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On Henstock Method to Stratonovich Integral with Respect to Continuous Semimartingale 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 Integral with respect to Continuous Semimartingale 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 space. Our definition of Stratonovich integral encompasses the classical definition of Stratonovich integral. 1. Introduction Further, to consider continuous local martingale as an integrator, we will show that a (, )-fine belated partial Classically, it has been emphasized that defining stochastic stochastic division ={(,)}=1 of [0, ] in [12, 16]isalso integrals using Riemann approach is impossible [1,p.40], a (, )-fine belated partial stochastic division of [0, ∧ ], since the integrators have paths of unbounded variation. where 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 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 quadratic variation 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 -semimartingale. We will develop the definition of 2 define Stratonovich integral with respect to continuous - Stratonovich-Henstock belated integral where the integrator martingale. isaBrownianmotionin[13]. Let (Ω, F,)be a probability 2 International Journal of Stochastic Analysis (Ω, F,) (R) ( ) ∫ ( ) space such that is complete. Also, let denote Then is a semimartingale, 0 is a local the class of functions which have continuous derivatives. ∫ ( ) (1/2) ∫ ( )⟨⟩ martingale, and 0 and 0 are bounded continuous processes. By definition of a semi- Definition 1. Let ={, F;0≤<∞}be a process. If ∞ martingale, there exists a nondecreasing sequence {}=1 of the stopping () times of the filtration {F},suchthat{ ≜∧ , F;0 ≤ ∫ ( )∘ = ∫ ( ) <∞}is a martingale for each ≥1and [lim→∞ = ∞] = 1 0 0 ,thenonesaysthat is local martingale. If, in (8) =0 ∈Mloc ∈M,loc 1 addition, 0 a.s., one writes (or ,if + ⟨, ∫ ( )⟩ . is continuous). 2 0 Definition 2. A continuous semimartingale ={, F;0≤ By the property of cross variation (see [18,p.143]) <∞} is an adapted process which has the decomposition ⟨, ∫ ( )⟩ = ∫ ( )⟨⟩ . =0 + +, 0≤<∞a.s., (9) (1) 0 0 ={, F ;0≤<∞}∈M,loc where and has finite Hence, variation. 1 ()=(0)+∫ () + ∫ ()⟨⟩ Definition 3. Let ={, F;0≤<∞}and = 0 2 0 { , F ;0≤<∞} be two continuous semimartingales such that =(0)+∫ ()∘. = + +;= + +, 0 0 0 (2) (10) ,loc where and are in M and and are adapted, continuous process of bounded variation with 0 =0 =0 a.s. The Stratonovich integral of with respect to is 3. Henstock Approach 1 ∫ ∘ ≜ ∫ + ∫ + ⟨, ⟩ , (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 on [0, ], 0≤<∞,where does not integral, the second one is pathwise integral of the Riemann- depend on ∈Ω, was used. From the proof of the main ⟨, ⟩ Stieltjes type, and is the cross variation process of results in [13, 14, 16, 20], () is deterministic, because the = ⟨, ⟩ =⟨⟩ and .If ,then is the quadratic quadratic variation of a Brownian motion is deterministic variation of (see [18,p.31]). and Fubini’s theorem can be applied in switching the order Proposition 4. :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 3(R) ={, F ;0 ≤ < ∞} and let be a continuous sums, the -function needs to depend on Ω and stochastic semimartingale with decomposition in (2).Then intervals are needed. Therefore, we need to redefine as also Ω dependent on (see [12]). ()=(0)+∫ ()∘ a.s. (4) 0 Definition 5. Let : [0, ] × Ω → (0,, ) 0≤<∞,be Proof. By Itoformula[ˆ 18,p.149],wehave a measurable function with respect to the two variables ∈ [0, ] and ∈Ω.Then is called a locally stopping process 1 ()=()+∫ ( ) + ∫ ( )⟨⟩ . [12] if, for each ∈[0,], +is a stopping time. 0 2 0 0 For each (,] ⊂ [0,],wedenotethemeasure[15] (5) induced by the quadratic variation process of an adapted We only have to prove that process ={,0≤<∞}by 1 (,] ≜(⟨⟩ − ⟨⟩ ) . ∫ ( )∘ = ∫ ( ) + ∫ ( )⟨⟩ . (11) 2 0 0 0 = (6) Definition 6. Let be a locally stopping process and {, F;0≤<∞}. A finite collection of stochastic interval- 2 Since ()∈(R),then,byItoformula,ˆ point pairs (((,], ), = 1,2,...,),where and are stopping times of for all =1,2,...,,issaidtobea(, )- [0, ] ()= (0)+∫ () + ∫ () finebelatedpartialstochasticdivisionof if 0 0 ( ,] (7) (1) for each is a stochastic interval and, for each 1 ∈ Ω, ( ,] = 1,2,..., + ∫ ()⟨⟩ . , , are disjoint left-open 2 0 subintervals of [0, 1]; International Journal of Stochastic Analysis 3 (2) each ((,], ) is a -fine belated division; that is, for + and are WSHB integrable with respect to stochastic each ∈Ω,onehas process .Furthermore, ( () , ()]⊂[, +(, )] ; (12) (WSH) ∫ ( + )∘ = (WSH) ∫ ∘ 0 0 >0| (0, ] − ∑ ( ,]| < (3) for any =1 . + (WSH) ∫ ∘; (17) For the case () = (), ((), ()] denotes {()}. 0 ∈[0,]+ Note that, for each , is a stopping time. Thus, ( (), ()] for each , there exists a stochastic interval with (WSH) ∫ ∘ =(WSH) ∫ ∘. () < () such that ((), ()] ⊂ [, +(,)].For 0 0 each ∈Ω,namely, Proposition 10. If stochastic process is WSHB integrable () =() ,() = { +( ,) ,} . inf (13) with respect to stochastic process ,thensoisitonsubinterval [, ] ⊂ [0,. ] In addition, by Vitali’s covering theorem, the (, )-fine belated partial stochastic division could cover [0, ] except for Proposition 11. Let the adapted process be WSHB integrable an arbitrarily small -measure [19,p.44][12, 16]. on [0, ] and [, ] with respect to stochastic process ,where 0<<.Then is WSHB integrable on [0, ] and, ={, F ;0 ≤ < Definition 7. An adapted process furthermore, ∞} in (Ω, F,)is said to be weakly Stratonovich-Henstock belated (denoted by WSHB) integrable to a process in (Ω, F,{F}≥0,) on [0, ], 0≤<∞,withrespecttoa (WSH) ∫ ∘ = (WSH) ∫ ∘ stochastic process if, for every >0, there exists a locally 0 0 [0, ] × Ω stopping process on for which (18) + ( ) ∫ ∘. WSH ( ,)+(,) ∑ { +1 ((() ,)−(() , )) 2 =1 The proofs of Propositions 8 to 11 are similar to the results 2 of classical Henstock integral [17]; hence we omit these proofs −[( () ,)−( () , )] } ≤, in this paper. (14) Next, we will consider the relationship between WSHB integral and Stratonovich integral with respect to continuous where +1 =. 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 ( ,)+(,) local martingale. Note that we can choose the stopping time +1 (∑ ( − )−( − ) ) such that a continuous local martingale becomes a bounded 2 =1 continuous martingale up to any finite stopping time [18,p. 44]. ≤, 2 (15) Theorem 12. If () ∈ (R) and ={, F;0 ≤ < ∞} is bounded continuous martingale, then () is WSHB for every (, )-fine belated partial stochastic division = integrable with respect to
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