Hindawi Publishing Corporation International Journal of Stochastic Analysis Volume 2014, Article ID 534864, 7 pages http://dx.doi.org/10.1155/2014/534864

Research Article On Henstock Method to Stratonovich with respect to Continuous

Haifeng Yang and Tin Lam Toh

National Institute of Education, Nanyang Technological University, Singapore 637616

Correspondence should be addressed to Tin Lam Toh; [email protected]

Received 15 September 2014; Revised 26 November 2014; Accepted 26 November 2014; Published 18 December 2014

Academic Editor: Fred Espen Benth

Copyright © 2014 H. Yang and T. L. Toh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We will use the Henstock (or generalized Riemann) approach to define the Stratonovich integral with respect to continuous 2 semimartingale in 𝐿 space. Our definition of Stratonovich integral encompasses the classical definition of Stratonovich integral.

1. Introduction Further, to consider continuous as an integrator, we will show that a (𝛿, 𝜂)-fine belated partial 𝑛 Classically, it has been emphasized that defining stochastic stochastic division 𝐷={(𝐼𝑘,𝜉𝑘)}𝑘=1 of [0, 𝑡] in [12, 16]isalso using Riemann approach is impossible [1,p.40], a (𝛿, 𝜂)-fine belated partial stochastic division of [0, 𝑡 ∧ 𝑆], since the integrators have paths of unbounded variation. where 𝑆 is a stopping time. Moreover, the integrands are usually highly oscillatory [1– Lastly, the definition of the Stratonovich integral with 3]. The uniform mesh in classical Riemann setting is unable respect to continuous semimartingale will be further refined. to handle highly oscillatory integrators and integrands. We divide the continuous semimartingale into two parts: a Kurzweil and Henstock [4–6] independently modified the continuous local martingale and the difference of continu- Riemann approach by using nonuniform mesh. This modi- ous, nondecreasing, and adapted processes. The former is fication leads to a larger class of integrals being studied (see discussed in Section 3.2.Thelatterhasfinitetotalvariation [4, 7–12]). on each bounded interval. Hence, integration with respect to The generalized Riemann approach, using nonuniform the latter process is classical Riemann-Stieltjes. This part has meshes, has been successful in giving an alternative definition already been addressed in [17]. to the Stratonovich integral with respect to Brownian motion We give a new definition of the Stratonovich integral (see [13]). This paper attempts to further generalize the result 2 with continuous semimartingale in 𝐿 space. This integrand of [13] to define the Stratonovich integral with respect to is weaker than the classical case. continuous semimartingale. The difficulty to extend from Brownian motion tosemi- martingale is that the of the continuous martingale may not be absolutely continuous. Hence, the 2. Stratonovich Integral definition of the Stratonovich-Henstock belated integral in [13]maynotbevalid.In[12, 14, 15], the definition of In this section, we will generalize the definition of Straton- weakly Henstock variation belated integral was used to ovich integral to include the case where the integrator is con- 2 address this difficulty. Similarly, we tap on this approach to tinuous 𝐿 -semimartingale. We will develop the definition of 2 define Stratonovich integral with respect to continuous 𝐿 - Stratonovich-Henstock belated integral where the integrator martingale. isaBrownianmotionin[13]. Let (Ω, F,𝑃)be a probability 2 International Journal of Stochastic Analysis

𝑡 (Ω, F,𝑃) 𝐶𝑛(R) 𝑓󸀠(𝑋 ) ∫ 𝑓󸀠󸀠(𝑋 )𝑑𝑀 space such that is complete. Also, let denote Then 𝑠 is a semimartingale, 0 𝑠 𝑠 is a local 𝑛 𝑡 𝑡 the class of functions which have continuous derivatives. ∫ 𝑓󸀠󸀠(𝑋 )𝑑𝐴 (1/2) ∫ 𝑓󸀠󸀠󸀠(𝑋 )𝑑⟨𝑀⟩ martingale, and 0 𝑠 𝑠 and 0 𝑠 𝑠 are bounded continuous processes. By definition of a semi- Definition 1. Let 𝑋={𝑋𝑡, F𝑡;0≤𝑡<∞}be a process. If ∞ martingale, there exists a nondecreasing sequence {𝑇𝑛}𝑛=1 of the stopping (𝑛) 𝑡 𝑡 times of the filtration {F𝑡},suchthat{𝑋𝑡 ≜𝑋𝑡∧𝑇 , F𝑡;0 ≤ 𝑛 ∫ 𝑓󸀠 (𝑋 )∘𝑑𝑋 = ∫ 𝑓󸀠 (𝑋 )𝑑𝑋 𝑡<∞}is a martingale for each 𝑛≥1and 𝑃[lim𝑛→∞𝑇𝑛 = 𝑠 𝑠 𝑠 𝑠 ∞] = 1 𝑋 0 0 ,thenonesaysthat is local martingale. If, in (8) 𝑋 =0 𝑋∈Mloc 𝑋∈M𝑐,loc 1 𝑡 addition, 0 a.s., one writes (or ,if + ⟨𝑀, ∫ 𝑓󸀠󸀠 (𝑋 )𝑑𝑀⟩ . 𝑋 𝑠 𝑠 is continuous). 2 0 𝑡

