# On Henstock Method to Stratonovich Integral with respect to Continuous Semimartingale

Authors: Haifeng Yang; Tin Lam Toh
Journal: International Journal of Stochastic Analysis
 (2014)

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## Abstract

We will use the Henstock (or generalized Riemann) approach to define the Stratonovich integral with respect to continuous semimartingale inL2 space. Our definition of Stratonovich integral encompasses the classical definition of Stratonovich integral.


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## Body

## 1. Introduction

Classically, it has been emphasized that defining stochastic integrals using Riemann approach is impossible [1, p. 40], since the integrators have paths of unbounded variation. Moreover, the integrands are usually highly oscillatory [1–3]. The uniform mesh in classical Riemann setting is unable to handle highly oscillatory integrators and integrands. Kurzweil and Henstock [4–6] independently modified the Riemann approach by using nonuniform mesh. This modification leads to a larger class of integrals being studied (see [4, 7–12]).The generalized Riemann approach, using nonuniform meshes, has been successful in giving an alternative definition to the Stratonovich integral with respect to Brownian motion (see [13]). This paper attempts to further generalize the result of [13] to define the Stratonovich integral with respect to continuous semimartingale.The difficulty to extend from Brownian motion to semimartingale is that the quadratic variation of the continuous martingale may not be absolutely continuous. Hence, the definition of the Stratonovich-Henstock belated integral in [13] may not be valid. In [12, 14, 15], the definition of weakly Henstock variation belated integral was used to address this difficulty. Similarly, we tap on this approach to define Stratonovich integral with respect to continuous L2-martingale.Further, to consider continuous local martingale as an integrator, we will show that a(δ,η)-fine belated partial stochastic division D=Ik,ξkk=1n of [0,t] in [12, 16] is also a (δ,η)-fine belated partial stochastic division of [0,t∧S], where S is a stopping time.Lastly, the definition of the Stratonovich integral with respect to continuous semimartingale will be further refined. We divide the continuous semimartingale into two parts: a continuous local martingale and the difference of continuous, nondecreasing, and adapted processes. The former is discussed in Section3.2. The latter has finite total variation on each bounded interval. Hence, integration with respect to the latter process is classical Riemann-Stieltjes. This part has already been addressed in [17].We give a new definition of the Stratonovich integral with continuous semimartingale inL2 space. This integrand is weaker than the classical case.


## 2. Stratonovich Integral

In this section, we will generalize the definition of Stratonovich integral to include the case where the integrator is continuousL2-semimartingale. We will develop the definition of Stratonovich-Henstock belated integral where the integrator is a Brownian motion in [13]. Let (Ω,F,P) be a probability space such that (Ω,F,P) is complete. Also, let Cn(R) denote the class of functions which have continuous n derivatives.Definition 1.
LetX={Xt,Ft;0≤t<∞} be a process. If there exists a nondecreasing sequence Tnn=1∞ of the stopping times of the filtration {Ft}, such that {Xtn≜Xt∧Tn,Ft;0≤t<∞} is a martingale for each n≥1 and P[lim⁡n→∞Tn=∞]=1, then one says that X is local martingale. If, in addition, X0=0 a.s., one writes X∈Mloc⁡ (or X∈Mc,loc⁡, if X is continuous).Definition 2.
A continuous semimartingaleX={Xt,Ft;0≤t<∞} is an adapted process which has the decomposition
(1)Xt=X0+Mt+Ct,0≤t<∞a.s.,
where M={Mt,Ft;0≤t<∞}∈Mc,loc⁡ and Ct has finite variation.Definition 3.
LetX={Xt,Ft;0≤t<∞} and Y={Yt,Ft;0≤t<∞} be two continuous semimartingales such that
(2)Xt=X0+Mt+Ct;Yt=Y0+Nt+Dt,
where M and N are in Mc,loc⁡ and Ct and Dt are adapted, continuous process of bounded variation with C0=D0=0 a.s. The Stratonovich integral of Yt with respect to Xt is
(3)∫0tYs∘dXs≜∫0tYsdMs+∫0tYsdCs+12M,Nt,
where the first integral on the right-hand side of (3) is an Itô integral, the second one is pathwise integral of the Riemann-Stieltjes type, and 〈M,N〉t is the cross variation process of M and N. If M=N, then M,Nt=Mt is the quadratic variation of M (see [18, p. 31]).Proposition 4.
Letf:R→R be a function of class C3(R) and let X={Xt,Ft;0≤t<∞} be a continuous semimartingale with decomposition in (2). Then
(4)fXt=fX0+∫0tf′Xs∘dXsa.s.Proof.
By Itô formula [18, p. 149], we have
(5)fXt=fX0+∫0tf′XsdXs+12∫0tf′′XsdMs.
We only have to prove that
(6)∫0tf′Xs∘dXs=∫0tf′XsdXs+12∫0tf′′XsdMs.
Since f′Xs∈C2(R), then, by Itô formula,
(7)f′Xs=f′X0+∫0tf′′(Xs)dMs+∫0tf′′(Xs)dAs+12∫0tf′′′(Xs)dMs.
Then f′Xs is a semimartingale, ∫0tf′′XsdMs is a local martingale, and ∫0tf′′XsdAs and 1/2∫0tf′′′XsdMs are bounded continuous processes. By definition of a semimartingale,
(8)∫0tf′Xs∘dXs=∫0tf′XsdXs+12M,∫0tf′′XsdMst.
By the property of cross variation (see [18, p. 143])
(9)M,∫0tf′′XsdMst=∫0tf′′(Xs)dMs.
Hence,
(10)fXt=fX0+∫0tf′XsdXs+12∫0tf′′(Xs)dMs=fX0+∫0tf′Xs∘dXs.


## 3. Henstock Approach

### 3.1. Henstock Approach to Define Stratonovich Integral with respect to Bounded Continuous Martingale

In [13–16, 19], a positive function δ on [0,t], 0≤t<∞, where δ does not depend on ω∈Ω, was used. From the proof of the main results in [13, 14, 16, 20], δ(ξ) is deterministic, because the quadratic variation of a Brownian motion is deterministic and Fubini’s theorem can be applied in switching the order of the two integrals. In this section, we consider martingales as integrators. To define stochastic integrals using Riemann sums, the δ-function needs to depend on Ω and stochastic intervals are needed. Therefore, we need to redefine δ as also dependent on Ω (see [12]).Definition 5.
Letδ:[0,t]×Ω→(0,t), 0≤t<∞, be a measurable function with respect to the two variables ξ∈[0,t] and ω∈Ω. Then δ is called alocally stopping process [12] if, for each ξ∈[0,t], ξ+δ is a stopping time.
For each(c,d]⊂[0,t], we denote the measure [15] induced by the quadratic variation process of an adapted process X={Xt,0≤t<∞} by
(11)μX(c,d]≜EXd-Xc.Definition 6.
Letδ be a locally stopping process and X={Xt,Ft;0≤t<∞}. A finite collection of stochastic interval-point pairs (((Si,Ti],ξi),i=1,2,…,n),  where Si and Ti are stopping times of X for all i=1,2,…,n, is said to be a (δ,η)-fine belated partial stochastic division of [0,t] if(1)
for eachi(Si,Ti] is a stochastic interval and, for each ω∈Ω,(Si,Ti],  i=1,2,…,n, are disjoint left-open subintervals of [0,1];(2)
each((Si,Ti],ξi) is a δ-fine belated division; that is, for each ω∈Ω, one has
(12)(Si(ω),Ti(ω)]⊂[ξi,ξi+δ(ξi,ω)];(3)
for anyη>0|μX(0,t]-∑i=1n‍μX(Si,Ti]|<η.
For the caseSi(ω)=Ti(ω), (Si(ω),Ti(ω)] denotes {Ti(ω)}. Note that, for each ξ∈[0,t], ξ+δ is a stopping time. Thus, for each ξi, there exists a stochastic interval (Si(ω),Ti(ω)] with Si(ω)<Ti(ω) such that (Si(ω),Ti(ω)]⊂[ξi,ξi+δ(ξi,ω)]. For each ω∈Ω, namely,
(13)Si(ω)=ξi(ω),Ti(ω)=inf⁡{ξi+δ(ξi,ω),t}.
In addition, by Vitali’s covering theorem, the (δ,η)-fine belated partial stochastic division could cover [0,t] except for an arbitrarily small μX-measure [19, p. 44] [12, 16].Definition 7.
An adapted processY={Yt,Ft;0≤t<∞} in (Ω,F,P) is said to be weakly Stratonovich-Henstock belated (denoted by WSHB) integrable to a process A in (Ω,F,Ftt≥0,P) on [0,t], 0≤t<∞, with respect to a stochastic process X if, for every ε>0, there exists a locally stopping process δ on [0,t]×Ω for which
(14)E∑i=1nYξi+1,ω+Yξi,ω2XTiω,ω-XSiω,ω-ATiω,ω-ASiω,ωYξi+1,ω+Yξi,ω22≤ε,
where ξn+1=t. For succinctness, one may write
(15)E∑i=1nYξi+1,ω+Yξi,ω2XTi-XSi-ATi-ASi2≤ε,
for every (δ,η)-fine belated partial stochastic division D={((Si,Ti],ξi),i=1,2,…,n} of [0,t].Proposition 8.
If an adapted processY={Yt,Ft;0≤t<∞} is WSHB integrable with respect to stochastic process X, then the weakly Henstock variational belated integral of Y is unique a.s.In light of Proposition8, we will denote the integral of the process Y with respect to stochastic process X by the notation
(16)WSH∫0tYs∘dWs≜At.Proposition 9.
LetY and Y′={Yt′,Ft;0≤t<∞} be WSHB integrable with respect to stochastic process X and α∈R. Then Y+Y′ and αY are WSHB integrable with respect to stochastic process X. Furthermore,
(17)WSH∫0tYs+Ys′∘dXs=WSH∫0tYs∘dXs+WSH∫0tYs′∘dXs;WSH∫0tαYs∘dXs=αWSH∫0tYs∘dXs.Proposition 10.
If stochastic processY is WSHB integrable with respect to stochastic process X, then so is it on subinterval [c,d]⊂[0,t].Proposition 11.
Let the adapted processY be WSHB integrable on [0,a] and [a,t] with respect to stochastic process X, where 0<a<t. Then Y is WSHB integrable on [0,t] and, furthermore,
(18)WSH∫0tYs∘dXs=WSH∫0aYs∘dXs+WSH∫atYs∘dXs.The proofs of Propositions8 to 11 are similar to the results of classical Henstock integral [17]; hence we omit these proofs in this paper.Next, we will consider the relationship between WSHB integral and Stratonovich integral with respect to continuous martingale. Here, we assume that the continuous martingale is bounded. In the next section we will address the continuous local martingale. Note that we can choose the stopping time such that a continuous local martingale becomes a bounded continuous martingale up to any finite stopping time [18, p. 44].Theorem 12.
Iff(x)∈C2(R) and X={Xt,Ft;0≤t<∞} is bounded continuous martingale, then f(Xt) is WSHB integrable with respect to bounded continuous martingale X on [0,t]. Furthermore,
(19)WSH∫0tf(Xs)∘dXs=∫0tf(Xs)∘dXs.Proof.
SinceX is a bounded continuous martingale and f(x)∈C2(R), then f(Xt), f′Xt, and f′′(Xt) are bounded and continuous with respect to t. By Definition 1, the classical Stratonovich integral with respect to bounded continuous martingale X is
(20)∫0tfXs∘dXs=∫0tfXsdXs+∫0tf′XsdXs.
Let D1=si,ti,ξii=1n be a (δ,η)-fine belated partial stochastic division of [0,t]. Then,
(21)E∑i=1n-112fXξi+fXξi+1Xti-Xsi-∫sitifXs∘dXs2=E∑i=1n-1f(Xξi)(Xti-Xsi)+∑i=1n-112fXξi+1-fXξiXti-Xsi-∫sitif(Xs)∘dXs2=E∑i=1n-1fXξiXti-Xsi+∑i=1n-112fXξi+1-fXξ1Xti-Xsi-∫sitifXsdXs-∫siti∂fXs∂xdXs2≤2E∑i=1n-1fXξiXti-Xsi-∫sitifXsdXs2+E∑i=1n-1fXξi+1-fXξiXti-Xsi-∫siti∂fXs∂xXs2=2R1+R2,
where
(22)R1=E∑i=1n-1fXξiXti-Xsi-∫sitifXsdXs2,R2=E∑i=1n-1fXξi+1-fXξiXti-Xsi-∫siti∂fXs∂xdXs2.
From [12], given ε1>0, there exists a (δ1,η1)-fine belated partial division of [0,t], D1=si,ti,ξii=1n, such that R1<ε1. We just need to consider R2. By mean value theorem, fXξi+1-fXξi=∂f/∂x(Xi~)(Xξi+1-Xξi), where Xi~∈[Xξi,Xξi+1]. Since Xt is continuous with respect to t, there exists a ξi^∈[ξi,ξi+1] such that Xξi^=Xi~ a.s. Similarly, there exists ξi¯∈[si,ti] such that ∫siti∂f(Xt)/∂xdXt=f′(Xξi¯)Xti-Xsi2 in L2(Ω) space. Then,
(23)R2=E∑i=1n-1(fXξi+1-f(Xξi))(Xti-Xsi)-∫siti∂fXt∂xdXt2=E∑i=1n-1∂fXξi¯∂xf′Xξi^Xξi+1-XξiXti-Xsi-∂fXξi¯∂xXti-Xsi22≤2E∑i=1n-1Xti-Xsi2f′Xξi^(Xξi+1-Xξi)(Xti-Xsi)-f′Xξi^Xti-Xsi2∑i=1n-12+2E∑i=1n-1f′Xξi^Xti-Xsi2-f′Xξi¯Xti-Xsi2∑i=1n-12.
Let
(24)r1=E∑i=1n-1f′Xξi^Xξi+1-XξiXti-Xsi-f′Xξi^(Xti-Xsi)2∑i=1n-12,r2=E∑i=1n-1f′Xξi^Xti-Xsi2-f′Xξi¯Xti-Xsi22,r1≤KE∑i=1n-1Xξi+1-XξiXti-Xsi-Xti-Xsi22≤KE∑i=1n-1Xti-XsiXti-Xsi-Xξi+1-Xξi2≤KE∑i=1n-1‍Xti-XsiXξi+1-Xti+Xsi-Xξi2≤K∑i=1n-1‍EXξi+1-Xti+Xsi-Xξi2EXti-Xsi2,
where K=sup⁡{f′Wt:0≤t≤1&ω∈Ω}. Given that Xt is a bounded martingale,
(25)EXξi+1-Xti+Xsi-Xξi2=EXξi+1-Xti2+EXsi-Xξi2=EXξi+1-EXti+EXsi-EXξi.
However, there exists a (δ2,η2)-fine belated partial stochastic division D2=ξi,ti,ξi:i=1,2,…,n} of [0,t] such that |μX(0,t]-∑i=1nμX(si,ti]|<η2. Then we have
(26)EXξi+1-EXti+EXsi-EXξi<η2.
Hence,
(27)r1≤Kε2∑i=1n-1EXti-Xsi2≤KEXtη2.
Since X is bounded continuous martingale, by Lemma 5.10 in [18, p. 33], we have
(28)E∑i=1n-1Xti-Xsi4⟶0asn⟶∞.
Hence, given ε2>0,
(29)r2≤sup⁡1≤i≤n&ω∈Ωf′Xξi^-f′Xξi¯E∑i=1n-1‍Xti-Xsi4≤2Kε2.
Now take η=min⁡{η1,η2} and let δ(ξi)=min⁡{δ1(ξi),δ2,ξi+1-ξi}; then a (δ,η)-fine belated partial division D=ξi,ti,ξi:i=1,2,…,n} of [0,t] is both (δ1,η1)-fine belated partial stochastic division and (δ2,η2)-fine belated partial stochastic division. Hence, for ε=max⁡{ε1,η2,ε2}, we have
(30)E∑i=1n-1‍fξi+fξi+12Wti-Wsi-∫sitiXt∘dWt2≤R1+r1+r2≤K¯ε,
where K¯ is a constant. Hence, fXt is WSHB integrable on [0,t] and, by the unique property of WSHB,
(31)WSH∫0tfXs∘dXs=∫0tfXs∘dXs.


### 3.2.  Henstock Approach to Define Stratonovich Integral with respect to Continuous Local Martingale

Lemma 13.
LetX={Xt,Ft;0≤t<∞} be a continuous local martingale. Suppose f is WSHB integrable to a process At with respect to local continuous martingale X. Let T be a stopping time; then f is WSHB integrable to a process AT with respect to XT, where XT={Xt∧T;0≤t<∞} and AT={At∧T;0≤t<∞}.Proof.
Letε>0 and δ(ξ,ω)>0 be given as in Definition 6 for the WSHB integral of f with respect to continuous local martingale X. Define
(32)δTξ,ω≜Tω-ξ,ifξ<Tω≤ξ+δξ,ωδξ,ω,otherwise.
Then δT(ξ,ω) is locally stopping process, since (T(ω)∧ξ)∧(ξ+δ(ξ,ω)) is a stopping time and {ω:T(ω)<ξ}∈Ft if ξ<t. Let ((U,V],ξ) be a (δT(ξ,ω),η)-fine belated stochastic interval-point pair.(i)
IfT(w)≤ξ, then
(33)XT(V(ω),ω)-XT(U(ω),ω)=XT(T(ω),ω)-XT(T(ω),ω)=0.(ii)
Ifξ<T(ω)≤ξ+δ(ξ,ω), then [U,V]⊂[ξ,T]⊂[ξ,ξ+δ(ξ,ω)]. Consequently, ((U,V],ξ) will be a (δ(ξ,ω),η)-fine belated stochastic interval-point pair, U∧T=U and V∧T=V.(iii)
IfT(ω)>ξ+δ(ξ,ω), then
(34)XT(U(ω),ω)-XTVω,ω=XVω∧Tω,ω-XUω∧Tω,ω=X(U(ω),ω)-XVω,ω.It is easy to verify that D={((Ui,Vi],ξi),i=1,2,…,n} is also a (δT(ξ,ω),η)-fine belated stochastic division of [0,t∧T(w)] only if D={((Ui,Vi],ξi),i=1,2,…,n} is a (δ(ξ,ω),η)-fine belated stochastic division of [0,t].
Similar results hold forAT. Since the D={((Ui,Vi],ξi),i=1,2,…,n} is also a (δT(ξ,ω),η)-fine belated stochastic division of [0,t∧T(w)], by the property of being WSHB integrable, we have that f is WSHB integrable to a process AT with respect to XT.Lemma 14.
LetX={Xt,Ft;0≤t<∞} be a continuous local martingale with the corresponding nondecreasing sequence of stopping times Tnn=1∞. Suppose that f is WSHB integrable to a process A with respect to X; then f is WSHB integrable to a process ATn with respect to XTn for each n=1,2,…. Furthermore, for each t∈[0,t],lim⁡n→∞ATn=A a.s.Proof.
This follows directly form Lemma13 and the uniqueness of the WSHB integral. Then we have the last statement
(35)lim⁡n→∞ATn=Aa.s.Theorem 15.
Iff(x)∈C2(R) and X={Xt,Ft;0≤t<∞} is a continuous local martingale, then f(Xt) is WSHB integrable with respect to X on [0,t]. Furthermore,
(36)(WSH)∫0tfXs∘dXs=∫0tfXs∘dXs.Proof.
First we introduce, for eachm≥1, the stopping time
(37)Tm=inf⁡t≥0;Xt≥morXt≥m.
The resulting sequence {Tm} is nondecreasing with P[lim⁡m→∞Tm=∞]=1. Thus, the stopping process Xt∧Tm is bounded; that is, Xt∧Tm is a bounded martingale for each m=1,2,…. Under this situation, the values of f outside [-m,m] are irrelevant. We may assume without loss of generality that f has compact support, so f, f′, and f′′ are bounded.
LetD={((Ui,Vi],ξi),i=1,2,…,n} be a (δ(ξ,ω),η)-fine belated stochastic division of [0,1]. From Lemma 13, we have that, for each m≥1, Dm={((Ui,Vi],ξi),i=1,2,…,n} is a (δTm(ξ,ω),η)-fine belated stochastic division of [0,t∧Tm(w)]. From Lemma 13, there is a process ATm={At∧Tm;0≤t<∞} such that
(38)E∑i=1n-112fXξi+1+XξiXVi∧Tm-XUi∧Tm∑i=1n-1-AVi∧Tm-AUi∧Tm2<ε,
for every (δ,η)-fine belated partial stochastic division D={((Ui,Vi],ξi):i=1,2,…,n} of [0,t]. Since
(39)lim⁡m→∞∑i=1n-1fXξi+1+XξiXVi∧Tm-XUi∧Tm-AVi∧Tm-AUi∧Tm∑i=1n-1=∑i=1n-1f(Xξi+1+Xξi)(XVi-XUi)-AVi-AUi∑i=1n-1a.s.,
we must have
(40)E∑i=1n-1fXξi+1+XξiXVi-XUi-AVi-AUi=Elim⁡m→∞∑i=1n-112fXξi+1+XξiXVi∧Tm-XUi∧Tm-AVi∧Tm-AUi∧Tm∑i=1n-12=Eliminf⁡m→∞∑i=1n-112fXξi+1+XξiXVi∧Tm-XUi∧Tm-AVi∧Tm-AUi∧Tm∑i=1n-12≤liminf⁡m→∞E∑i=1n-112fXξi+1+XξiXVi∧Tm-XUi∧Tm-AVi∧Tm-AUi∧Tm∑i=1n-12<liminf⁡m→∞ε≤ε.
This shows that f(Xt) is WSHB integrable on [0,t] and
(41)(WSH)∫0tf(Xs)∘dXs=∫0tf(Xs)∘dXs.


### 3.3. Henstock Approach to Define Stratonovich Integral with respect to Continuous Semimartingale

LetX={Xt,Ft;0≤t<∞} be a continuous semimartingale such that
(42)Xt=X0+Mt+Ct,
where Mt is a continuous local martingale and Ct is adapted, continuous process of bounded variation with C0=0 a.s. We have
(43)12fXξi+1+XξiXVi-XUi=12fXξi+1+XξiMVi-MUi+12fXξi+1+XξiCVi-CUi.
Now we consider a stopping time
(44)Tm=0;vvivifX0≥minf⁡C˘tt≥0;Mt≥morMt≥morC˘t≥m;vvviivifX0<m∞;ifX0<m&vvvivinf⁡C˘tt≥0;Mt≥morvvvvvvvvMt≥morC˘t≥m=∞,
where C˘t is the total variation of Ct on [0,t]. Then the sequence {Tm} is nondecreasing with P[lim⁡m→∞Tm=∞]=1. Thus, the stopping process Mt∧Tm is a bounded martingale for each m=1,2,…. In addition, Ct1{t≤Tm(w)} is of bounded variation. Under this situation, the values of f outside [-3m,3m] are irrelevant. We may assume without loss of generality that f has compact support, so f, f′, and f′′ are bounded. Given that the ∑i=1n‍1/2f(Xξi+1+Xξi)(CVi-CUi) is classic Riemann-Stieltjes integral, 1/2f(Xξi+1+Xξi)(AVi-AUi) is WSHB integrable for every (δ,ε)-fine belated partial stochastic division D={((Ui,Vi],ξi):i=1,2,…,n} of [0,t] (see [17]). Since the WSHB integral satisfies the linear property, we obtain the following result.Theorem 16.
Iff(x)∈C2(R) and X is a continuous semimartingale, then f(Xt) is WSHB integrable with respect to Xt on [0,t]. Furthermore,
(45)WSH∫0tf(Xs)∘dXs=∫0tf(Xs)∘dXs.


