownian–Poisson multiple integrals. Product formulas for multiple integrals have been obtained in the literature by several methods. For example, in [22,29,33]diagram formulae are used. In [25]and[18], the formulas are obtained by induction on the order of the multiple integrals. In [24]and[21], the relation between the boson Fock space and the representation of the stochastic exponential of I1(f)is exploited. In [1], the properties of the stochastic exponential are used to derive a formal expression for the product formula for N≥2 multiple Poisson stochastic integrals. Some papers, as e.g. [19]or[2], obtain explicit moment and cumulant formulas for Poisson integrals relying on Mecke’s formula rather than on product formulas. In [19], the authors study multiple Poisson integrals and in [2] the main goal is to consider stochastic integrands instead of deterministic ones (see also [30]). In particular, Theorem 3.1 of [2] provides a joint moment formula for single Poisson stochastic integrals with random integrands, whereas here the moment formula in Sect.5.2 considers deterministic integrands. In the present paper we use, like in [7], the properties of compensated-covariation stable families of martingales. This allows us to directly show a product formula for N≥2multiple Brownian-Poisson stochastic integrals associated with arbitrary Lévy processes. In [7]a product formula for iterated integrals with respect to integrators belonging to certain families of Lévy martingales was shown. Since the multiple integrals considered here exhibit better symmetry properties that allow, e.g. the definition of the star operator l lo(see Sect.4.2 below), to our opinion it is more effective to study multiple integrals independently, that is without using the results from [7] on iterated integrals. Moreover, to obtain the product formulas for the multiple integrals from the one obtained in [7], one has to use a dominated convergence theorem. For this, we have to require integrability conditions for the kernels, that would turn out to be more restrictive than those we use here. Obviously, a moment formula can be immediately obtained by taking the expectation in our product formula. The present paper has the following structure: Sect. 2is devoted to multiple BrownianPoisson stochastic integrals. Section3recalls a representation formula for products of compensated-covariation stable families of martingales that we apply in Sect.4to obtain a first product formula for iterated integrals. This product formula is then transformed into one for multiple integrals and extended to the general setting of Theorem 4.8. The latter is applied in Sect.5to obtain moment and cumulant formulas for stochastic integrals. These formulas are then exploited to show a central limit theorem that describes the long time behaviour of multiple integrals of order one for a class of integrands. Finally, in Sect.6we provide several methods of algorithmic interest to count the number of summands contained in the product formula from Sect.4. 2 Multiple integrals Let L=(Lt)t≥0be a càdlàg one-dimensional Lévy process with characteristic triplet (γ, σ 2,ν)on a complete probability space (, F,P). The augmented natural filtration of L will be denoted by (Ft)t≥0.WenowfixT>0, restrict our analysis to [0,T]and assume that F=FT. 123 P. Di Tella et al. The Lévy-Itô decomposition of Lreads Lt=γt+σWt+ (0,t]×{|x|≤1} x˜ N(ds,dx)+ (0,t]×{|x|>1} xN(ds,dx), where γ∈R,σ ≥0, Wis a standard Brownian motion and N(˜ N) is the (compensated) Poisson random measure corresponding to L. The random measure Mgiven by M(dt,dx)=σdWtδ0(dx)+˜ N(dt,dx), where δ0denotes the Dirac measure concentrated in zero, is used to define the following iterated integrals:WesetR0:= R\{0}and m(dt,dx):= dt(σ 2δ0(dx)+ν(dx)). For k≥1 we put f∈Lp k:= Lp(([0,T]×R)k,B(([0,T]×R)k), m⊗k), for p≥1, and denote ·Lp kthe Lp k-norm of f.Wedenoteby ˆ f((t1,x1),...,(tk,xk)) the symmetrisation of the kernel f((t1,x1),...,(tk,xk)) w.r.t.the kpairs of variables: ˆ f((t1,x1),...,(tk,xk)) =1 k! π∈Sk f((tπ(1),xπ(1)),...,(tπ(k),xπ(k))), (2) where Skis the set of all permutations πof {1,...,k}.We introduce the iterated integrals as follows: J0:= the identity map on R J1(f)T:=  (0,T]×R f(t,x)M(dt,dx), f∈L2 1, J2(ˆ f)T:=  (0,T]×R (0,t2)×R ˆ f((t1,x1), (t2,x2))M(dt1,dx1)M(dt2,dx2), f∈L2 2, Jk(ˆ f)T:=  (0,T]×R Jk−1(ˆ f(...,(tk,xk)))tk−M(dtk,dxk), f∈L2 k. To define multiple integrals w.r.t. M, one considers the linear space Ekof simple functions of the form g(z1,...,zk)= n  j1,..., jk=1 aj1,..., jk1Aj1×···×Ajk(z1,...,zk), zi:= (tj,xj), (3) where the A1,...,An∈B([0,T]×R)are disjoint and such that m(Aj)<∞for all j=1,...,n.Moreover, aj1,..., jkis zero whenever two or more of the indices j1,..., jk coincide. We then put Ik(g)T:= n  j1,..., jk=1 aj1,..., jkM(Aj1)···M(Ajk). It holds that EM(Aj)=0andEM(Aj)2=m(Aj), for j=1,...,n,and the M(A1),...,M(An)are independent. In the next lemma we recall some properties of Jkand Ik. 123 Product formulas for multiple stochastic... Lemma 2.1 Let k,m∈Nand f ∈L2 k,g∈L2 m.Wethenhave: 1. The space Ekis dense in L2 k, and Ik:Ek→L2(P)is a linear operator. The map Ikhas a unique extension Ik:L2 k→L2(P). 2. EIk(f)T=0,Ik(f)TL2(P)=k! ˆ fL2 k≤k!fL2 k,and EIk(f)TIm(g)T=0if m= k. 3. Ik(ˆ f)T=Ik(f)Tand Ik(ˆ f)T=k!Jk(ˆ f)T. Proof (1) By [14, Theorem 2.1] we have that Ekis dense in L2 k.In [14] it is also shown that (2) holds for all f∈Ek, and therefore, by linearity and continuity, Ikhas a unique extension to L2 k. Also (3) is first shown for all f∈Ekand then extended to L2 k. For convenience and later use we recall the proof of the relation between Ikand Jkin (3). We follow [21]andstart assuming g(z1,...,zk)=N j=1aj1Aj 1×···×Aj k (z1,...,zk)for measurable and pairwise disjoint Aj 1,...,Aj ksatisfying m(Aj l)<∞for l=1,...,k. Moreover, we assume g(z1, ..., zk)=0for(z1,...,zk)/∈T(k),where T(k):= {((t1,x1),...,(tk,xk)) :0≤t1<···<tk<T,x1,...,xk∈R}(4) so that, if (t1,x1)∈Aj iand (t2,x2)∈Aj l,thent1<t2for i<l.By (2) and the definition of g,weget Ik(ˆg)T=Ik⎛ ⎝1 k! π∈Sk N  j=1 aj1Aj π(1)×...×Aj π(k)⎞ ⎠T =1 k! π∈Sk N  j=1 ajM(Aj π(1))···M(Aj π(k))= N  j=1 ajM(Aj 1)···M(Aj k)=Ik(g)T. On the other hand, the definition of the iterated integrals yields Jk(ˆg)T=1 k! N  j=1 ajM(Aj 1)···M(Aj k)=1 k!Ik(g)T. This extends to symmetric kernels ˆ f∈L2 k,hence the iterated integrals and the multiple integrals are related by Ik(ˆ f)T=k!Jk(ˆ f)T. We recall Itô’s chaos expansion result: Theorem 2.2 ([15]) There exists for any ξ∈L2(, F,P)a unique chaos expansion ξ=∞  k=0 Ik(ˆ f)T, where ˆ f∈L2 k,and I0is the identity map on R. Our aim is to show a product formula, that means we want to find conditions on the (f(j))N j=1, where f(j)∈L2 kj,such that the product N  j=1 Ikj(ˆ f(j))T can be written as a sum of square integrable multiple integrals. 123 P. Di Tella et al. 3 Compensated-covariation stability and martingale products To state a representation result for products of martingales, we first need to recall the concept of compensated covariation stable families (see [6, Definition 3.1]): Let be an arbitrary parameter set and X={Xα,α∈}a family of square integrable martingales. Set Xα1:2:= [Xα1,Xα2]−Xα1,Xα2,α 1,α 2∈. Then Xis called compensated-covariation stable if Xα1:2∈Xholds for any α1,α 2∈. Let Xbe such a family. For α1,...,α k∈we can recursively define Xαj1:k:= [Xαj1:(k−1),Xαjk]−Xαj1:(k−1),Xαjk.(5) For a compensated-covariation stable family Xthe following representation formula holds: Proposition 3.1 ([6, Proposition 3.3]) Let {Xα,α∈},be a compensated-covariation stable family of quasi-left continuous square integrable martingales. For every N ≥1and α1,...,αN∈,wehave N  i=1 Xαi t= N  i=1 1≤j1<···<ji≤N t 0N  k=1 k=j1,..., ji Xαk s−dXαj1:i s + N  i=2 1≤j1<...< ji≤N t 0N  k=1 k=j1,..., ji Xαk s−dXαj1:i−1Xαjis.(6) Using the random measure Mfrom §2, we construct compensated-covariation stable families of martingales: For α∈L2 1, we define the square integrable martingale Yα t:= σ t 0 α(s,0)dWs+ [0,t]×R0 α(s,x)˜ N(ds,dx), t∈[0,T]. We will call these processes Engelbert martingales, they are a generalisation of the processes introduced in [5, Equation (39)]. It holds [Yα1,Yα2]t=σ2 t 0 α1(s,0)α2(s,0)ds + [0,t]×R0 α1(s,x)α2(s,x)N(ds,dx) and Yα1,Yα2t=σ2 t 0 α1(s,0)α2(s,0)ds + [0,t]×R0 α1(s,x)α2(s,x)dsν(dx). For =p≥2Lp 1, it is clear that the family X:= {Yα:α∈}is compensatedcovariation stable. Note that, for Yα1,...,Yαk∈X,k≥2, we get the following identity Yα1:k t:= [Yα1:(k−1),Yαk]t−Yα1:(k−1),Yαkt= [0,t]×R0k  j=1 αj(s,x)˜ N(ds,dx). 