Definition 2. A continuous semimartingale 𝑋={𝑋𝑡, F𝑡;0≤ By the property of cross variation (see [18,p.143]) 𝑡<∞} is an adapted process which has the decomposition 𝑡 𝑡 ⟨𝑀, ∫ 𝑓󸀠󸀠 (𝑋 )𝑑𝑀⟩ = ∫ 𝑓󸀠󸀠 (𝑋 )𝑑⟨𝑀⟩ . 𝑋𝑡 =𝑋0 +𝑀𝑡 +𝐶𝑡, 0≤𝑡<∞a.s., 𝑠 𝑠 𝑠 𝑠 (9) (1) 0 𝑡 0 𝑀={𝑀, F ;0≤𝑡<∞}∈M𝑐,loc 𝐶 where 𝑡 𝑡 and 𝑡 has finite Hence, variation. 𝑡 𝑡 󸀠 1 󸀠󸀠 𝑓(𝑋𝑡)=𝑓(𝑋0)+∫ 𝑓 (𝑋𝑠)𝑑𝑋𝑠 + ∫ 𝑓 (𝑋𝑠)𝑑⟨𝑀⟩𝑠 Definition 3. Let 𝑋={𝑋𝑡, F𝑡;0≤𝑡<∞}and 𝑌= 0 2 0 {𝑌 , F ;0≤𝑡<∞} 𝑡 𝑡 be two continuous semimartingales such 𝑡 that 󸀠 =𝑓(𝑋0)+∫ 𝑓 (𝑋𝑠)∘𝑑𝑋𝑠. 𝑋 =𝑋 +𝑀 +𝐶;𝑌=𝑌 +𝑁 +𝐷, 0 𝑡 0 𝑡 𝑡 𝑡 0 𝑡 𝑡 (2) (10) 𝑐,loc where 𝑀 and 𝑁 are in M 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. Henstock Approach 𝑡 𝑡 𝑡 1 ∫ 𝑌𝑠 ∘𝑑𝑋𝑠 ≜ ∫ 𝑌𝑠𝑑𝑀𝑠 + ∫ 𝑌𝑠𝑑𝐶𝑠 + ⟨𝑀, 𝑁⟩𝑡 , (3) 0 0 0 2 3.1. Henstock Approach to Define Stratonovich Integral with respect to Bounded Continuous Martingale. In [13–16, 19], a where the first integral on the right-hand side of (3) is an Itoˆ positive function 𝛿 on [0, 𝑡], 0≤𝑡<∞,where𝛿 does not integral, the second one is pathwise integral of the Riemann- depend on 𝜔∈Ω, was used. From the proof of the main ⟨𝑀, 𝑁⟩ 𝑀 Stieltjes type, and 𝑡 is the cross variation process of results in [13, 14, 16, 20], 𝛿(𝜉) is deterministic, because the 𝑁 𝑀=𝑁 ⟨𝑀, 𝑁⟩ =⟨𝑀⟩ and .If ,then 𝑡 𝑡 is the quadratic quadratic variation of a Brownian motion is deterministic 𝑀 variation of (see [18,p.31]). and Fubini’s theorem can be applied in switching the order Proposition 4. 𝑓:R → R of the two integrals. In this section, we consider martingales Let be a function of class as integrators. To define stochastic integrals using Riemann 𝐶3(R) 𝑋={𝑋, F ;0 ≤ 𝑡 < ∞} and let 𝑡 𝑡 be a continuous sums, the 𝛿-function needs to depend on Ω and stochastic semimartingale with decomposition in (2).Then intervals are needed. Therefore, we need to redefine 𝛿 as also 𝑡 Ω 󸀠 dependent on (see [12]). 𝑓(𝑋𝑡)=𝑓(𝑋0)+∫ 𝑓 (𝑋𝑠)∘𝑑𝑋𝑠 a.s. (4) 0 Definition 5. Let 𝛿 : [0, 𝑡] × Ω → (0,, 𝑡) 0≤𝑡<∞,be Proof. By Itoformula[ˆ 18,p.149],wehave a measurable function with respect to the two variables 𝜉∈ 𝑡 𝑡 [0, 𝑡] and 𝜔∈Ω.Then𝛿 is called a locally stopping process 󸀠 1 󸀠󸀠 𝑓(𝑋)=𝑓(𝑋)+∫ 𝑓 (𝑋 )𝑑𝑋 + ∫ 𝑓 (𝑋 )𝑑⟨𝑀⟩ . [12] if, for each 𝜉∈[0,𝑡], 𝜉+𝛿is a stopping time. 𝑡 0 𝑠 𝑠 2 𝑠 𝑠 0 0 For each (𝑐,𝑑] ⊂ [0,𝑡],wedenotethemeasure[15] (5) induced by the quadratic variation process of an adapted We only have to prove that process 𝑋={𝑋𝑡,0≤𝑡<∞}by 𝑡 𝑡 1 𝑡 𝜇 (𝑐, 𝑑] ≜𝐸(⟨𝑋⟩ − ⟨𝑋⟩ ) . ∫ 𝑓󸀠 (𝑋 )∘𝑑𝑋 = ∫ 𝑓󸀠 (𝑋 )𝑑𝑋 + ∫ 𝑓󸀠󸀠 (𝑋 )𝑑⟨𝑀⟩ . 𝑋 𝑑 𝑐 (11) 𝑠 𝑠 𝑠 𝑠 2 𝑠 𝑠 0 0 0 𝛿 𝑋= (6) Definition 6. Let be a locally stopping process and {𝑋𝑡, F𝑡;0≤𝑡<∞}. A finite collection of stochastic interval- 󸀠 2 Since 𝑓 (𝑋𝑠)∈𝐶(R),then,byItoformula,ˆ point pairs (((𝑆𝑖,𝑇𝑖], 𝜉𝑖),𝑖 = 1,2,...,𝑛),where𝑆𝑖 and 𝑇𝑖 are 𝑡 𝑡 stopping times of 𝑋 for all 𝑖=1,2,...,𝑛,issaidtobea(𝛿, 𝜂)- 󸀠 󸀠 󸀠󸀠 󸀠󸀠 [0, 𝑡] 𝑓 (𝑋𝑠)=𝑓 (𝑋0)+∫ 𝑓 (𝑋𝑠)𝑑𝑀𝑠 + ∫ 𝑓 (𝑋𝑠)𝑑𝐴𝑠 finebelatedpartialstochasticdivisionof if 0 0 𝑖(𝑆 ,𝑇] 𝑡 (7) (1) for each 𝑖 𝑖 is a stochastic interval and, for each 1 󸀠󸀠󸀠 𝜔 ∈ Ω, (𝑆 ,𝑇] 𝑖 = 1,2,...,𝑛 + ∫ 𝑓 (𝑋𝑠)𝑑⟨𝑀⟩𝑠 . 𝑖 𝑖 , , are disjoint left-open 2 0 subintervals of [0, 1]; International Journal of Stochastic Analysis 3

󸀠 (2) each ((𝑆𝑖,𝑇𝑖], 𝜉𝑖) is a 𝛿-fine belated division; that is, for 𝑌+𝑌 and 𝛼𝑌 are WSHB integrable with respect to stochastic each 𝜔∈Ω,onehas process 𝑋.Furthermore,

