## Abstract

This paper studies the weak and strong solutions to the stochastic differential equation dX(t)=−[fomula presented]Ẇ(X(t))dt+dB(t), where (B(t),t≥0) is a standard Brownian motion and W(x) is a two sided Brownian motion, independent of B. It is shown that the Itô–McKean representation associated with any Brownian motion (independent of W) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term ∫_{0}^{T}Ẇ(X(t))dt, the main idea is to use the concept of local time together with the polygonal approximation W_{π}. Some new results on the local time of Brownian motion needed in our proof are established.

Original language | English |
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Pages (from-to) | 2281-2315 |

Number of pages | 35 |

Journal | Stochastic Processes and their Applications |

Volume | 127 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2017 |

## Keywords

- Brox diffusion
- Itô formula
- Local time
- Random environment
- Strong solution
- Uniqueness
- White noise drift

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics