The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space

Badri Mamporia

Niko Muskhelishvili Institute of Computational Mathematics, Technical University of Georgia, Tbilisi, Georgia

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Badri Mamporia. The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space. Pure and Applied Mathematics Journal. Vol. 4, No. 4, 2015, pp. 164-171. doi: 10.11648/j.pamj.20150404.15

Abstract: Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

Keywords: Wiener Process in a Banach Space, Covariance Operators, Ito Stochastic Integrals and Ito Processes, the Ito Formula, Stochastic Differential Equations in a Banach Space

1. Introduction and Preliminaries

As in the finite dimensional case, the Ito formula plays an important role in the infinite dimensional stochastic analysis. For the cases when the Banach spaces have special geometrical properties, the Ito formula was proved in [1] and [2]. For the Wiener process in an arbitrary separable Banach space, the Ito formula was proved in [3]. The Ito formula for the case, when the stochastic integral that appears in the Ito process is taken from Banach space valued non-anticipating process by the one dimensional Wiener process was proved in [4]. In this paper we prove the Ito formula for the Ito processes when the stochastic integral is taken from the operator valued non-anticipating random processes by the Wiener processes in a Banach space. The main unsolved problem to prove this formula in an arbitrary Banach space is to find such a sequence of step functions converging to the integrand function that their stochastic integrals converge to the stochastic integral from the integrand function. We use the concept of the generalized random element; we consider the space of generalized random processes and introduce the generalized Ito process there. Firstly we prove the Ito formula for the generalized random processes. Then from the initial Ito process in a Banach space we receive a generalized Ito process and write the Ito formula there. Afterward, In the obtained equality we check decomposability of the members of the equality; we found that all of them are Banach space valued. Therefore, we get the Ito formula for the Banach space valued random process. Now we give, some definitions and preliminary results.

Let  be a real separable Banach space, - its conjugate, () – the Borel -algebra of , () - a probability space. The continuous linear operator :L(P) is called a generalized random element (GRE) (sometimes it is used the terms: random linear function or cylindrical random element). Denote by :=L(, L(B,P)) the Banach space of GRE with the norm. A random element (measurable map)  is said to have a weak second order, if, for all , E<. We can realize the random element  as an element of : x=, but not conversely: in infinite dimensional Banach space for all :L(P) , there does not always exists the random element such that x= for all . The problem of existence such random element is the well known problem of decompasibility of the GRE. Denote by  the linear normed space of all random elements of the weak second order with the norm . Thus, we can assume M.

Let (W be the Wiener process in a Banach space. That is 1) W=0 almost surely (a.s.); 2. for all, the random elements in ,  ,  are independent; 3. for all   is a Gaussian random element with a mean  and the covariance operator  , where  is a fixed Gaussian covariance; 4.  has continuous sample pats.

Let -be an increasing family of algebras such that a)  is -measurable for all  ; b)  -is independent of the -algebra  for all .  contains all P-null sets in . In this case we say that  is  -adapted.

If  is a covariance operator of the random element , and , (, is a representation of the operator  (see [5], Lemma 3.1.1), then there exists the sequence of independent, standard, real valued Wiener processes  such that and the convergence is a. s. uniformly for  in (see [6 ], Th. 1.4, [7], Th.1.4). We may choose ( and  such that  will be -adapted for all  (see [7], prop. 2.1).

Let (T) be a family of GRE. We call it a generalized random process (GRP). If we have a weak second order random process (, , it will be realized as a GRP: T.

Denote by  the linear space of random functions  such that for all   is measurable and .  is a pseudonorm in .

Definition 1. A function  is called non-anticipating with respect to  if the function  from ([0,1] into () is measurable for all , and the function  is -measurable for all .

By  we denote the class of non-anticipating random functions , for which .  is a linear space and  is a pseudonorm in it.

If  is a step-function , , , then the stochastic integral of  with respect to , is naturally defined by the equality

.

The following lemma is true:

Lemma 1 ([7]). For an arbitrary  there exists a sequence of step-functions  such that  and  converges in .

Definition 2 ([7]). Let  and  be step-functions such that  and  converges in . The limit of the sequence is called the stochastic integral of a random function  with respect to the Wiener process  and is denoted by .

Proposition 1. Let  be such a representation of  that  is -adapted for all , then = a. s. for all

Proof. As  is -adapted for all , the real valued stochastic integrals  exist. The sum  converges in  and . Let  be step-functions such that  and  converges in . It is easy to see, that the above equality holds for step-functions  . That is = a. s. for all   converges also to  for all fixed . Therefore, , when  and   when .

Now consider the linear bounded operator  for all fixed , . Denote by  the space of such operators with the property:  is a pseudonorm in . Consider now the family of linear bounded operators ,, such that for all , the random process  is non -anticipating and . Denote by  the space of such family of operators.

Afterward, we will consider the family ,  with the property .

Proposition 2. If the family of linear bounded operators ,  are such that , then , that is .

Proof. . As  is a Gaussian covariance operator, by the Kwapien-Szymanski’s theorem (see [5] p. 262, [8]) there exist the sequences  and such that , , , for all ,  and . Then

We can naturally define the stochastic integral from which is the GRE, defined by the equality . Accordingly, we have the isometrical operator  ,  .

Lemma 2. For an arbitrary separable-valued  there exists a sequence of step-functions such that  in  and  converges in .

Proof. Analogous to the case of proposition 2, consider the sequences  and such that , , , for all ,  and . Let and denote by , . the functions from .

.

For any fixed , consider now the GRP , , so we have the map ,  and  is separable-valued, therefore, by the lemma 1 from [9], there exists the sequence of non-anticipating step functions :[0,1] such that . Afterward for , the sequence  is such that , when . Therefore, we may choose a sequence of step-functions such that . It is easy to see, that  converges in  to the .

If we have  non-anticipating operator valued random process and non-anticipating , then we can define the generalized stochastic integral ,

,

as  is a generalized random process. Here we use the representation of the Wiener process in a Banach space by the sum of one dimensional, independent, non-anticipating Wiener processes, and  is such that is the representation of the covariance operator of . Let now be a separable Banach space and be the space of bounded linear operators from  to . We will consider the random processes  such that  for all and .

Proposition 2. If the random process  is such, that  for all and , then , where the operator is conjugate of the operator .

Proof. Firstly we prove that  Consider the family of linear operators  , . From the closed graph theorem it follows that  is a continuous operator for all fixed . That is  is a collection of continuous linear operators from  to . For all fix , if we consider the linear operator , , by the closed graph theorem, we can proof boundedness of the operator . That is, for all fixed , . Then, by the uniform boundedness principle, .

Let now the sequences  and be such that , , , for all ,  and . Then we have

.

Definition 3. The random process  is non-anticipating with respect to the family of the -algebra  if, for all ,  is measurable and, for all , the random element  is -measurable.

Definition 4. We say that the non-anticipating random process ,  belongs to the class  if

,

where  is the linear operator, conjugate to the operator.  is a linear space with the pseudonorm

Let  and . Then  is non-anticipating and . We can define the stochastic integral , which is the random variable with a mean 0 and variance . Therefore, we can consider the GRE ,

Definition 5. The generalized random element is called the generalized stochastic integral from the random process . If there exists the random element  such that  for all , then we say that there exists the stochastic integral from the operator-valued non-anticipating random process ,  by the Wiener process in a Banach space  and then we write .

2. The Ito Formula

We will prove the Ito formula for the generalized Ito processes and, as a consequence, we receive the Ito formula for the Banach space valued Ito processes, where the stochastic integral is taken from the operator-valued random process by the Wiener processes in a Banach space.

Definition 4. A non-anticipating GRP is called the Generalized Ito process, if there exist non-anticipating GRP , , and non-anticipating ,  , such that, for all ,

T a.s.

Lemma 3. Let the generalized Ito process  be such that  and  are separable-valued, then there exist the sequences of non-anticipating step functions  and  such that ,  and  uniformly for , where

T .

Proof. The existence of the sequences of non-anticipating step functions is proved in [ 3 ] ( lemma 1) and existence of the sequence of non-anticipating step functions  follows from the lemma 2. Uniformness of the convergence for  of  follows from the inequality

.

Theorem 1(Formula Ito). Let T be a generalized Ito process, where  and  are separable-valued non-anticipating GRP such that , . Let :[0,1] be a continuous function such that the derivatives,  and (are continuous. Then

+ .

Proof. As in the finite dimensional case, we show that it is enough to prove this theorem for the step functions and : let  and  be the sequences of step functions such that  and 0, then ,  uniformly for t, where

.

Let the Ito formula be true for the step functions:

+ ..

As  are continuous functions on [0,1] converging to the continuous function , then they are bounded. Thereby, by the Lebesgue theorem, we have the convergence . Furthermore, we have

++C++() (;

In principle, we can similarly prove

and

  . Therefore, it is enough to prove the Ito lemma for the step functions and, by additivity of integrals, we need to prove it when , where  and  are the elements of  and  correspondingly. For simplicity, we can assume that . Then the function  has the same smoothness as and so, it is enough to prove the Ito formula for function . Let  , , . Then, by the Taylor’s formula, we have

+=++++=+++++,

where

,

=,

 ,

and is such, that there exists , , , , for all ,  and .

Using the technique developed in [ 3], (theorem 1) it is not difficult to prove, that ,  and for all in . Therefore, we have

++

for all . That is, we have the equality in :

++.

Now let us return to the function and remember that , then . Therefore, we have

,

where is such, that there exists , , , , for all

,  and .

Let the generalized Ito process T be such that there exists the X-valued random process  with propertyfor all  and , where , are -measurable, F-adapted and , . Let also :[0,1], be such that :[0,1],   and  are continuous by the norm of  (). Then, taking into consideration that the step functions for  we can take -valued, we have

++ .

The first five members of the aforementioned equality are functionals from the-valued processes. Therefore, the stochastic integral  as the-valued random process exists. Consequently, we have received the Ito formula for the Banach space-valued Ito process.

Theorem 2. Let , where ,  be  measurable, F-adapted and , . Let :[0,1] be such that :[0,1],   and  are, continuous then

 .

We have chosen  such that , , and . Analogously of proposition 3 from [3], we can prove, that the expression  is the same for all  such that

3. Conclusion

The theory of stochastic differential equations in Banach space is develops in three directions. The first direction is the case, when the stochastic integral of the equation is taken from Banach space valued non-anticipating random function by the real-valued Wiener process; the second direction is the case, when the stochastic integral of the equation is taken from operator-valued non-anticipating random function by the Wiener process in a Banach space; the third direction is the case, when the stochastic integral of the equation is taken from operator-valued (from Hilbert space to Banach space) non-anticipating random function by the canonical generalized Wiener process in Hilbert space. Analogous to the finite dimensional case, the Ito formula is one of the main tools in stochastic analysis in Banach space. In this paper we consider the second case and prove the Ito Formula in this case. Existence and uniqueness of solutions is considered in [12] for this case. As we mentioned above the first case is considered in [4] and the third case, when the Banach space has special geometry are considered in [1] and [2] . in [10] is considered the case, when the function  maps from Banach space to real line.

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