(𝑆 (𝜔) ,𝑇 (𝜔)]⊂[𝜉,𝜉 +𝛿(𝜉, 𝜔)] ; 𝑖 𝑖 𝑖 𝑖 𝑖 (12) 𝑡 𝑡 󸀠 (WSH) ∫ (𝑌𝑠 +𝑌𝑠 )∘𝑑𝑋𝑠 = (WSH) ∫ 𝑌𝑠 ∘𝑑𝑋𝑠 0 0 𝜂>0|𝜇 (0, 𝑡] − ∑𝑛 𝜇 (𝑆 ,𝑇]| < 𝜂 (3) for any 𝑋 𝑖=1 𝑋 𝑖 𝑖 . 𝑡 󸀠 + (WSH) ∫ 𝑌𝑠 ∘𝑑𝑋𝑠; (17) For the case 𝑆𝑖(𝜔) =𝑖 𝑇 (𝜔), (𝑆𝑖(𝜔),𝑖 𝑇 (𝜔)] denotes {𝑇𝑖(𝜔)}. 0 𝜉∈[0,𝑡]𝜉+𝛿 Note that, for each , is a stopping time. Thus, 𝑡 𝑡 𝜉 (𝑆 (𝜔), 𝑇 (𝜔)] for each 𝑖, there exists a stochastic interval 𝑖 𝑖 with (WSH) ∫ 𝛼𝑌𝑠 ∘𝑑𝑋𝑠 =𝛼(WSH) ∫ 𝑌𝑠 ∘𝑑𝑋𝑠. 𝑆𝑖(𝜔) <𝑖 𝑇 (𝜔) such that (𝑆𝑖(𝜔),𝑖 𝑇 (𝜔)] ⊂ [𝜉𝑖,𝜉𝑖 +𝛿(𝜉𝑖,𝜔)].For 0 0 each 𝜔∈Ω,namely, Proposition 10. If 𝑌 is WSHB integrable 𝑆 (𝜔) =𝜉(𝜔) ,𝑇(𝜔) = {𝜉 +𝛿(𝜉 ,𝜔) ,𝑡} . 𝑖 𝑖 𝑖 inf 𝑖 𝑖 (13) with respect to stochastic process 𝑋,thensoisitonsubinterval [𝑐, 𝑑] ⊂ [0, 𝑡]. In addition, by Vitali’s covering theorem, the (𝛿, 𝜂)-fine belated partial stochastic division could cover [0, 𝑡] except for Proposition 11. Let the adapted process 𝑌 be WSHB integrable an arbitrarily small 𝜇𝑋-measure [19,p.44][12, 16]. on [0, 𝑎] and [𝑎, 𝑡] with respect to stochastic process 𝑋,where 0<𝑎<𝑡.Then𝑌 is WSHB integrable on [0, 𝑡] and, 𝑌={𝑌, F ;0 ≤ 𝑡 < Definition 7. An adapted process 𝑡 𝑡 furthermore, ∞} in (Ω, F,𝑃)is said to be weakly Stratonovich-Henstock belated (denoted by WSHB) integrable to a process 𝐴 in 𝑡 𝑎 (Ω, F,{F𝑡}𝑡≥0,𝑃) on [0, 𝑡], 0≤𝑡<∞,withrespecttoa (WSH) ∫ 𝑌𝑠 ∘𝑑𝑋𝑠 = (WSH) ∫ 𝑌𝑠 ∘𝑑𝑋𝑠 stochastic process 𝑋 if, for every 𝜀>0, there exists a locally 0 0 𝛿 [0, 𝑡] × Ω stopping process on for which 𝑡 (18) + ( ) ∫ 𝑌 ∘𝑑𝑋. 󵄨 𝑛 WSH 𝑠 𝑠 󵄨 𝑌(𝜉 ,𝜔)+𝑌(𝜉,𝜔) 𝑎 𝐸 󵄨∑ { 𝑖+1 𝑖 (𝑋(𝑇(𝜔) ,𝜔)−𝑋(𝑆(𝜔) , 𝜔)) 󵄨 2 𝑖 𝑖 󵄨𝑖=1 The proofs of Propositions 8 to 11 are similar to the results 󵄨2 󵄨 󵄨 of classical Henstock integral [17]; hence we omit these proofs −[𝐴(𝑇𝑖 (𝜔) ,𝜔)−𝐴(𝑆𝑖 (𝜔) , 𝜔)]󵄨 } ≤𝜀, 󵄨 in this paper. (14) Next, we will consider the relationship between WSHB integral and Stratonovich integral with respect to continuous where 𝜉𝑛+1 =𝑡. For succinctness, one may write martingale. Here, we assume that the continuous martingale is bounded. In the next section we will address the continuous 󵄨 󵄨2 󵄨 𝑛 𝑌(𝜉 ,𝜔)+𝑌(𝜉,𝜔) 󵄨 local martingale. Note that we can choose the stopping time 󵄨 𝑖+1 𝑖 󵄨 𝐸(󵄨∑ (𝑋𝑇 −𝑋𝑆 )−(𝐴𝑇 −𝐴𝑆 )󵄨 ) such that a continuous local martingale becomes a bounded 󵄨 2 𝑖 𝑖 𝑖 𝑖 󵄨 󵄨𝑖=1 󵄨 continuous martingale up to any finite stopping time [18,p. 44]. ≤𝜀, 2 (15) Theorem 12. If 𝑓(𝑥) ∈𝐶 (R) and 𝑋={𝑋𝑡, F𝑡;0 ≤ 𝑡 < ∞} is bounded continuous martingale, then 𝑓(𝑋𝑡) is WSHB for every (𝛿, 𝜂)-fine belated partial stochastic division 𝐷= integrable with respect to bounded continuous martingale 𝑋 on {((𝑆𝑖,𝑇𝑖], 𝜉𝑖),𝑖=1,2,...,𝑛}of [0, 𝑡]. [0, 𝑡].Furthermore, Proposition 8. 𝑌={𝑌, F ;0≤𝑡<∞} If an adapted process 𝑡 𝑡 𝑡 𝑡 𝑋 is WSHB integrable with respect to stochastic process ,then (WSH) ∫ 𝑓 (𝑋𝑠) ∘𝑑𝑋𝑠 = ∫ 𝑓 (𝑋𝑠) ∘𝑑𝑋𝑠. (19) the weakly Henstock variational belated integral of 𝑌 is unique 0 0 a.s. Proof. Since 𝑋 is a bounded continuous martingale and In light of Proposition 8,wewilldenotetheintegralofthe 𝑓(𝑥)2 ∈𝐶 (R) 𝑓(𝑋 ) 𝑓󸀠(𝑋 ) 𝑓󸀠󸀠(𝑋 ) process 𝑌 with respect to stochastic process 𝑋 by the notation ,then 𝑡 , 𝑡 ,and 𝑡 are bounded and continuous with respect to 𝑡.ByDefinition1,theclassical 𝑡 Stratonovich integral with respect to bounded continuous (WSH) ∫ 𝑌𝑠 ∘𝑑𝑊𝑠 ≜𝐴𝑡. (16) martingale 𝑋 is 0

󸀠 󸀠 𝑡 𝑡 𝑡 Proposition 9. Let 𝑌 and 𝑌 ={𝑌𝑡 , F𝑡;0≤𝑡<∞}be WSHB 󸀠 𝑋 𝛼∈R ∫ 𝑓(𝑋𝑠)∘𝑑𝑋𝑠 = ∫ 𝑓(𝑋𝑠)𝑑𝑋𝑠 + ∫ 𝑓 (𝑋𝑠)𝑑⟨𝑋⟩𝑠 . (20) integrable with respect to stochastic process and .Then 0 0 0 4 International Journal of Stochastic Analysis

