Positivity of the Density for Rough Differential Equations

Abstract

Small noise problems are quite important for all types of stochastic differential equations. In this paper we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter H ∈ (1/4, 1/2]. We prove a moderate deviation principle for this equation as the scale parameter tends to zero.

Support Theorem for Pinned Diffusion Processes

“…Technically, this section is the core of this paper. These properties were already proved in [30] for Gaussian rough paths with respect to the p -variation topology under the condition called the complementary Young regularity. In this section, we will show these properties for Brownian rough path with respect to the Besov rough path topology and also clean up arguments in [30].…”