123 Product formulas for multiple stochastic... Lemma 3.2 The linear hull of the iterated integrals (Jk(ˆ f)t)t∈[0,T]:f=α1⊗···⊗αkwith α1,...,α k∈ p≥2 Lp 1,k=0,1,... is compensated-covariation stable. Proof For symmetric kernels fand gwe have [Jn(f), Jk(g)]T= (0,T]×R Jn−1(f(...,(t,x)))t−Jk−1(g(...,(t,x)))t− ×(σ2dtδ0(dx)+N(dt,dx)) (7) and Jn(f), Jk(g)T= (0,T]×R Jn−1(f(...,(t,x)))t−Jk−1(g(...,(t,x)))t− ×(σ2dtδ0(dx)+dtdν(x)) (8) Now we use the product formula for two factors from Lee and Shih [21, Theorem 3.5] (or [9] for the pure jump case) which states that the product of two multiple integrals of order n and kis equal to a linear combination of multiple integrals of order less or equal to n+k. Moreover, from [21, equation (21)] it can be seen that the kernels of these multiple integrals are built from tensor products of functions from p≥2Lp 1. 4 Product formulas 4.1 Product formula for iterated integrals The representation formula (6) will be used to derive a product formula for N j=1Jmj(f(j))T. First we consider a special case assuming that the kernels are given by tensor products: f(j)=α⊗mj jwith αj∈p≥2Lp 1.We introduce some notation. Let S:= {s={j1,..., ji}:1≤j1<···<ji≤Nfor i=1,...,N}.(9) Then S=2{1,...,N}\{∅}has ηN:= 2N−1 elements which we denote by S={e1,...,eηN}.(10) Proposition 4.1 (Product formula for iterated integrals) Let αj∈p≥2Lp 1for j = 1,...,N. Then N  j=1 Jmj(α⊗mj j)T= k≤m1+···+mN (s1,...,sk)∈Ak  T(k) α(s1)(z1)M(s1)(dz1)...α (sk)(zk)M(sk)(dzk) (11) 123 P. Di Tella et al. where zn:= (tn,xn), and α(sn)(zn):=  ∈sn α(zn), Ak:= (s1,...,sk)∈Sk: k  n=1 1sn(j)=mj,∀j=1, ..., N,(12) and T(k)stands for the ’simplex’ defined in (4). For any sn∈Sthe integrators are given by (here |sn|denotes the cardinality of the set sn) M(sn)(dzn):= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ M(dtn,dxn)if |sn|=1, m(dtn,dxn)+˜ N(dtn,dxn)if |sn|=2, dtnν(dxn)+˜ N(dtn,dxn)if |sn|≥3. (13) The splitting into the cases |sn|=1,|sn|=2and|sn|≥3in(13) can be explained by (7)and (8). Note that the terms related to the Brownian part vanish for |sn|≥3 due to the iterated use of (5). Proof We use (6)for Xαj T:= Jmjα⊗mj jT= (0,T]×R Jmj−1α⊗(mj−1) jt−αj(t,x)M(dt,dx). Then Xαj1:2 T= (0,T]×R0 Jmj1−1α⊗(mj1−1) j1t− Jmj2−1α⊗(mj2−1) j2t− αj1(t,x)α j2(t,x)˜ N(dt,dx). (14) Further Xαj1,Xαj2T= (0,T]×R Jmj1−1α⊗(mj1−1) j1t− Jmj2−1α⊗(mj2−1) j2t− αj1(t,x)α j2(t,x) ×m(dt,dx), (15) and for more than two processes we have Xαj1:i T= (0,T]×R0 i  k=1Jmjk−1α⊗(mjk−1) jkt−αjk(t,x)˜ N(dt,dx), (16) Xαj1:(i−1),XαjiT= (0,T]×R0 i  k=1Jmjk−1α⊗(mjk−1) jkt−αjk(t,x)dtν(dx). (17) Then (6) implies that N  j=1 Jmj(α⊗mj j)T= 2N−1  i=1 (0,T]×RN  n=1 n/∈ei Jmnα⊗mn nt−  jk∈ei Jmjk−1α⊗(mjk−1) jkt−α(ei)(t,x)M(ei)(dt,dx). (18) 123 Product formulas for multiple stochastic... We note that M i=11Rj,i∈Hmj,where Hmjdenotes the linear hull of the set ⊗mj i=1αi:αi∈∩p≥2Lp 1. Moreover, 0 ≤M i=11Rj,i≤1. So we have shown that there exists a sequence f(j) n∞ n=1⊆ Hmjwith |f(j) n|≤1and f(j) n→1Ajin L2 mj,as n→∞,for all j=1...,N.(29) For each nwe have that N j=1Imjf(j) nT∈L2(P)and satisfies (27). To show that the product N j=1Imjf(j) nTis a Cauchy sequence in L2(P)for n→∞we estimate (abbreviating mN:= m1+···+mN) &&&&&& N  j=1 Imjf(j) nT− N  j=1 Imjf(j) mT&&&&&&L2(P) ≤ N  r=1&&&&&& r−1  j=1 Imjf(j) nTImrf(r) n−f(r)T N  j=r+1 Imjf(j) mT&&&&&&L2(P) ≤ N  r=1 k≤mN |l|=k, (l,lo)∈DN m1!···mN! l!lo!&&&Ik(! l lof(1) n,..., f(r−1) n,f(r) n−f(r) m,f(r+1) m, ..., f(N) mT&&&L2(P) = N  r=1 k≤mN |l|=k, (l,lo)∈DN m1!···mN! l!lo!k!&&&! l lof(1) n, ..., f(r−1) n,f(r) n−f(r) m,f(r+1) m, ..., f(N) m&&&L2 k ≤ N  r=1 k≤mN |l|=k, (l,lo)∈DN m1!...mN! l!lo!k!