𝑛 𝑡𝑖 Let 𝐷1 = {([𝑠𝑖,𝑡𝑖], 𝜉𝑖)}𝑖=1 be a (𝛿, 𝜂)-fine belated partial there exists 𝜉𝑖 ∈[𝑠𝑖,𝑡𝑖] such that ∫ (𝜕𝑓(𝑋𝑡)/𝜕𝑥)𝑑⟨𝑋⟩𝑡 = 𝑠𝑖 stochastic division of [0, 𝑡].Then, 󸀠 2 2 𝑓 (𝑋 )(𝑋𝑡 −𝑋𝑠 ) in 𝐿 (Ω) space. Then, 𝜉𝑖 𝑖 𝑖 󵄨 󵄨𝑛−1 1 󵄨 󵄨𝑛−1 𝐸 󵄨∑ (𝑓 𝜉(𝑋 )+𝑓(𝑋𝜉 )) (𝑋𝑡 −𝑋𝑠 ) 󵄨 󵄨 2 𝑖 𝑖+1 𝑖 𝑖 󵄨 󵄨𝑖=1 𝑅2 =𝐸󵄨∑ (𝑓 𝜉(𝑋 )−𝑓(𝑋𝜉 )) (𝑋𝑡 −𝑋𝑠 ) 󵄨 󵄨 𝑖+1 𝑖 𝑖 𝑖 󵄨𝑖=1 󵄨2 𝑡𝑖 󵄨 󵄨 󵄨2 − ∫ 𝑓(𝑋)∘𝑑𝑋󵄨 𝑡𝑖 󵄨 𝑠 𝑠󵄨 𝜕𝑓 (𝑋 𝑡) 󵄨 𝑠 󵄨 − ∫ 𝑑 ⟨𝑋⟩ 󵄨 𝑖 󵄨 𝑡󵄨 𝑠 𝜕𝑥 󵄨 󵄨 𝑖 󵄨 󵄨𝑛−1 󵄨 󵄨 =𝐸󵄨∑ 𝑓(𝑋𝜉 )(𝑋𝑡 −𝑋𝑠 ) 󵄨𝑛−1 󵄨 𝑖 𝑖 𝑖 󵄨 { 󵄨𝑖=1 󵄨 󸀠 =𝐸󵄨∑ {𝑓 (𝑋𝜉̂)(𝑋𝜉 −𝑋𝜉 )(𝑋𝑡 −𝑋𝑠 ) 󵄨 𝑖 𝑖+1 𝑖 𝑖 𝑖 󵄨𝑖=1 𝑛−1 1 󵄨 { + ∑ (𝑓 𝜉(𝑋 )−𝑓(𝑋𝜉 )) (𝑋𝑡 −𝑋𝑠 ) 2 𝑖+1 𝑖 𝑖 𝑖 󵄨2 𝑖=1 𝜕𝑓 (𝑋 ) 󵄨 𝜉 2}󵄨 − 𝑖 (𝑋 −𝑋 ) 󵄨 𝑡 󵄨2 𝑡𝑖 𝑠𝑖 }󵄨 𝑖 󵄨 𝜕𝑥 󵄨 󵄨 󵄨 − ∫ 𝑓(𝑋𝑠)∘𝑑𝑋𝑠󵄨 }󵄨 (23) 𝑠 󵄨 𝑖 󵄨 󵄨 󵄨𝑛−1 󵄨𝑛−1 󵄨 󸀠 󵄨 ≤2𝐸󵄨∑ {𝑓 (𝑋̂)(𝑋𝜉 −𝑋𝜉 )(𝑋𝑡 −𝑋𝑠 ) 󵄨 󵄨 𝜉𝑖 𝑖+1 𝑖 𝑖 𝑖 =𝐸󵄨∑ 𝑓(𝑋𝜉 )(𝑋𝑡 −𝑋𝑠 ) 󵄨𝑖=1 󵄨 𝑖 𝑖 𝑖 (21) 󵄨𝑖=1 󵄨2 󵄨 𝑛−1 󸀠 2 󵄨 1 −𝑓 (𝑋𝜉̂)(𝑋𝑡 −𝑋𝑠 ) } 󵄨 + ∑ (𝑓 (𝑋 )−𝑓(𝑋 )) (𝑋 −𝑋 ) 𝑖 𝑖 𝑖 󵄨 𝜉𝑖+1 𝜉1 𝑡𝑖 𝑠𝑖 󵄨 𝑖=1 2 󵄨𝑛−1 󵄨2 󵄨 󸀠 2 𝑡𝑖 𝑡𝑖 󵄨 +2𝐸󵄨∑ 𝑓 (𝑋 )(𝑋 −𝑋 ) 𝜕𝑓 (𝑋 𝑠) 󵄨 󵄨 𝜉̂ 𝑡 𝑠 − ∫ 𝑓(𝑋)𝑑𝑋 − ∫ 𝑑 ⟨𝑋⟩ 󵄨 󵄨 𝑖 𝑖 𝑖 𝑠 𝑠 𝜕𝑥 𝑠󵄨 󵄨𝑖=1 𝑠𝑖 𝑠𝑖 󵄨 󵄨2 2 󵄨 󵄨𝑛−1 𝑡 󵄨 󸀠 2 󵄨 󵄨 𝑖 󵄨 −𝑓 (𝑋 )(𝑋 −𝑋 ) 󵄨 . 󵄨 󵄨 𝜉 𝑡 𝑠 󵄨 ≤2𝐸󵄨∑ 𝑓(𝑋𝜉 )(𝑋𝑡 −𝑋𝑠 )−∫ 𝑓(𝑋𝑠)𝑑𝑋𝑠󵄨 𝑖 𝑖 𝑖 󵄨 󵄨 𝑖 𝑖 𝑖 󵄨 󵄨 󵄨𝑖=1 𝑠𝑖 󵄨 󵄨 󵄨𝑛−1 󵄨 Let +𝐸󵄨∑ (𝑓 𝜉(𝑋 )−𝑓(𝑋𝜉 )) (𝑋𝑡 −𝑋𝑠 ) 󵄨 𝑖+1 𝑖 𝑖 𝑖 󵄨𝑖=1 󵄨 󵄨𝑛−1 󵄨 󸀠 𝑡 󵄨2 𝑟1 =𝐸󵄨∑ {𝑓 (𝑋̂)(𝑋𝜉 −𝑋𝜉 )(𝑋𝑡 −𝑋𝑠 ) 𝑖 𝜕𝑓 (𝑋 ) 󵄨 󵄨 𝜉𝑖 𝑖+1 𝑖 𝑖 𝑖 𝑠 󵄨 󵄨𝑖=1 − ∫ ⟨𝑋⟩𝑠󵄨 =2𝑅1 +𝑅2, 𝑠 𝜕𝑥 󵄨 𝑖 󵄨 󵄨2 󵄨 󸀠 2 󵄨 −𝑓 (𝑋̂)(𝑋𝑡 −𝑋𝑠 ) }󵄨 , 𝜉𝑖 𝑖 𝑖 󵄨 where 󵄨

2 󵄨𝑛−1 󵄨2 󵄨𝑛−1 𝑡 󵄨 󵄨 2 2󵄨 󵄨 𝑖 󵄨 󵄨 󸀠 󸀠 󵄨 󵄨 󵄨 𝑟2 =𝐸󵄨∑ 𝑓 (𝑋̂)(𝑋𝑡 −𝑋𝑠 ) −𝑓 (𝑋 )(𝑋𝑡 −𝑋𝑠 ) 󵄨 , 𝑅1 =𝐸󵄨∑ 𝑓(𝑋𝜉 )(𝑋𝑡 −𝑋𝑠 )−∫ 𝑓(𝑋𝑠)𝑑𝑋𝑠󵄨 , 󵄨 𝜉𝑖 𝑖 𝑖 𝜉𝑖 𝑖 𝑖 󵄨 󵄨 𝑖 𝑖 𝑖 󵄨 󵄨𝑖=1 󵄨 󵄨𝑖=1 𝑠𝑖 󵄨 󵄨 󵄨𝑛−1 󵄨2 󵄨𝑛−1 󵄨 2󵄨 󵄨 𝑟 ≤𝐾𝐸󵄨∑ (𝑋 −𝑋 )(𝑋 −𝑋 )−(𝑋 −𝑋 ) 󵄨 𝑅2 =𝐸󵄨∑ 𝑓(𝑋𝜉 )−𝑓(𝑋𝜉 )(𝑋𝑡 −𝑋𝑠 ) (22) 1 󵄨 𝜉 𝜉 𝑡 𝑠 𝑡 𝑠 󵄨 󵄨 𝑖+1 𝑖 𝑖 𝑖 󵄨 𝑖+1 𝑖 𝑖 𝑖 𝑖 𝑖 󵄨 󵄨𝑖=1 󵄨𝑖=1 󵄨

