As for an analysis of stochastic differential equations driven by extended Wiener process in the sense of nonstandard analysis, we need to extend “Ito’s formula” for Wiener process or Ito’s process. In the previous paper, we extended a concept of convolution in Fourier series to the case of nonstandard analysis. According to the result, we shall extend some theorems in probability theory, for example, the law of large numbers and the central limit theorem, and shall reconstruct Ito’s formula by using nonstandard analysis. We shall give the proof of the reconstruction of Ito’s formula in the case that the convolution of probability density which functions in a nonstandard extension is convergent for some functional of Ito’s process . The problem was not solved still now.
If the convolution is not convergent, what kind of problem does it occur? In Taylor expansion of , the higher terms may not vanish. Then, Ito’s formula does not be established. As to what we shall give extended law of large numbers and extended central limit theorem, they will be provided precisely in the next paper.
2. Ito’s Integral for Extended Wiener Process in Nonstandard
In our previous paper  , we showed that Fourier series can be described by the convolution in nonstandard analysis, then the series of i.i.d. random variables using Loeb measure  converges in sense under some moment condition. Therefore, the definition of stochastic integral in classical probability theory can be extended by the way of nonstandard analysis  ,  .
Furthermore, we need to prove some laws of large numbers for i.i.d. random variables to show the convergence to a stochastic integral.
In fact, we use an extended concept of the convolution to investigate the expectation or the distribution of series of i.i.d. random variables for the nonstandardization of the law of large numbers.
In order to prove the convergence of sums of higher order of such as in the proof of Ito formula, we need to extend the law of large numbers for in the sense of nonstandard.
From the above discussion, we shall define the stochastic integral in nonstandard analysis.
Let be the infinitesimal and . The extended Winer process is defined as follows.
Definition 2.1. Let and . Assume that a sequence of i.i.d. random variables has the distribution
for each . An extended Wiener process is defined by
Ito’s integral (stochastic integral) in the nonstandard sense is defined as follows.
Definition 2.2. Let be an extended Wiener process. Assume that an adapted process with respect the Wiener process is defined by
where and each is measurable with respect to . Assume that
A stochastic integral in nonstandard analysis is defined by
where is independent of is a sequence of i.i.d. random variables with the distribution
Remark 1. In classical (standard) probability theory, Ito’s integral is well defined under the condition of the existence of the variance of for each . In nonstandard analysis, the convergence of the series in (5) may not be ensured. On the other hand, take note that we have already given some sufficient conditions for the convergence of the convolution in Fourier series. See  .
3. Proof of Ito’s Formula for Extended Wiener Process in Nonstandard
From the concept of Ito’s integral for the extended Wiener process, we provide Ito’s formula for the extended Wiener process.
Theorem 3.1. Let be of . Assume that the condition (4) is satisfied, then we have the following for the extended Wiener Process . For any ,
4. Proof of Ito’s Formula for Ito’s Process in Nonstandard
Let be Ito’s process defined by
where and are adapted processes with respect to a Wiener process . Then, we have Ito’s formula for the Ito’s process.
Theorem 4.1. Let be of , then we have the following. For any ,
Proof. We provide the proof by using nonstandard analysis.
From the Taylor expansion for the two-dimensional function , we have the following.
In nonstandard analysis, we can represent the Ito’s process for the extended Wiener process by
where for infinitesimal
Therefore, the difference of can be represented by the following,
On the other hand,
for each from (1).
Thus, we have the following.
Thus we prove Ito’s formula for the extended Ito’s process.
Remark 2. Let be independent random variables with density functions , respectively. Then, the distribution of can be represented by convolution . In standard analysis, for the Fourier transform of the convolution
is established, where is the Fourier transform of .
From our previous paper  , the result can be extended in the sense of nonstandard. See pp.976. Therefore, it is applied for the extension of limit theorems as like central limit theorem, law of large numbers and so on.
In classical (standard) probability theory, the stochastic integral
is defined under the condition of the existence of the variance of for each . In nonstandard analysis, the convergence of the series in (5) is proved from the above arguments. On the proof of Ito’s formula, it can be applied for other estimations as the same way.
Furthermore, the proof of Ito’s formula in nonstandard analysis becomes simple rather than the proof in standard one.
The first author is supported in part by Grant-in-Aid Scientific Research (C), No.18K03431, Ministry of Education, Science and Culture, Japan.