&&&! l lo|f(1) n|,...,|f(r−1) n|,|f(r) n−f(r) m|,|f(r+1) m|,...,|f(N) m|&&&L2 k (30) which converges to zero as n,m→∞by dominated convergence since the integrands are bounded by ! l lo(1A1,...,1AN)and converge in measure to zero. Hence N j=1Imj(f(j) n)Tis a Cauchy sequence in L2(P). On the other hand, (29) implies that Imj(f(j) n)T→Imj(1Aj)T in L2(P)for j=1, ..., N,and consequently N j=1Imj(f(j) n)T→N j=1Imj(1Aj)Tin probability. This gives N j=1Imj(1Aj)T∈L2(P), as it is also the limit of the Cauchy sequence in L2(P). It remains to show that the r.h.s. of (27) (used now for the ˆ f(1) n,..., ˆ f(N) n) converges in L2(P)if (29) holds. This follows from the convergence ! l lo(ˆ f(1) n,..., ˆ f(N) n)−! l lo(ˆ 1A1, ..., ˆ 1AN) in L2 k. Step 2. Let f(i)∈L2 mifor i=1,...,Nsatisfy (26). Especially, then the r.h.s. of (27) is well-defined and in L2(P). We show (27) by approximation as follows. Since (27) holds for indicator functions, it holds for simple functions. We approximate any f(j) by simple functions (g(j) n)∞ n=1such that 0 ≤g(j),± n↑f(j),±.Applying now (30)to &&&N j=1Imj(g(j) n)T−N j=1Imj(g(j) m)T&&&L2(P)we have again convergence to zero by dom123 P. Di Tella et al. inated convergence which holds thanks to (26). Repeating the arguments of Step 1 for the present situation it follows that N j=1Imj(f(j))T∈L2(P)and that (27) holds. The relation 28 follows immediately from 27 by taking k=0. Similar to [9, Theorem 2.2.] which concerns the product of 2 multiple integrals, we expect that there exists an if and only if relation between N j=1Imj(f(j))∈L2(P)and the existence of the product formula. This was already addressed by Surgailis in [33]. Conjecture 4.9 Let f (j)∈L2 mjfor j =1, ..., N. N  j=1 Imj(f(j))Tis square integrable ⇐⇒  |l|=k, (l,lo)∈DN 1 l!lo!! l lo(f(1), ..., f(N))∈L2 k,for each k =1, ..., m1+... +mN, ! l lo(f(1), ..., f(N))is well defined in the sense of Definition 4.5, and the expression is finite for k =0. Moreover, the product formula (27)holds. 5 Applications 5.1 The expectation of a power of a stochastic integral We apply now (27) to obtain explicit formulas for moments and cumulants of I1(f)T.The approach in [29, Chapter 7], considers the Brownian and Poisson setting separately and uses diagram formulae to treat all kinds of power moments and products. This program can of course be conducted here as well – however, for this application, we follow a more elementary way. To point out the connection between the sets enand the numbers lnand lo nmore clearly, we will use the notation lsor leninstead of lnif s=en, and similarly for lo n. First we observe that to compute EI1(f)N T,forsomeN≥2 (which means we have m1= ··· = mN=1), we need to extract the term for k=0in(27). This has the consequence that: •The identity ! 0 lo(f,..., f)=0 lo(f,..., f)holds, since no z-variables are involved in these expressions. •It is only necessary to consider tuples (0,...,0,lo 1,...,lo sN)in DNso that for each j∈{1,...,N},weneedsjlo s=mj=1. The latter point implies that one can establish a bijection between the tuples in D∗ N, i.e. those tuples from DNsuch that l1= ··· = lsN=lo {1}= ··· = lo {N}=0, and the partitions of {1,...,N}of block size ≥2. (We call the elements of the partition blocks and their cardinality block size.) We consider this set of partitions, since by (24), blocks of size 1 do not yield a term in the product formula for k=0. Note that this bijection is one particular case of those that are analyzed and numbered by diagrams and multigraphs in [29]. 123 Product formulas for multiple stochastic... We thus get {Partitions of block size ≥2}→D∗ N,P→ 1P= s∈P 1{s}. For one such partition consisting of qsets excluding singletons, {s1,...,sq}, we denote the block sizes by pi:= |si|. To compute 0 lo(f,..., f),wehavetoinvestigatethevariables (zen,0 1:lo n)that arise in the product N j=1f(zen,0 1:lo n)1≤n≤sN,j∈en. For a given partition {s1,...,sq},if jis contained in si, then the same variable zsi,0 jappears in piof the product’s factors, and altogether there can be only qdifferent variables. Hence we obtain 0 lo(f,..., f)= q  i=1 (0,T]×R f(z)picsi(z)m(dz), and, using (24) again for the sets of block size 2, 0 lo(f,..., f)= (0,T]×R f(z)2m(dz)|{i:pi=2}| · pi>2 (0,T]×R0 f(z)pim(dz). (31) To find how many partitions deliver such a term, we first consider the number of all partitions of {1,...,N}into qblocks of size p1,...,pq(that is, without the restriction pi≥2). If we assume that there are jablocks of size ain the partition, we have the relationships j1+j2+···+jN−q+1=q(and there cannot be more than N−q+1), ja=|{i:pi=a}| and p1+···+pq=j1+2j2+···+(N−q+1)jN−q+1=N. Then the number of such partitions is given by the coefficient bN,( j1,..., jN−q+1)of the monomial Xj1 1···XjN−q+1 N−q+1 in the N-th partial exponential Bell polynomial BN,q(X1,...,XN−q+1)(see, for example, [10, Definition 11.2] or [29, Definition 2.2.1]). In particular, the coefficient is bN,( j1,..., jN−q+1)=N! j1!