󵄨2 󵄨𝑛−1 󵄨2 𝑡𝑖 𝜕𝑓 (𝑋 ) 󵄨 󵄨 󵄨 𝑠 󵄨 󵄨 󵄨 − ∫ 𝑑 ⟨𝑋⟩𝑠󵄨 . ≤𝐾𝐸󵄨∑ (𝑋𝑡 −𝑋𝑠 )[(𝑋𝑡 −𝑋𝑠 )−(𝑋𝜉 −𝑋𝜉 )]󵄨 𝜕𝑥 󵄨 󵄨 𝑖 𝑖 𝑖 𝑖 𝑖+1 𝑖 󵄨 𝑠𝑖 󵄨 󵄨𝑖=1 󵄨 󵄨 󵄨2 󵄨𝑛−1 󵄨 From [12], given 𝜀1 >0, there exists a (𝛿1,𝜂1)-fine belated 󵄨 󵄨 𝑛 ≤𝐾𝐸󵄨∑ (𝑋𝑡 −𝑋𝑠 )[(𝑋𝜉 −𝑋𝑡 )+(𝑋𝑠 −𝑋𝜉 )]󵄨 [0, 𝑡] 𝐷 = {([𝑠 ,𝑡], 𝜉 )} 𝑅 < 󵄨 𝑖 𝑖 𝑖+1 𝑖 𝑖 𝑖 󵄨 partial division of , 1 𝑖 𝑖 𝑖 𝑖=1,suchthat 1 󵄨𝑖=1 󵄨 𝜀1. We just need to consider 𝑅2.Bymeanvaluetheorem, ̃ ̃ 𝑛−1 𝑓(𝑋𝜉 )−𝑓(𝑋𝜉 ) = (𝜕𝑓/𝜕𝑥)(𝑋𝑖)(𝑋𝜉 −𝑋𝜉 ),where𝑋𝑖 ∈ 󵄨 󵄨2 2 𝑖+1 𝑖 𝑖+1 𝑖 ≤𝐾∑ 𝐸 󵄨(𝑋 −𝑋 )+(𝑋 −𝑋 )󵄨 𝐸(𝑋 −𝑋 ) , [𝑋 ,𝑋 ] 𝑋 𝑡 󵄨 𝜉𝑖+1 𝑡𝑖 𝑠𝑖 𝜉𝑖 󵄨 𝑡𝑖 𝑠𝑖 𝜉𝑖 𝜉𝑖+1 .Since 𝑡 is continuous with respect to ,there ̂ ̃ 𝑖=1 𝜉𝑖 ∈[𝜉𝑖,𝜉𝑖+1] 𝑋̂ = 𝑋𝑖 exists a such that 𝜉𝑖 a.s. Similarly, (24) International Journal of Stochastic Analysis 5

󸀠 where 𝐾=sup{𝑓 (𝑊𝑡):0≤𝑡≤1& 𝜔∈Ω}.Giventhat𝑋𝑡 Proof. Let 𝜀>0and 𝛿(𝜉, 𝜔) >0 begivenasinDefinition6 is a bounded martingale, for the WSHB integral of 𝑓 with respect to continuous local 󵄨 󵄨2 martingale 𝑋.Define 𝐸 󵄨(𝑋 −𝑋 )+(𝑋 −𝑋 )󵄨 󵄨 𝜉𝑖+1 𝑡𝑖 𝑠𝑖 𝜉𝑖 󵄨 𝑇 𝑇 (𝜔) −𝜉, if 𝜉<𝑇(𝜔) ≤𝜉+𝛿(𝜉,) 𝜔 2 2 𝛿 (𝜉,) 𝜔 ≜{ =𝐸(𝑋 −𝑋 ) +𝐸(𝑋 −𝑋 ) (25) 𝛿 (𝜉,) 𝜔 , . (32) 𝜉𝑖+1 𝑡𝑖 𝑠𝑖 𝜉𝑖 otherwise =𝐸⟨𝑋⟩ −𝐸⟨𝑋⟩ +𝐸⟨𝑋⟩ −𝐸⟨𝑋⟩ . 𝑇 𝜉𝑖+1 𝑡𝑖 𝑠𝑖 𝜉𝑖 Then 𝛿 (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 However, there exists a (𝛿2,𝜂2)-fine belated partial stochastic 𝑇 𝜉<𝑡.Let((𝑈, 𝑉], 𝜉) be a (𝛿 (𝜉, 𝜔),-fine 𝜂) belated stochastic division 𝐷2 ={[𝜉𝑖,𝑡𝑖], 𝜉𝑖) : 𝑖 = 1,2,...,𝑛}of [0, 𝑡] such that 𝑛 interval-point pair. |𝜇𝑋(0, 𝑡] − ∑𝑖=1 𝜇𝑋(𝑠𝑖,𝑡𝑖]| < 𝜂2.Thenwehave 𝐸 ⟨𝑋⟩ −𝐸⟨𝑋⟩ +𝐸⟨𝑋⟩ −𝐸⟨𝑋⟩ <𝜂. (i) If 𝑇(𝑤) ≤𝜉,then 𝜉𝑖+1 𝑡𝑖 𝑠𝑖 𝜉𝑖 2 (26) 𝑇 𝑇 Hence, 𝑋 (𝑉 (𝜔) ,𝜔) −𝑋 (𝑈 (𝜔) ,𝜔) 𝑛−1 (33) 2 =𝑋𝑇 (𝑇 (𝜔) ,𝜔) −𝑋𝑇 (𝑇 (𝜔) ,𝜔) =0. 𝑟 ≤𝐾𝜀 ∑ 𝐸(𝑋 −𝑋 ) ≤𝐾𝐸⟨𝑋⟩ 𝜂 . 1 2 𝑡𝑖 𝑠𝑖 𝑡 2 (27) 𝑖=1 𝜉 < 𝑇(𝜔) ≤ 𝜉 + 𝛿(𝜉,𝜔) [𝑈, 𝑉] ⊂ [𝜉, 𝑇]⊂ 𝑋 (ii) If ,then Since is bounded continuous martingale, by Lemma 5.10 in [𝜉, 𝜉 + 𝛿(𝜉,.Consequently, 𝜔)] ((𝑈, 𝑉], 𝜉) will be a [18,p.33],wehave (𝛿(𝜉, 𝜔),-fine 𝜂) belated stochastic interval-point pair, 󵄨 󵄨 󵄨𝑛−1 󵄨 𝑈∧𝑇=𝑈and 𝑉∧𝑇=𝑉. 󵄨 4󵄨 𝐸 󵄨∑ (𝑋𝑡 −𝑋𝑠 ) 󵄨 󳨀→ 0 as 𝑛󳨀→∞. (28) 󵄨 𝑖 𝑖 󵄨 (iii) If 𝑇(𝜔) > 𝜉 + 𝛿(𝜉, 𝜔),then 󵄨𝑖=1 󵄨 𝑇 𝑇 Hence, given 𝜀2 >0, 𝑋 (𝑈 (𝜔) ,𝜔) −𝑋 (𝑉 (𝜔) ,𝜔) 󵄨 󵄨 󵄨 󵄨 󵄨𝑛−1 󵄨 =𝑋(𝑉 (𝜔) ∧𝑇(𝜔) ,𝜔) −𝑋(𝑈 (𝜔) ∧𝑇(𝜔) ,𝜔) 󵄨 󸀠 󸀠 󵄨 󵄨 4󵄨 (34) 𝑟2 ≤ sup 󵄨𝑓 (𝑋̂)−𝑓 (𝑋 )󵄨 𝐸 󵄨∑ (𝑋𝑡 −𝑋𝑠 ) 󵄨 󵄨 𝜉𝑖 𝜉𝑖 󵄨 󵄨 𝑖 𝑖 󵄨 1≤𝑖≤𝑛 & 𝜔∈Ω 󵄨𝑖=1 󵄨 =𝑋(𝑈 (𝜔) ,𝜔) −𝑋(𝑉 (𝜔) ,𝜔) .

