# On a class of stochastic differential equations with random and H\"older continuous coefficients arising in biological modeling

## TL;DR

This paper studies a broad class of 2D stochastic differential equations with random and H"older continuous coefficients, proving existence and uniqueness of solutions relevant to biological epidemic models.

## Contribution

It extends prior models by establishing existence and uniqueness results for SDEs with less regular coefficients using a Cauchy-Euler-Peano approximation scheme.

## Key findings

- Proved existence of a unique strong solution for the class of SDEs.
- Demonstrated convergence of the approximation scheme to the solution.
- Applicable to biological models with stochastic and irregular coefficients.

## Abstract

Inspired by the paper Greenhalgh et al. [5] we investigate a class of two dimensional stochastic differential equations related to susceptible-infected-susceptible epidemic models with demographic stochasticity. While preserving the key features of the model considered in [5], where an ad hoc approach has been utilized to prove existence, uniqueness and non explosivity of the solution, we consider an encompassing family of models described by a stochastic differential equation with random and H\"older continuous coefficients. We prove the existence of a unique strong solution by means of a Cauchy-Euler-Peano approximation scheme which is shown to converge in the proper topologies to the unique solution