# Stochastic differential equation in a random environment

## Abstract

Solutions of the Itô stochastic differential equation in a random environment are considered. The random environment is formed by the generalized telegraph process. It is proved that the initial problem is equivalent to a system of two stochastic differential equations with nonrandom coefficients. The first equation is the Itô equation, and the initial process is its solution. The second equation is an equation with Poisson process, and its solution is a generalized telegraph process. The theorems of existence and uniqueness of strong and weak solutions are proved.

- Authors:

- Institute of Applied Mathematics and Mechanics of the NAS of Ukraine (Ukraine)

- Publication Date:

- OSTI Identifier:
- 22771218

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 231; Journal Issue: 1; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; DIFFERENTIAL EQUATIONS; MATHEMATICAL SOLUTIONS; RANDOMNESS; STOCHASTIC PROCESSES

### Citation Formats

```
Makhno, Sergei Ya., E-mail: smahmo@gmail.com, and Mel’nik, Sergei A., E-mail: melniks1953@gmail.com.
```*Stochastic differential equation in a random environment*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3805-1.

```
Makhno, Sergei Ya., E-mail: smahmo@gmail.com, & Mel’nik, Sergei A., E-mail: melniks1953@gmail.com.
```*Stochastic differential equation in a random environment*. United States. doi:10.1007/S10958-018-3805-1.

```
Makhno, Sergei Ya., E-mail: smahmo@gmail.com, and Mel’nik, Sergei A., E-mail: melniks1953@gmail.com. Tue .
"Stochastic differential equation in a random environment". United States. doi:10.1007/S10958-018-3805-1.
```

```
@article{osti_22771218,
```

title = {Stochastic differential equation in a random environment},

author = {Makhno, Sergei Ya., E-mail: smahmo@gmail.com and Mel’nik, Sergei A., E-mail: melniks1953@gmail.com},

abstractNote = {Solutions of the Itô stochastic differential equation in a random environment are considered. The random environment is formed by the generalized telegraph process. It is proved that the initial problem is equivalent to a system of two stochastic differential equations with nonrandom coefficients. The first equation is the Itô equation, and the initial process is its solution. The second equation is an equation with Poisson process, and its solution is a generalized telegraph process. The theorems of existence and uniqueness of strong and weak solutions are proved.},

doi = {10.1007/S10958-018-3805-1},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 1,

volume = 231,

place = {United States},

year = {2018},

month = {5}

}

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