≤2𝐾𝜀2. It is easy to verify that 𝐷 = {((𝑈𝑖,𝑉𝑖], 𝜉𝑖),𝑖=1,2,...,𝑛}is also (29) 𝑇 a (𝛿 (𝜉, 𝜔),-fine 𝜂) belated stochastic division of [0, 𝑡 ∧ 𝑇(𝑤)] Now take 𝜂=min{𝜂1,𝜂2} and let 𝛿(𝜉𝑖)=min{𝛿1(𝜉𝑖), 𝛿2,𝜉𝑖+1 − only if 𝐷 = {((𝑈𝑖,𝑉𝑖], 𝜉𝑖),𝑖 = 1,2,...,𝑛}is a (𝛿(𝜉, 𝜔),-fine 𝜂) 𝜉𝑖};thena(𝛿, 𝜂)-fine belated partial division 𝐷={[𝜉𝑖,𝑡𝑖], 𝜉𝑖): belated stochastic division of [0, 𝑡]. 𝑇 𝑖 = 1,2,...,𝑛} of [0, 𝑡] is both (𝛿1,𝜂1)-fine belated partial Similar results hold for 𝐴 .Sincethe𝐷= 𝑇 stochastic division and (𝛿2,𝜂2)-fine belated partial stochastic {((𝑈𝑖,𝑉𝑖], 𝜉𝑖),𝑖 = 1,2,...,𝑛} is also a (𝛿 (𝜉, 𝜔),-fine 𝜂) division. Hence, for 𝜀=max{𝜀1,𝜂2,𝜀2},wehave belated stochastic division of [0, 𝑡 ∧ 𝑇(𝑤)],bythepropertyof 2 being WSHB integrable, we have that 𝑓 is WSHB integrable 󵄨𝑛−1 𝑡 󵄨 󵄨 𝑓𝜉 +𝑓𝜉 𝑖 󵄨 𝑇 𝑇 󵄨 𝑖 𝑖+1 󵄨 to a process 𝐴 with respect to 𝑋 . 𝐸 󵄨∑ (𝑊𝑡 −𝑊𝑠 )−∫ 𝑋𝑡 ∘𝑑𝑊𝑡󵄨 󵄨 2 𝑖 𝑖 𝑠 󵄨 󵄨𝑖=1 𝑖 󵄨 (30) Lemma 14. Let 𝑋={𝑋𝑡, F𝑡;0≤𝑡<∞}be a contin- ≤𝑅 +𝑟 +𝑟 ≤ 𝐾𝜀, uous local martingale with the corresponding nondecreasing 1 1 2 ∞ sequence of stopping times {𝑇𝑛}𝑛=1.Supposethat𝑓 is WSHB where 𝐾 is a constant. Hence, 𝑓(𝑋𝑡) is WSHB integrable on integrable to a process 𝐴 with respect to 𝑋;then𝑓 is WSHB 𝑇 𝑇 [0, 𝑡] and, by the unique property of WSHB, integrable to a process 𝐴 𝑛 with respect to 𝑋 𝑛 for each 𝑛= 𝑇𝑛 𝑡 𝑡 1, 2, . . ..Furthermore,foreach𝑡 ∈ [0, 𝑡], lim𝑛→∞𝐴 =𝐴a.s. (WSH) ∫ 𝑓 (𝑋𝑠) ∘𝑑𝑋𝑠 = ∫ 𝑓 (𝑋𝑠) ∘𝑑𝑋𝑠. (31) 0 0 Proof. This follows directly form Lemma 13 and the unique- ness of the WSHB integral. Then we have the last statement

𝑇 lim 𝐴 𝑛 =𝐴a.s. 3.2. Henstock Approach to Define Stratonovich Integral with 𝑛→∞ (35) respect to Continuous Local Martingale

Lemma 13. Let 𝑋={𝑋𝑡, F𝑡;0≤𝑡<∞}be a continuous 2 Theorem 15. If 𝑓(𝑥) ∈𝐶 (R) and 𝑋={𝑋𝑡, F𝑡;0≤𝑡<∞}is local martingale. Suppose 𝑓 is WSHB integrable to a process a continuous local martingale, then 𝑓(𝑋𝑡) is WSHB integrable 𝐴𝑡 with respect to local continuous martingale 𝑋.Let𝑇 be a 𝑇 with respect to 𝑋 on [0, 𝑡].Furthermore, stopping time; then 𝑓 is WSHB integrable to a process 𝐴 with 𝑇 𝑇 𝑇 𝑡 𝑡 respect to 𝑋 ,where𝑋 ={𝑋𝑡∧𝑇;0≤𝑡<∞}and 𝐴 = {𝐴 ;0≤𝑡<∞} (WSH) ∫ 𝑓(𝑋𝑠)∘𝑑𝑋𝑠 = ∫ 𝑓(𝑋𝑠)∘𝑑𝑋𝑠. (36) 𝑡∧𝑇 . 0 0 6 International Journal of Stochastic Analysis