## 3.1. Henstock Approach to Define Stratonovich Integral with respect to Bounded Continuous Martingale

In [13–16, 19], a positive function δ on [0,t], 0≤t<∞, where δ does not depend on ω∈Ω, was used. From the proof of the main results in [13, 14, 16, 20], δ(ξ) is deterministic, because the quadratic variation of a Brownian motion is deterministic and Fubini’s theorem can be applied in switching the order of the two integrals. In this section, we consider martingales as integrators. To define stochastic integrals using Riemann sums, the δ-function needs to depend on Ω and stochastic intervals are needed. Therefore, we need to redefine δ as also dependent on Ω (see [12]).Definition 5.
Letδ:[0,t]×Ω→(0,t), 0≤t<∞, be a measurable function with respect to the two variables ξ∈[0,t] and ω∈Ω. Then δ is called alocally stopping process [12] if, for each ξ∈[0,t], ξ+δ is a stopping time.
For each(c,d]⊂[0,t], we denote the measure [15] induced by the quadratic variation process of an adapted process X={Xt,0≤t<∞} by
(11)μX(c,d]≜EXd-Xc.Definition 6.
Letδ be a locally stopping process and X={Xt,Ft;0≤t<∞}. A finite collection of stochastic interval-point pairs (((Si,Ti],ξi),i=1,2,…,n),  where Si and Ti are stopping times of X for all i=1,2,…,n, is said to be a (δ,η)-fine belated partial stochastic division of [0,t] if(1)
for eachi(Si,Ti] is a stochastic interval and, for each ω∈Ω,(Si,Ti],  i=1,2,…,n, are disjoint left-open subintervals of [0,1];(2)
each((Si,Ti],ξi) is a δ-fine belated division; that is, for each ω∈Ω, one has
(12)(Si(ω),Ti(ω)]⊂[ξi,ξi+δ(ξi,ω)];(3)
for anyη>0|μX(0,t]-∑i=1n‍μX(Si,Ti]|<η.
For the caseSi(ω)=Ti(ω), (Si(ω),Ti(ω)] denotes {Ti(ω)}. Note that, for each ξ∈[0,t], ξ+δ is a stopping time. Thus, for each ξi, there exists a stochastic interval (Si(ω),Ti(ω)] with Si(ω)<Ti(ω) such that (Si(ω),Ti(ω)]⊂[ξi,ξi+δ(ξi,ω)]. For each ω∈Ω, namely,
(13)Si(ω)=ξi(ω),Ti(ω)=inf⁡{ξi+δ(ξi,ω),t}.
In addition, by Vitali’s covering theorem, the (δ,η)-fine belated partial stochastic division could cover [0,t] except for an arbitrarily small μX-measure [19, p. 44] [12, 16].Definition 7.
An adapted processY={Yt,Ft;0≤t<∞} in (Ω,F,P) is said to be weakly Stratonovich-Henstock belated (denoted by WSHB) integrable to a process A in (Ω,F,Ftt≥0,P) on [0,t], 0≤t<∞, with respect to a stochastic process X if, for every ε>0, there exists a locally stopping process δ on [0,t]×Ω for which
(14)E∑i=1nYξi+1,ω+Yξi,ω2XTiω,ω-XSiω,ω-ATiω,ω-ASiω,ωYξi+1,ω+Yξi,ω22≤ε,
where ξn+1=t. For succinctness, one may write
(15)E∑i=1nYξi+1,ω+Yξi,ω2XTi-XSi-ATi-ASi2≤ε,
for every (δ,η)-fine belated partial stochastic division D={((Si,Ti],ξi),i=1,2,…,n} of [0,t].Proposition 8.
If an adapted processY={Yt,Ft;0≤t<∞} is WSHB integrable with respect to stochastic process X, then the weakly Henstock variational belated integral of Y is unique a.s.In light of Proposition8, we will denote the integral of the process Y with respect to stochastic process X by the notation
(16)WSH∫0tYs∘dWs≜At.Proposition 9.
LetY and Y′={Yt′,Ft;0≤t<∞} be WSHB integrable with respect to stochastic process X and α∈R. Then Y+Y′ and αY are WSHB integrable with respect to stochastic process X. Furthermore,
(17)WSH∫0tYs+Ys′∘dXs=WSH∫0tYs∘dXs+WSH∫0tYs′∘dXs;WSH∫0tαYs∘dXs=αWSH∫0tYs∘dXs.Proposition 10.
If stochastic processY is WSHB integrable with respect to stochastic process X, then so is it on subinterval [c,d]⊂[0,t].Proposition 11.
Let the adapted processY be WSHB integrable on [0,a] and [a,t] with respect to stochastic process X, where 0<a<t. Then Y is WSHB integrable on [0,t] and, furthermore,
(18)WSH∫0tYs∘dXs=WSH∫0aYs∘dXs+WSH∫atYs∘dXs.The proofs of Propositions8 to 11 are similar to the results of classical Henstock integral [17]; hence we omit these proofs in this paper.Next, we will consider the relationship between WSHB integral and Stratonovich integral with respect to continuous martingale. Here, we assume that the continuous martingale is bounded. In the next section we will address the continuous local martingale. Note that we can choose the stopping time such that a continuous local martingale becomes a bounded continuous martingale up to any finite stopping time [18, p. 44].Theorem 12.
Iff(x)∈C2(R) and X={Xt,Ft;0≤t<∞} is bounded continuous martingale, then f(Xt) is WSHB integrable with respect to bounded continuous martingale X on [0,t]. Furthermore,
(19)WSH∫0tf(Xs)∘dXs=∫0tf(Xs)∘dXs.Proof.
SinceX is a bounded continuous martingale and f(x)∈C2(R), then f(Xt), f′Xt, and f′′(Xt) are bounded and continuous with respect to t. By Definition 1, the classical Stratonovich integral with respect to bounded continuous martingale X is
(20)∫0tfXs∘dXs=∫0tfXsdXs+∫0tf′XsdXs.
Let D1=si,ti,ξii=1n be a (δ,η)-fine belated partial stochastic division of [0,t]. Then,
(21)E∑i=1n-112fXξi+fXξi+1Xti-Xsi-∫sitifXs∘dXs2=E∑i=1n-1f(Xξi)(Xti-Xsi)+∑i=1n-112fXξi+1-fXξiXti-Xsi-∫sitif(Xs)∘dXs2=E∑i=1n-1fXξiXti-Xsi+∑i=1n-112fXξi+1-fXξ1Xti-Xsi-∫sitifXsdXs-∫siti∂fXs∂xdXs2≤2E∑i=1n-1fXξiXti-Xsi-∫sitifXsdXs2+E∑i=1n-1fXξi+1-fXξiXti-Xsi-∫siti∂fXs∂xXs2=2R1+R2,
where
(22)R1=E∑i=1n-1fXξiXti-Xsi-∫sitifXsdXs2,R2=E∑i=1n-1fXξi+1-fXξiXti-Xsi-∫siti∂fXs∂xdXs2.
From [12], given ε1>0, there exists a (δ1,η1)-fine belated partial division of [0,t], D1=si,ti,ξii=1n, such that R1<ε1. We just need to consider R2. By mean value theorem, fXξi+1-fXξi=∂f/∂x(Xi~)(Xξi+1-Xξi), where Xi~∈[Xξi,Xξi+1]. Since Xt is continuous with respect to t, there exists a ξi^∈[ξi,ξi+1] such that Xξi^=Xi~ a.s. Similarly, there exists ξi¯∈[si,ti] such that ∫siti∂f(Xt)/∂xdXt=f′(Xξi¯)Xti-Xsi2 in L2(Ω) space. Then,
(23)R2=E∑i=1n-1(fXξi+1-f(Xξi))(Xti-Xsi)-∫siti∂fXt∂xdXt2=E∑i=1n-1∂fXξi¯∂xf′Xξi^Xξi+1-XξiXti-Xsi-∂fXξi¯∂xXti-Xsi22≤2E∑i=1n-1Xti-Xsi2f′Xξi^(Xξi+1-Xξi)(Xti-Xsi)-f′Xξi^Xti-Xsi2∑i=1n-12+2E∑i=1n-1f′Xξi^Xti-Xsi2-f′Xξi¯Xti-Xsi2∑i=1n-12.
Let
(24)r1=E∑i=1n-1f′Xξi^Xξi+1-XξiXti-Xsi-f′Xξi^(Xti-Xsi)2∑i=1n-12,r2=E∑i=1n-1f′Xξi^Xti-Xsi2-f′Xξi¯Xti-Xsi22,r1≤KE∑i=1n-1Xξi+1-XξiXti-Xsi-Xti-Xsi22≤KE∑i=1n-1Xti-XsiXti-Xsi-Xξi+1-Xξi2≤KE∑i=1n-1‍Xti-XsiXξi+1-Xti+Xsi-Xξi2≤K∑i=1n-1‍EXξi+1-Xti+Xsi-Xξi2EXti-Xsi2,
where K=sup⁡{f′Wt:0≤t≤1&ω∈Ω}. Given that Xt is a bounded martingale,
(25)EXξi+1-Xti+Xsi-Xξi2=EXξi+1-Xti2+EXsi-Xξi2=EXξi+1-EXti+EXsi-EXξi.
However, there exists a (δ2,η2)-fine belated partial stochastic division D2=ξi,ti,ξi:i=1,2,…,n} of [0,t] such that |μX(0,t]-∑i=1nμX(si,ti]|<η2. Then we have
(26)EXξi+1-EXti+EXsi-EXξi<η2.
Hence,
(27)r1≤Kε2∑i=1n-1EXti-Xsi2≤KEXtη2.
Since X is bounded continuous martingale, by Lemma 5.10 in [18, p. 33], we have
(28)E∑i=1n-1Xti-Xsi4⟶0asn⟶∞.
Hence, given ε2>0,
(29)r2≤sup⁡1≤i≤n&ω∈Ωf′Xξi^-f′Xξi¯E∑i=1n-1‍Xti-Xsi4≤2Kε2.
Now take η=min⁡{η1,η2} and let δ(ξi)=min⁡{δ1(ξi),δ2,ξi+1-ξi}; then a (δ,η)-fine belated partial division D=ξi,ti,ξi:i=1,2,…,n} of [0,t] is both (δ1,η1)-fine belated partial stochastic division and (δ2,η2)-fine belated partial stochastic division. Hence, for ε=max⁡{ε1,η2,ε2}, we have
(30)E∑i=1n-1‍fξi+fξi+12Wti-Wsi-∫sitiXt∘dWt2≤R1+r1+r2≤K¯ε,
where K¯ is a constant. Hence, fXt is WSHB integrable on [0,t] and, by the unique property of WSHB,
(31)WSH∫0tfXs∘dXs=∫0tfXs∘dXs.