(1!)j1j2!(2!)j2... jN−q+q!((N−q+1)!)N−q+1. The polynomial containing the information about all partitions i.e.with arbitrarily many blocks is the (exponential) Bell polynomial BN(X1,...,XN):= N q=1BN,q(X1,..., XN−q+1). Since we exclude partitions with block size 1, we only have to consider the polynomial BN(0,X2,...,XN).In(31), each pi-integral factor appears jpitimes. Therefore, we get the following relations: Proposition 5.1 . 1. Let f ∈L2 1∩LN 1.Then EI1(f)N T=BN⎛ ⎜ ⎝0, (0,T]×R f(z)2m(dz),  (0,T]×R0 f(z)3m(dz),...,  (0,T]×R0 f(z)Nm(dz)⎞ ⎟ ⎠, 123 P. Di Tella et al. or more in detail, EI1(f)N T= N  q=1 N−q+1 i=2ji=q N−q+1 i=2ij i=N bN,( j1,..., jN−q+1)⎛ ⎜ ⎝ (0,T]×R f(z)2m(dz)⎞ ⎟ ⎠ j2 ···⎛ ⎜ ⎝ (0,T]×R0 f(z)N−q+1m(dz)⎞ ⎟ ⎠ jN−q+1 . 2. For the Brownian case we recover the known relation EI1(f)N T=BN⎛ ⎜ ⎝0, (0,T] f(s)2σ2ds,0,...,0⎞ ⎟ ⎠ =(N−1)!! (0,T] f(s)2σ2dsN 21{N∈2N}. 3. If f ∈L2 1∩LN 1for all N ∈Nthen the cumulants κNof I1(f)Tare given by κ1=0,κ 2= (0,T]×R f(z)2m(dz), κN= (0,T]×R0 f(z)Nm(dz), N≥3. Proof (1) is clear from the above considerations, (2) is obvious. (3) one gets from (1) by the formula relating cumulants and moments (see [29, Corollary 3.2.2] or [30]).  5.2 Expectations of products of stochastic integrals In the same way as before, we may consider the product of integrals of different functions, EI1(f(1))T···I1(f(N))T. Using partitions as above, we can compute 0 lo(f(1),..., f(N)) by relating lowith the according partition {s1,...,sq}without singletons and get 0 lo(f(1),..., f(N))= q  i=1 (0,T]×R j∈si f(j)(z)m(dz). Taking all partitions into account, we obtain EI1(f(1))T...I1(f(N))T = P∈P∗(N) P={sq,...,sq} ⎛ ⎜ ⎜ ⎝ q  i=1 |si|=2 (0,T]×R j∈si f(j)(z)m(dz)⎞ ⎟ ⎟ ⎠⎛ ⎜ ⎜ ⎝ q  i=1 |si|>2 (0,T]×R0 j∈si f(j)(z)m(dz)⎞ ⎟ ⎟ ⎠. 123 Product formulas for multiple stochastic... 5.3 Long time behaviour and limit theorems In this section, which is inspired by [19, §4], we are going to address the long time behaviour of integrals of the form I1(f)Tand, combining (27) with the method of moments and cumulants (see [26, §A.3]), we deduce a central limit theorem (from now on CLT) for T→+∞in some special cases. Using the properties of the cumulants and setting ' I1(f)T:= I1(f)T E[I2 1(f)T]1/2we get from Proposition 5.1 (3) that κN' I1(f)T=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1,N=2, ( (0,T]×R0 fN(z)m(dz) "( (0,T]×R f2(z)m(dz)#N/2,N≥3. So, if fis measurable on [0,∞)×Rand f∈LN([0,T]×R,m)for all N≥1andfor every T>0 satisfies also lim T→+∞ ( (0,T]×R0 fN(z)m(dz) "( (0,T]×R f2(z)m(dz)#N/2=0,N≥3,(32) then, by [16, Theorem 1] we get that ' I1(f)Tconverges in distribution to X∼N(0,1)as T→+∞. Note that for f≥0 condition (32) can be reformulated in terms of norms. We now consider functions fof the form f(z)=f(t,x)=g(t)h(x),whereg: [0,∞)→Rsatisfies T (0|g(t)|Ndt <∞for all T>0andh∈LN(ρ) for every N≥1.We set ρ(dx)=σ2δ0(dx)+ν(dx). We assume hL2(ρ) =1 and denote ν(hN):= ( R0 hN(x)ν(dx). In this special case, we have κN' I1(f)T=ν(hN)(T 0gN(t)dt (T 0g2(t)dtN/2,N≥3. Therefore, a sufficient condition for the CLT for ' I1(f)T,is lim T→+∞ (T 0gN(t)dt (T 0g2(t)dtN/2=0,N≥3.(33) Condition (33) is trivially satisfied if gis a polynomial function or if it is the product of a polynomial function with a rapidly decaying function as, for example, if g(t)=tme−t2/2. Moreover, if gis constant, which means that (' I1(f)T)T≥0is a Lévy process, (33)isalways satisfied. More interesting is the case g(t)=etα, with α>0, that we illustrate in the following lemma. 