Research Article | Open Access
On Henstock Method to Stratonovich Integral with respect to Continuous Semimartingale
We will use the Henstock (or generalized Riemann) approach to define the Stratonovich integral with respect to continuous semimartingale in space. Our definition of Stratonovich integral encompasses the classical definition of Stratonovich integral.
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 ). This paper attempts to further generalize the result of  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  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 -martingale.
Further, to consider continuous local martingale as an integrator, we will show that a -fine belated partial stochastic division of in [12, 16] is also a -fine belated partial stochastic division of , where 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 Section 3.2. 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 .
We give a new definition of the Stratonovich integral with continuous semimartingale in 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 continuous -semimartingale. We will develop the definition of Stratonovich-Henstock belated integral where the integrator is a Brownian motion in . Let be a probability space such that is complete. Also, let denote the class of functions which have continuous derivatives.
Definition 1. Let be a process. If there exists a nondecreasing sequence of the stopping times of the filtration , such that is a martingale for each and , then one says that is local martingale. If, in addition, a.s., one writes (or , if is continuous).
Definition 2. A continuous semimartingale is an adapted process which has the decomposition where and has finite variation.
Definition 3. Let and be two continuous semimartingales such that where and are in and and are adapted, continuous process of bounded variation with a.s. The Stratonovich integral of with respect to is 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 is the cross variation process of and . If , then is the quadratic variation of (see [18, p. 31]).
Proposition 4. Let be a function of class and let be a continuous semimartingale with decomposition in (2). Then
Proof. By Itô formula [18, p. 149], we have We only have to prove that Since , then, by Itô formula, Then is a semimartingale, is a local martingale, and and are bounded continuous processes. By definition of a semimartingale, By the property of cross variation (see [18, p. 143]) Hence,
3. Henstock Approach
3.1. Henstock Approach to Define Stratonovich Integral with respect to Bounded Continuous Martingale
In [13–16, 19], a positive function on , , where does not depend on , was used. From the proof of the main results in [13, 14, 16, 20], 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 -function needs to depend on and stochastic intervals are needed. Therefore, we need to redefine as also dependent on (see ).
Definition 5. Let , , be a measurable function with respect to the two variables and . Then is called a locally stopping process  if, for each , is a stopping time.
For each , we denote the measure  induced by the quadratic variation process of an adapted process by
Definition 6. Let be a locally stopping process and . A finite collection of stochastic interval-point pairs , where and are stopping times of for all , is said to be a -fine belated partial stochastic division of if(1)for each is a stochastic interval and, for each , , are disjoint left-open subintervals of ;(2)each is a -fine belated division; that is, for each , one has
(3)for any .
For the case , denotes . Note that, for each , is a stopping time. Thus, for each , there exists a stochastic interval with such that . For each , namely, In addition, by Vitali’s covering theorem, the -fine belated partial stochastic division could cover except for an arbitrarily small -measure [19, p. 44] [12, 16].
Definition 7. An adapted process in is said to be weakly Stratonovich-Henstock belated (denoted by WSHB) integrable to a process in on , , with respect to a stochastic process if, for every , there exists a locally stopping process on for which where . For succinctness, one may write for every -fine belated partial stochastic division of .
Proposition 8. If an adapted process is WSHB integrable with respect to stochastic process , then the weakly Henstock variational belated integral of is unique a.s.
In light of Proposition 8, we will denote the integral of the process with respect to stochastic process by the notation
Proposition 9. Let and be WSHB integrable with respect to stochastic process and . Then and are WSHB integrable with respect to stochastic process . Furthermore,
Proposition 10. If stochastic process is WSHB integrable with respect to stochastic process , then so is it on subinterval .
Proposition 11. Let the adapted process be WSHB integrable on and with respect to stochastic process , where . Then is WSHB integrable on and, furthermore,
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. If and is bounded continuous martingale, then is WSHB integrable with respect to bounded continuous martingale on . Furthermore,
Proof. Since is a bounded continuous martingale and , then , , and are bounded and continuous with respect to . By Definition 1, the classical Stratonovich integral with respect to bounded continuous martingale is Let be a -fine belated partial stochastic division of . Then, where From , given , there exists a -fine belated partial division of , , such that . We just need to consider . By mean value theorem, , where . Since is continuous with respect to , there exists a such that a.s. Similarly, there exists such that in space. Then, Let where . Given that is a bounded martingale, However, there exists a -fine belated partial stochastic division of such that . Then we have Hence, Since is bounded continuous martingale, by Lemma 5.10 in [18, p. 33], we have Hence, given , Now take and let ; then a -fine belated partial division of is both -fine belated partial stochastic division and -fine belated partial stochastic division. Hence, for , we have where is a constant. Hence, is WSHB integrable on and, by the unique property of WSHB,
3.2. Henstock Approach to Define Stratonovich Integral with respect to Continuous Local Martingale
Lemma 13. Let be a continuous local martingale. Suppose is WSHB integrable to a process with respect to local continuous martingale . Let be a stopping time; then is WSHB integrable to a process with respect to , where and .
Proof. Let and be given as in Definition 6 for the WSHB integral of with respect to continuous local martingale . Define
Then is locally stopping process, since is a stopping time and if . Let be a -fine belated stochastic interval-point pair.(i)If , then
(ii)If , then . Consequently, will be a -fine belated stochastic interval-point pair, and .(iii)If , then
It is easy to verify that is also a -fine belated stochastic division of only if is a -fine belated stochastic division of .
Similar results hold for . Since the is also a -fine belated stochastic division of , by the property of being WSHB integrable, we have that is WSHB integrable to a process with respect to .
Lemma 14. Let be a continuous local martingale with the corresponding nondecreasing sequence of stopping times . Suppose that is WSHB integrable to a process with respect to ; then is WSHB integrable to a process with respect to for each . Furthermore, for each a.s.
Proof. This follows directly form Lemma 13 and the uniqueness of the WSHB integral. Then we have the last statement
Theorem 15. If and is a continuous local martingale, then is WSHB integrable with respect to on . Furthermore,
Proof. First we introduce, for each , the stopping time
The resulting sequence is nondecreasing with . Thus, the stopping process is bounded; that is, is a bounded martingale for each . Under this situation, the values of outside are irrelevant. We may assume without loss of generality that has compact support, so , , and are bounded.
Let be a -fine belated stochastic division of . From Lemma 13, we have that, for each , is a -fine belated stochastic division of . From Lemma 13, there is a process such that for every -fine belated partial stochastic division of . Since we must have This shows that is WSHB integrable on and
3.3. Henstock Approach to Define Stratonovich Integral with respect to Continuous Semimartingale
Let be a continuous semimartingale such that where is a continuous local martingale and is adapted, continuous process of bounded variation with a.s. We have Now we consider a stopping time where is the total variation of on . Then the sequence is nondecreasing with . Thus, the stopping process is a bounded martingale for each . In addition, is of bounded variation. Under this situation, the values of outside are irrelevant. We may assume without loss of generality that has compact support, so , , and are bounded. Given that the is classic Riemann-Stieltjes integral, is WSHB integrable for every -fine belated partial stochastic division of (see ). Since the WSHB integral satisfies the linear property, we obtain the following result.
Theorem 16. If and is a continuous semimartingale, then is WSHB integrable with respect to on . Furthermore,
From Theorem 16, if is of class and is a continuous semimartingale, is WSHB integrable to on . Now we consider the Itô formula for Stratonovich integral; if , as shown in Proposition 4, We substitute with . Then, 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).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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