## 3.2.  Henstock Approach to Define Stratonovich Integral with respect to Continuous Local Martingale

Lemma 13.
LetX={Xt,Ft;0≤t<∞} be a continuous local martingale. Suppose f is WSHB integrable to a process At with respect to local continuous martingale X. Let T be a stopping time; then f is WSHB integrable to a process AT with respect to XT, where XT={Xt∧T;0≤t<∞} and AT={At∧T;0≤t<∞}.Proof.
Letε>0 and δ(ξ,ω)>0 be given as in Definition 6 for the WSHB integral of f with respect to continuous local martingale X. Define
(32)δTξ,ω≜Tω-ξ,ifξ<Tω≤ξ+δξ,ωδξ,ω,otherwise.
Then δT(ξ,ω) is locally stopping process, since (T(ω)∧ξ)∧(ξ+δ(ξ,ω)) is a stopping time and {ω:T(ω)<ξ}∈Ft if ξ<t. Let ((U,V],ξ) be a (δT(ξ,ω),η)-fine belated stochastic interval-point pair.(i)
IfT(w)≤ξ, then
(33)XT(V(ω),ω)-XT(U(ω),ω)=XT(T(ω),ω)-XT(T(ω),ω)=0.(ii)
Ifξ<T(ω)≤ξ+δ(ξ,ω), then [U,V]⊂[ξ,T]⊂[ξ,ξ+δ(ξ,ω)]. Consequently, ((U,V],ξ) will be a (δ(ξ,ω),η)-fine belated stochastic interval-point pair, U∧T=U and V∧T=V.(iii)
IfT(ω)>ξ+δ(ξ,ω), then
(34)XT(U(ω),ω)-XTVω,ω=XVω∧Tω,ω-XUω∧Tω,ω=X(U(ω),ω)-XVω,ω.It is easy to verify that D={((Ui,Vi],ξi),i=1,2,…,n} is also a (δT(ξ,ω),η)-fine belated stochastic division of [0,t∧T(w)] only if D={((Ui,Vi],ξi),i=1,2,…,n} is a (δ(ξ,ω),η)-fine belated stochastic division of [0,t].
Similar results hold forAT. Since the D={((Ui,Vi],ξi),i=1,2,…,n} is also a (δT(ξ,ω),η)-fine belated stochastic division of [0,t∧T(w)], by the property of being WSHB integrable, we have that f is WSHB integrable to a process AT with respect to XT.Lemma 14.
LetX={Xt,Ft;0≤t<∞} be a continuous local martingale with the corresponding nondecreasing sequence of stopping times Tnn=1∞. Suppose that f is WSHB integrable to a process A with respect to X; then f is WSHB integrable to a process ATn with respect to XTn for each n=1,2,…. Furthermore, for each t∈[0,t],lim⁡n→∞ATn=A a.s.Proof.
This follows directly form Lemma13 and the uniqueness of the WSHB integral. Then we have the last statement
(35)lim⁡n→∞ATn=Aa.s.Theorem 15.
Iff(x)∈C2(R) and X={Xt,Ft;0≤t<∞} is a continuous local martingale, then f(Xt) is WSHB integrable with respect to X on [0,t]. Furthermore,
(36)(WSH)∫0tfXs∘dXs=∫0tfXs∘dXs.Proof.
First we introduce, for eachm≥1, the stopping time
(37)Tm=inf⁡t≥0;Xt≥morXt≥m.
The resulting sequence {Tm} is nondecreasing with P[lim⁡m→∞Tm=∞]=1. Thus, the stopping process Xt∧Tm is bounded; that is, Xt∧Tm is a bounded martingale for each m=1,2,…. Under this situation, the values of f outside [-m,m] are irrelevant. We may assume without loss of generality that f has compact support, so f, f′, and f′′ are bounded.
LetD={((Ui,Vi],ξi),i=1,2,…,n} be a (δ(ξ,ω),η)-fine belated stochastic division of [0,1]. From Lemma 13, we have that, for each m≥1, Dm={((Ui,Vi],ξi),i=1,2,…,n} is a (δTm(ξ,ω),η)-fine belated stochastic division of [0,t∧Tm(w)]. From Lemma 13, there is a process ATm={At∧Tm;0≤t<∞} such that
(38)E∑i=1n-112fXξi+1+XξiXVi∧Tm-XUi∧Tm∑i=1n-1-AVi∧Tm-AUi∧Tm2<ε,
for every (δ,η)-fine belated partial stochastic division D={((Ui,Vi],ξi):i=1,2,…,n} of [0,t]. Since
(39)lim⁡m→∞∑i=1n-1fXξi+1+XξiXVi∧Tm-XUi∧Tm-AVi∧Tm-AUi∧Tm∑i=1n-1=∑i=1n-1f(Xξi+1+Xξi)(XVi-XUi)-AVi-AUi∑i=1n-1a.s.,
we must have
(40)E∑i=1n-1fXξi+1+XξiXVi-XUi-AVi-AUi=Elim⁡m→∞∑i=1n-112fXξi+1+XξiXVi∧Tm-XUi∧Tm-AVi∧Tm-AUi∧Tm∑i=1n-12=Eliminf⁡m→∞∑i=1n-112fXξi+1+XξiXVi∧Tm-XUi∧Tm-AVi∧Tm-AUi∧Tm∑i=1n-12≤liminf⁡m→∞E∑i=1n-112fXξi+1+XξiXVi∧Tm-XUi∧Tm-AVi∧Tm-AUi∧Tm∑i=1n-12<liminf⁡m→∞ε≤ε.
This shows that f(Xt) is WSHB integrable on [0,t] and
(41)(WSH)∫0tf(Xs)∘dXs=∫0tf(Xs)∘dXs.