󵄨 2 𝑚≥1 󵄨 Proof. First we introduce, for each ,thestoppingtime 󵄨 −(𝐴𝑉∧𝑇 −𝐴𝑈 ∧𝑇 ) 󵄨) 𝑖 𝑚 𝑖 𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 𝑇 = {𝑡 ≥ 0; 󵄨𝑋 󵄨 ≥𝑚 ⟨𝑋⟩ ≥𝑚}. 󵄨𝑛−1 𝑚 inf 󵄨 𝑡󵄨 or 𝑡 (37) 󵄨 1 =𝐸(lim inf 󵄨∑ 𝑓(𝑋𝜉 +𝑋𝜉 )(𝑋𝑉∧𝑇 −𝑋𝑈 ∧𝑇 ) 𝑚→∞ 󵄨 2 𝑖+1 𝑖 𝑖 𝑚 𝑖 𝑚 󵄨𝑖=1 󵄨 2 {𝑇 } 󵄨 The resulting sequence 𝑚 is nondecreasing with 󵄨 −(𝐴𝑉∧𝑇 −𝐴𝑈 ∧𝑇 ) 󵄨) 𝑃[lim𝑚→∞𝑇𝑚 =∞]=1. Thus, the stopping process 𝑖 𝑚 𝑖 𝑚 󵄨 󵄨 𝑋𝑡∧𝑇 is bounded; that is, 𝑋𝑡∧𝑇 is a bounded martingale 𝑚 𝑚 󵄨 for each 𝑚 = 1, 2, . . Under this situation, the values of 𝑓 󵄨𝑛−1 󵄨 1 outside [−𝑚, 𝑚] are irrelevant. We may assume without loss ≤ lim inf𝐸(󵄨∑ 𝑓(𝑋𝜉 +𝑋𝜉 )(𝑋𝑉∧𝑇 −𝑋𝑈 ∧𝑇 ) 󸀠 󸀠󸀠 𝑚→∞ 󵄨 2 𝑖+1 𝑖 𝑖 𝑚 𝑖 𝑚 of generality that 𝑓 has compact support, so 𝑓, 𝑓 ,and𝑓 󵄨𝑖=1 are bounded. 󵄨 2 󵄨 Let 𝐷 = {((𝑈𝑖,𝑉𝑖], 𝜉𝑖),𝑖=1,2,...,𝑛}be a (𝛿(𝜉, 𝜔),-fine 𝜂) 󵄨 −(𝐴𝑉∧𝑇 −𝐴𝑈 ∧𝑇 ) 󵄨) 𝑖 𝑚 𝑖 𝑚 󵄨 belated stochastic division of [0, 1].FromLemma13,wehave 󵄨 that, for each 𝑚≥1, 𝐷𝑚 = {((𝑈𝑖,𝑉𝑖], 𝜉𝑖),𝑖 = 1,2,...,𝑛}is a 𝑇𝑚 < 𝜀≤𝜀. (𝛿 (𝜉, 𝜔),-fine 𝜂) belated stochastic division of [0, 𝑡∧𝑇𝑚(𝑤)]. lim𝑚→∞ inf 𝑇𝑚 𝐴 ={𝐴𝑡∧𝑇 ;0≤𝑡<∞} From Lemma 13, there is a process 𝑚 (40) such that This shows that 𝑓(𝑋𝑡) is WSHB integrable on [0, 𝑡] and

󵄨𝑛−1 𝑡 𝑡 󵄨 1 󵄨 ( ) ∫ 𝑓 (𝑋𝑠) ∘𝑑𝑋𝑠 = ∫ 𝑓 (𝑋𝑠) ∘𝑑𝑋𝑠. (41) 𝐸(󵄨∑ 𝑓(𝑋𝜉 +𝑋𝜉 )(𝑋𝑉∧𝑇 −𝑋𝑈 ∧𝑇 ) WSH 󵄨 2 𝑖+1 𝑖 𝑖 𝑚 𝑖 𝑚 0 0 󵄨𝑖=1 󵄨2 (38) 󵄨 󵄨 −(𝐴𝑉∧𝑇 −𝐴𝑈 ∧𝑇 )󵄨 )<𝜀, 𝑖 𝑚 𝑖 𝑚 󵄨 󵄨 3.3. Henstock Approach to Define Stratonovich Integral with respect to Continuous Semimartingale. Let 𝑋={𝑋𝑡, F𝑡;0 ≤ 𝑡<∞}be a continuous semimartingale such that for every (𝛿, 𝜂)-fine belated partial stochastic division 𝐷= 𝑋𝑡 =𝑋0 +𝑀𝑡 +𝐶𝑡, (42) {((𝑈𝑖,𝑉𝑖], 𝜉𝑖):𝑖=1,2,...,𝑛}of [0, 𝑡].Since where 𝑀𝑡 is a continuous local martingale and 𝐶𝑡 is adapted, continuous process of bounded variation with 𝐶0 =0a.s. We 󵄨 󵄨𝑛−1 have 󵄨 lim 󵄨∑ 𝑓(𝑋𝜉 +𝑋𝜉 )(𝑋𝑉∧𝑇 −𝑋𝑈 ∧𝑇 ) 𝑚→∞󵄨 𝑖+1 𝑖 𝑖 𝑚 𝑖 𝑚 1 󵄨𝑖=1 𝑓(𝑋𝜉 +𝑋𝜉 )(𝑋𝑉 −𝑋𝑈 ) 󵄨 2 𝑖+1 𝑖 𝑖 𝑖 󵄨 󵄨 1 −(𝐴𝑉∧𝑇 −𝐴𝑈 ∧𝑇 ) 󵄨 𝑖 𝑚 𝑖 𝑚 󵄨 = 𝑓(𝑋𝜉 +𝑋𝜉 )(𝑀𝑉 −𝑀𝑈 ) (43) 󵄨 2 𝑖+1 𝑖 𝑖 𝑖 (39) 󵄨𝑛−1 1 󵄨 󵄨 + 𝑓(𝑋𝜉 +𝑋𝜉 )(𝐶𝑉 −𝐶𝑈 ). = 󵄨∑ 𝑓(𝑋𝜉 +𝑋𝜉 )(𝑋𝑉 −𝑋𝑈 ) 𝑖+1 𝑖 𝑖 𝑖 󵄨 𝑖+1 𝑖 𝑖 𝑖 2 󵄨𝑖=1 󵄨 Now we consider a stopping time 󵄨 󵄨 −(𝐴𝑉 −𝐴𝑈 ) 󵄨 . ., 󵄨 󵄨 𝑖 𝑖 󵄨 a s 󵄨 󵄨 󵄨 {0; if 󵄨𝑋0󵄨 ≥𝑚 { 󵄨 󵄨 {inf {𝑡 ≥ 0; 󵄨𝑀𝑡󵄨 ≥𝑚or { 󵄨 󵄨 { ⟨𝑀⟩ ≥𝑚 𝐶̆ ≥𝑚}; { 𝑡 or 𝑡 󵄨 󵄨 we must have 𝑇𝑚 = { if 󵄨𝑋0󵄨 <𝑚 (44) { 󵄨 󵄨 {∞; 󵄨𝑋 󵄨 <𝑚 { if 󵄨 0󵄨 & { 󵄨 󵄨 󵄨 󵄨 { inf {𝑡 ≥ 0; 󵄨𝑀𝑡󵄨 ≥𝑚or 󵄨𝑛−1 󵄨 { 󵄨 󵄨 ⟨𝑀⟩ ≥𝑚 𝐶̆ ≥𝑚}=∞, 𝐸 󵄨∑ 𝑓(𝑋𝜉 +𝑋𝜉 )(𝑋𝑉 −𝑋𝑈 )−(𝐴𝑉 −𝐴𝑈 )󵄨 { 𝑡 or 𝑡 󵄨 𝑖+1 𝑖 𝑖 𝑖 𝑖 𝑖 󵄨 󵄨𝑖=1 󵄨 𝐶̆ 𝐶 [0, 𝑡] 󵄨𝑛−1 where 𝑡 is the total variation of 𝑡 on .Thenthe 󵄨 1 󵄨 sequence {𝑇𝑚} is nondecreasing with 𝑃[lim𝑚→∞𝑇𝑚 =∞]= =𝐸( lim 󵄨∑ 𝑓(𝑋𝜉 +𝑋𝜉 )(𝑋𝑉∧𝑇 −𝑋𝑈 ∧𝑇 ) 𝑚→∞󵄨 2 𝑖+1 𝑖 𝑖 𝑚 𝑖 𝑚 1 𝑀 󵄨𝑖=1 . Thus, the stopping process 𝑡∧𝑇𝑚 is a bounded martingale International Journal of Stochastic Analysis 7