123 P. Di Tella et al. Lemma 5.2 Let g(t)=etα,α>0. We then have lim T→+∞ (T 0eNtαdt (T 0e2tαdtN/2=⎧ ⎪ ⎨ ⎪ ⎩ 0,α<1, 2N/2 N,α=1, +∞,α>1, N≥3. Proof We use the variable transform u=tαto rewrite the expression (T 0eNtαdt (T 0e2tαdtN/2= 1 α(Tα 0eNuu1 α−1du 1 α(Tα 0e2uu1 α−1duN/2. Since for all k∈Nit holds limx→+∞ k(x 0ekuu1 α−1du ekxx1 α−1=1,we get lim T→+∞ 1 α(Tα 0eNuu1 α−1du 1 α(Tα 0e2uu1 α−1dtN/2=lim T→+∞ 1 NαeNTαT1−α 1 2αe2TαT1−αN/2 =⎧ ⎪ ⎨ ⎪ ⎩ 0,α<1, 2N/2 N,α=1, +∞,α>1,  The following theorem is an immediate consequence of Lemma 5.2. Theorem 5.3 Let f (t,x)=h(x)etα,whereh∈LN(ρ) for every N ≥1and hL2(ρ) =1. We have: (i) If α<1,' I1(f)Tconverges in law as T →+∞to a standard normal distributed random variable. (ii) If α>1,' I1(f)Tcannot converge in law as T →+∞to a random variable whose distribution is determined by its moments. Remark 5.4 1. In [11, Thm 4.1 and Corollary 4.3] estimates for the Kolmogorov distance between multiple integrals Im(f)(if m≥2) with respect to the compensated Poisson random measure and a standard Gaussian random variable Zare shown by the Malliavin–Stein method. For our simple case of first order chaos with the special setting f(t,x)=h(x)etαwe get easily an estimate in Wasserstein distance from [27, Corollary 3.4] which states that (for σ=0) dW(' I1(f)T,Z)≤$$$$$$$ 1− T 0 R0 f(t,x)2dtν(dx)$$$$$$$+ T 0 R0 f(t,x)3dtν(dx) =|1−ν(h2)|+ν(|h|3)(T 0eNtαdt (T 0e2tαdtN/2. This provides us with a convergence rate for T→∞if α<1andν(h2)=1. 123 Product formulas for multiple stochastic... 2. In [28, Theorem 2.6] conditions on the kernels are given such that multiple integrals of order m≥2 with respect to the compensated Poisson random measure converge to a Gamma distribution. We will show in the next proposition that our first order chaos expression ˜ I1(f)Twith the kernel f(t,x)=h(x)etassuming h(0)=0 converges weakly to a Poisson distributed random variable if T→∞. Proposition 5.5 Let be a Poisson random measure on R+×R×R0with the compensator λ+⊗λ⊗ν(here we denote the Lebesgue measure on Rand on R+by λand λ+, respectively) and let the function h ≥0be such that ν(euh)<+∞ for every u ≤ε, for a fixed ε>0.We set a(x,y):= h(x)2y/2,E:= [0,1 2ln 2]×(−∞,1]×R0and define Y:= E a(x,y)( −λ+⊗λ⊗ν)(dt,dy,dx). We then have: (i) E[euY ]<+∞,foreveryu≤ε √2. Hence the distribution PYof Y is determined by its moments. (ii) The sequence of the cumulants of Y is given by kY 1=0and kY N=ν(hN)2N/2 N,N≥2. (iii) Let f (t,x)=h(x)etwith h(0)=0.Then' I1(f)Tconverges in distribution to Y as T→+∞.Forh(0)>0it holds that ' I1(f)Tconverges in distribution to Y +Gas T→+∞where G is a centered Gaussian random variable with variance σ2h(0)2, independent from Y . Proof Since the condition on the convergence of the cumulants is equivalent to the condition on the convergence of the moments, (iii) for h(0)=0 follows from (i), (ii) and Lemma 5.2 by [3, Theorem 30.2]. For h(0)>0 we simply take into consideration that one can decompose ˜ I1(f)Tinto an independent sum of Gaussian part and jump part, and h(0)σ (T 0g(t)dWt/(T 0g2(t)dt ∼G.To see (i) and (ii), we introduce the nonnegative random variable Z:= (Ea(x,y)(dt,dy,dx)and notice that, for every 0 <u≤ε √2, from [20, Theorem 3.9 and Exercise 3.4], we get E)euZ*=exp 1 2ln 2 0 1  −∞  R0 (eua(x,y)−1)ν(dx)dydt≤exp ν(e√2uh −1)<+∞, where, to get the first estimate, we expand the integrand as a power series. This together with the definition of Yyields +∞ >E)euY *=exp +∞  N=2 uN N!ν(hN)2N/2 N,0<u≤ε √2. Thus, (i) and (ii) are shown.  6 The number of elements in DNif |l|=k Without handling the enumeration of appearing terms, it is not clear how to perform the calculations on a computer. The formulas below allow for recursive as well as iterative implementations. Here we use again the notation lo {j}or lo enfor lo nif en={j}. Counting the 123 P. Di Tella et al. number of contributing terms in (27)foragiven|l|=k≤m1+···+mN, note that for any j∈{1,...,N}it holds lo {j}=0. This is because of definition (24) (stating that the c{j}are zero). Thus, we will omit those components. We will tackle the problem in the subsections below in 3 ways, allowing a variety of approaches to calculate the number on a computer: by deriving a recursion formula, by using weak compositions, and by applying generating functions. Of course, it is also possible to count and study the set DNthrough other combinatorial approaches like numbering diagrams and multigraphs. This has been done in [29] for the Brownian and pure-jump setting separately. The same program as there can be conducted for our case where the Brownian part and the jump part are combined and is subject to ongoing research. Studies in combinatorical details of this very direction ought to deliver further insights into the structure of integrands appearing in the product formula. 6.1 The number of elements by a recursion formula Proposition 6.1 We have the following recursion formula for |{(l,lo)∈DN:|l|=k and lo {1}=···=lo {N}=0}| =: C (k,N,(m1,...,mN,0,...)), where C (k,˜ N,(m1,...,mN,0,...)):= 0if ˜ N= N, C k,N,(m1,...,mN,0,...) := 0if ∃k∈{1, ..., N}with mk<0.(34) C 0,0,(0,0,...) := 1, and C k,N,(m1,...,mN,0,...)  = min{k,mN}  κ=0 l{N}+···+l{1,...,N} =κ lo {1,N}+···+lo {1,...,N} =mN−κ C k−κ, N−1,(ˆm1(l,lo), ..., ˆmN−1(l,lo), 0,...)  where ˆmr(l,lo):= mr− s∈P({1,...,N−1}) r∈{N}∪s (l{N}∪s+lo {N}∪s), and P({1,...,N−1})stands for the power set of {1,...,N−1}. Proof The definition of DN(see (21)) requires (we use here len:= ln) ηN  n=1 (len+lo en)1en(j)=mjfor all j=1, ..., N. Ordering the index set such that all encontaining Nare behind the other sets that do not contain N, we observe the tuples (l{1},...,l{1,...,N−1},l{N},...,l{1,...,N},lo eN+1,...,lo eηN). 123 Product formulas for multiple stochastic... We put κ:= |(l{N},...,l{1,...,N})|and know that κ+|(lo {1,N},...,lo {N−1,N}...,lo {1,...,N})|=mNand κ≤min{k,mN}. For every choice of (l{N},...,l{1,...,N})and (lo {1,N},...,lo {N−1,N},...,lo {1,...,N})the number of possible choices for the remaining (l{1},...,l{1,...,N−1},lo {1,2},...,lo {N−2,N−1},...,lo {1,...,N−1}) is determined by m1− s∈P({1,...,N−1}) 1∈{N}∪s (l{N}∪s+lo {N}∪s), ... mN−1− s∈P({1,...,N−1}) N−1∈{N}∪s (l{N}∪s+lo {N}∪s) =(ˆm1(l,lo), ..., ˆmN−1(l,lo)) with kreplaced by k−κ, which implies the claim. Clearly, only those (l,lo)can be chosen which satisfy ˆmr(l,lo)≥0forallr=1, ..., N−1 because ˆmr(l,lo)is the number of ’unused’ variables of the kernel in the r-th factor of the product. Therefore we use (34).  Concerning our special cases considered in Remark 4.7 the following holds Remark 6.2 1. For the jump case M(dt,dx)=˜ N(dt,dx)the number of appearing terms is C(k,N,(m1,...,mN,0, ...)). 2. For the Brownian motion case M(dt,dx)=σdWtδ0(dx)the number of appearing possible nonzero terms is significantly less and can be – in the same way as before – recursively enumerated by the formula CWk,N,(m1,...,mN,0,...)  = min{k,mN}  κ=0 lo {1,N}+···+lo {N−1,N}=mN−κ lo {i,j}≥0,i=j, CWk−κ, N−1,(m1−lo {1,N}, ... , mN−1−lo {N−1,N},0, ...), setting CWto zero or one for all other cases, as before. 6.2 The number of elements by counting weak compositions To count those tuples without using a recursion, the following formula for tuples (lu)u∈U where Uis a finite set and lu∈{0,1,...}, involving weak compositions is a help. A weak composition of r∈N∪{0}into n∈Nparts is any representation r=a1+... +anwith a1, ..., an∈N∪{0}.If we denote by r nthe number of weak compositions of rinto n parts, it holds that r n=r+n−1 n−1. For example, the weak compositions of r=5 into n=2 parts are given by 0+5,5+0,1+4,4+1,2+3,3+2. Lemma 6.3 Let A1,...,ANbe subsets of a finite set U such that +N i=1Ai=U, Aj∩ N i=1 i=j Ac i=∅, for all j =1,...,N, and let numbers m1,...,mN∈Nbe given. For 123 P. Di Tella et al. t=(t1, ..., tN)∈{0,1}Nwe set A(t1,...,tN):= ⎛ ⎝ N  i=1,ti=1 Ai⎞ ⎠∩⎛ ⎝ N  j=1,tj=0 Ac j⎞ ⎠ min tm:= min{mi:ti=1,i=1, ..., N}. Let n:{0,1}N\{(0,...,0)}↔{1,...,2N}be a bijection such that n(1,...,1)=1,n(0,...,0,1)=2N and n(t)<n(s)whenever s>t(where by s>twe mean that si≥tifor all i =1, ..., N, and there exists at least one i0such that si0=1and ti0=0). For shortness, with a slight abuse of notation we identify qn(t)and qt(so e.g. q(1,...,1)=q1), and we abbreviate, min tm−q>:= min ,mi− n(t)−1  ι=1 n−1(ι)(i)=1 qι:ti=1,i=1, ..., N-. Then the cardinality of ,(lu)u∈U: a∈Ai la=mi,i=1,...,Nis given by minn−1(1)m−q>  q1=0 q1 |An−1(1)|··· minn−1(j)m−q>  qj=0 qj |An−1(j)|··· ··· minn−1(2N−N)m−q>  q2N−N=0 q2N−N |An−1(2N−N)|· |t|=1mintm−q> |At|, and whenever a set Asis empty, the according summation does not appear in the above formula and qsis then set to zero. Proof The formula can be seen by partitioning Uinto all possible pieces emerging from intersecting the sets A1,...,AN. Provided that N i=1Ai=∅, one summand contained in each of the sums a∈Ailais a∈N i=1Aila=a∈A(1,...,1)la=q(1,...,1)=q1(an intersection with all sets involved). The possibilities for q1are 0, ..., min(1,...,1)m(which equals min(1,...,1)m−q>), and the number of tuples (la)a∈A(1,...,1)that sum up to q1are given by  q1 |A(1,...,1)|= q1 |An−1(1)|.For the second sum, as we already decided for a part of that sum to be q1,forq2=q(1,...,1,0,1,...,1)there are possibilities from 0 to min(1,...,1,0,1,...,1)m−q1, which equals min(1,...,1,0,1,...,1)m−q>again. For q3and the according tuple n−1(3),wehave the upper limit min{mi−q1−q2·1{n−1(2)(i)=1}:n−1(3)(i)=1,i=1,...,N},whichis minn−1(3)m−q>, and the number of tuples is given by  q3 |An−1(3)|. We proceed nesting the sums until we reach the number 2N−N, which means that from thereon, the corresponding tuples n−1(j), j>2N−N,have only one nonzero element. 123 Product formulas for multiple stochastic... Proof of Proposition 4.3 We have by Proposition 4.1 and (19) N  j=1 Jmj(α⊗mj j)T = k (s1,...,sk)∈Ak i∈{0,1}k T(k) αs1,i1(z1)Mi1(dz1)...α sk,ik(zk)Mik(dzk). We denote by [s1,i1, ..., sk,ik]/∼all equivalence classes [(s1,i1), ..., (sk,ik)]with respect to permutations from Skwhere (s1, ..., sk)∈Akand (i1, ..., ik)∈{0,1}k.Our intention is to replace the above expression by the following one (up to some factors which we want to determine next)  k [s1,i1,...,sk,ik]/∼ σ∈Sk T(k) k  j=1 αsσ(j),iσ(j)(zj)Miσ(1)(dz1)...Miσ(k)(dzk). Whenever we have the equality (defining ∼) ((s1,i1), ..., (sk,ik)) =((sσ(1),iσ(1)), ..., (sσ(k),iσ(k))), the same summand appears. Like in the multinomial theorem, also here the multiplicity of the summands equals the number of those permutations that do not change a tuple (in the multinomial theorem for (a1+···+aK)n, with pairwise disjoint ai, the multiplicity of the term K i=1aki iis n! k1!···kK!,where k1+...+kK=n). In our case, for any (s1, ..., sk)∈Ak and i∈{0,1}k, to count the multiplicity of the smin ((s1,i1), ..., (sk,ik)) paired either with im=1orim=0, we define for j=1, ..., ηN lj:= |{m:im=1andsm=ej}| and lo j:= |{m:im=0andsm=ej}|. Then  k (s1,...,sk)∈Ak i∈{0,1}k T(k) αs1,i1(z1)Mi1(dz1)...α sk,ik(zk)Mik(dzk) = k [s1,i1,...,sk,ik]/∼ 1 l!lo! σ∈Sk T(k) k  j=1 αsσ(j),iσ(j)(zj)Miσ(1)(dz1)...Miσ(k)(dzk). We denote by n:= |lo|=k−k j=1ijthe number of zeros in i∈{0,1}kand choose a permutation π∈Skfor which Miπ(1)(dz1)...Miπ(k)(dzk)=(m(dz1),...,m(dzn), M(dzn+1), ..., M(dzk)). 123 P. Di Tella et al. Then we have by Lemma A.3 that  σ∈SkT(k) k  j=1 αsσ(j),iσ(j)(zj)Miσ(1)(dz1)...Miσ(k)(dzk) =((0,T]×R)k k  j=1 αsπ(j),iπ(j)(zj)m(dz1)···m(dzn)M(dzn+1)···M(dzk) = ((0,T]×R)n n . j=1 αsπ(j),0(z1, ..., zn)m(dz1)...m(dzn)Ik−n⎛ ⎝ k . j=n+1 αsπ(j),1⎞ ⎠T . This yields N  j=1 Jmj(α⊗mj j)T = k [s1,i1,...,sk,ik]/∼ 1 l!lo! σ∈Sk T(k) k  j=1 αsσ(j),iσ(j)(zj)Miσ(1)(dz1)...Miσ(k)(dzk) = k |lo|+|l|=k, (l,lo)∈DN 1 l!lo!⎛ ⎜ ⎝ ((0,T]×R)|lo| α⊗lodm⊗|lo|⎞ ⎟ ⎠I|l|(α⊗l)T, where α⊗l=α⊗l1 e1,1⊗... ⊗α⊗lηN eηN,1and α⊗lo=α⊗lo 1 e1,0⊗... ⊗α⊗lo ηN eηN,0. Rewriting the condition in Akfrom Proposition 4.1 to this setting leads to the set DNgiven in 21. By rearranging the summands we get N  j=1 Jmj(α⊗mj j)T= k |l|=k, (l,lo)∈DN 1 l!lo!⎛ ⎜ ⎝ ((0,T]×R)|lo| α⊗lodm⊗|lo|⎞ ⎟ ⎠Ik(α⊗l)T. Finally, to get (23), we use Lemma 2.1 to write the iterated integrals on the l.h.s. as multiple integrals which implies on the r.h.s. the factor m1!···mN!. References 1. Agrawal, N., Hu, Y., Sharma, N.: General product formula of multiple integrals of Lévy process. J. Stoch. Anal. 1(3), 3–12 (2020) 2. Bogdan, K., Rosi´nski, J., Serafin, G., Wojciechowski, L.u.: Lévy systems and moment formulas for mixed Poisson integrals. 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