## 3.3. Henstock Approach to Define Stratonovich Integral with respect to Continuous Semimartingale

LetX={Xt,Ft;0≤t<∞} be a continuous semimartingale such that
(42)Xt=X0+Mt+Ct,
where Mt is a continuous local martingale and Ct is adapted, continuous process of bounded variation with C0=0 a.s. We have
(43)12fXξi+1+XξiXVi-XUi=12fXξi+1+XξiMVi-MUi+12fXξi+1+XξiCVi-CUi.
Now we consider a stopping time
(44)Tm=0;vvivifX0≥minf⁡C˘tt≥0;Mt≥morMt≥morC˘t≥m;vvviivifX0<m∞;ifX0<m&vvvivinf⁡C˘tt≥0;Mt≥morvvvvvvvvMt≥morC˘t≥m=∞,
where C˘t is the total variation of Ct on [0,t]. Then the sequence {Tm} is nondecreasing with P[lim⁡m→∞Tm=∞]=1. Thus, the stopping process Mt∧Tm is a bounded martingale for each m=1,2,…. In addition, Ct1{t≤Tm(w)} is of bounded variation. Under this situation, the values of f outside [-3m,3m] are irrelevant. We may assume without loss of generality that f has compact support, so f, f′, and f′′ are bounded. Given that the ∑i=1n‍1/2f(Xξi+1+Xξi)(CVi-CUi) is classic Riemann-Stieltjes integral, 1/2f(Xξi+1+Xξi)(AVi-AUi) is WSHB integrable for every (δ,ε)-fine belated partial stochastic division D={((Ui,Vi],ξi):i=1,2,…,n} of [0,t] (see [17]). Since the WSHB integral satisfies the linear property, we obtain the following result.Theorem 16.
Iff(x)∈C2(R) and X is a continuous semimartingale, then f(Xt) is WSHB integrable with respect to Xt on [0,t]. Furthermore,
(45)WSH∫0tf(Xs)∘dXs=∫0tf(Xs)∘dXs.


## 4. Conclusion

From Theorem16, if f is of class C2(R) and X is a continuous semimartingale, f(Xt) is WSHB integrable to ∫0t‍f(Xs)∘dXs on [0,t]. Now we consider the Itô formula for Stratonovich integral; if f∈C3(R), as shown in Proposition 4,
(46)fXt=fX0+∫0tf′Xs∘dXs.
We substitute WSH∫0t‍f′Xs∘dXs with ∫0t‍f′Xs∘dXs. Then,
(47)fXt=fX0+WSH∫0tf′Xs∘dXs.
In conclusion, from the definition of Stratonovich integral with respect to continuous semimartingale using Henstock method, we manage to keep the important properties of the classical Stratonovich integral and also probably enlarge the scope of the integrands which satisfy classical Stratonovich integral Itô formula (46).



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