𝑚 = 1, 2, . . 𝐶 1 for each . In addition, 𝑡 {𝑡≤𝑇𝑚(𝑤)} is of bounded [5] R. Henstock, “The efficiency of convergence factors for func- variation. Under this situation, the values of 𝑓 outside tions of a continuous real variable,” Journal of the London [−3𝑚, 3𝑚] are irrelevant. We may assume without loss of Mathematical Society,vol.30,pp.273–286,1955. 󸀠 󸀠󸀠 generality that 𝑓 has compact support, so 𝑓, 𝑓 ,and𝑓 are [6] J. Kurzweil, “Generalized ordinary differential equations and 𝑛 ∑ (1/2)𝑓(𝑋 +𝑋 )(𝐶 −𝐶 ) continuous dependence on a parameter,” Czechoslovak Mathe- bounded. Given that the 𝑖=1 𝜉𝑖+1 𝜉𝑖 𝑉𝑖 𝑈𝑖 is (1/2)𝑓(𝑋 +𝑋 )(𝐴 − matical Journal,vol.7,no.3,pp.418–449,1957. classic Riemann-Stieltjes integral, 𝜉𝑖+1 𝜉𝑖 𝑉𝑖 𝐴 ) (𝛿, 𝜀) [7] R. Henstock, The General Theory of Integration,OxfordUniver- 𝑈𝑖 is WSHB integrable for every -fine belated partial sity Press, Oxford, UK, 1991. stochastic division 𝐷 = {((𝑈𝑖,𝑉𝑖], 𝜉𝑖) : 𝑖 = 1,2,...,𝑛} of [0, 𝑡] (see [17]). Since the WSHB integral satisfies the linear [8] D. Nualart, The Malliavin and Related Topics,Springer, property, we obtain the following result. New York, NY, USA, 1995. [9] D. Nualart and E. Pardoux, “ with anticipat- 2 Theorem 16. If 𝑓(𝑥) ∈𝐶 (R) and 𝑋 is a continuous ing integrands,” Probability Theory and Related Fields,vol.78, no. 4, pp. 535–581, 1988. semimartingale, then 𝑓(𝑋𝑡) is WSHB integrable with respect to 𝑋𝑡 on [0, 𝑡].Furthermore, [10] P. Protter, “A comparison of stochastic integrals,” The Annals of Probability,vol.7,no.2,pp.276–289,1979. 𝑡 𝑡 [11] R. L. Stratonovich, “A new representation for stochastic inte- (WSH) ∫ 𝑓(𝑋𝑠)∘𝑑𝑋𝑠 = ∫ 𝑓(𝑋𝑠)∘𝑑𝑋𝑠. (45) 0 0 grals and equations,” SIAM Journal on Control and Optimiza- tion,vol.4,pp.362–371,1966. 4. Conclusion [12] T.-L. Toh and T.-S. Chew, “The Riemann approach to stochastic integration using non-uniform meshes,” Journal of Mathemati- 2 From Theorem 16,if𝑓 is of class 𝐶 (R) and 𝑋 is a continuous cal Analysis and Applications,vol.280,no.1,pp.133–147,2003. 𝑡 𝑓(𝑋 ) ∫ 𝑓(𝑋 )∘𝑑𝑋 [13] H. F. Yang and T. L. Toh, “On Henstock-Kurzweil method to semimartingale, 𝑡 is WSHB integrable to 0 𝑠 𝑠 Stratonovich integral,” Mathematical Bohemica, In press. on [0, 𝑡]. Now we consider the Itoˆ formula for Stratonovich 𝑓∈𝐶3(R) [14] T. L. Toh and T. S. Chew, “A variational approach to Ito’sˆ integral; if , as shown in Proposition 4, integral,” in Proceedings of SAP’s ’98, pp. 291–299, 1998. 𝑡 [15] T.-L. Toh and T.-S. Chew, “The Kurzweil-Henstock theory of 󸀠 𝑓(𝑋𝑡)=𝑓(𝑋0)+∫ 𝑓 (𝑋𝑠)∘𝑑𝑋𝑠. (46) stochastic integration,” Czechoslovak Mathematical Journal,vol. 0 62,no.3,pp.829–848,2012. 𝑡 𝑡 ( )∫ 𝑓󸀠(𝑋 )∘𝑑𝑋 ∫ 𝑓󸀠(𝑋 )∘𝑑𝑋 [16] T.-S. Chew, J.-Y.Tay, and T.-L. Toh, “The non-uniform Riemann We substitute WSH 0 𝑠 𝑠 with 0 𝑠 𝑠. approach to Ito’sˆ integral,” Real Analysis Exchange,vol.27,no.2, Then, pp.495–514,2001. 𝑡 [17] P. Y. Lee, LanzhouLecturesonHenstockIntegration,World 󸀠 𝑓 (𝑋𝑡) =𝑓(𝑋0) + (WSH) ∫ 𝑓 (𝑋𝑠) ∘𝑑𝑋𝑠. (47) Scientific, Singapore, 1989. 0 [18] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, NY, USA, 2nd edition, 2000. In conclusion, from the definition of Stratonovich integral with respect to continuous semimartingale using Henstock [19] E. J. McShane, Stochastic Calculus and Stochastic Models,Aca- method, we manage to keep the important properties of the demic Press, 1974. classical Stratonovich integral and also probably enlarge the [20] T.-L. Toh and T.-S. Chew, “On Io-Kurzweil-Henstockˆ integral scope of the integrands which satisfy classical Stratonovich and integration-by-part formula,” Czechoslovak Mathematical Journal,vol.55,no.3,pp.653–663,2005. integral Itoformulaˆ (46).

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

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