Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the definition of a solution to a rough differential equation
51
0
0.0
(
0
)
Authors
I. Bailleul
Ask ChatGPT about the research
No Arabic abstract
We give an elementary proof that Davies definition of a solution to a rough differential equation and the notion of solution given by Bailleul in (Flows driven by rough paths) coincide. This provides an alternative point on view on the deep algebraic insights of Cass and Weidner in their work (Tree algebras over topological vector spaces in rough path theory).
rate research
Read More
We prove that if $f:mathbb{R}tomathbb{R}$ is Lipschitz continuous, then for every $Hin(0,1/4]$ there exists a probability space on which we can construct a fractional Brownian motion $X$ with Hurst parameter $H$, together with a process $Y$ that: (i) is Holder-continuous with Holder exponent $gamma$ for any $gammain(0,H)$; and (ii) solves the differential equation $dY_t = f(Y_t) dX_t$. More significantly, we describe the law of the stochastic process $Y$ in terms of the solution to a non-linear stochastic partial differential equation.
We define in this work a notion of Young differential inclusion $$ dz_t in F(z_t)dx_t, $$ for an $alpha$-Holder control $x$, with $alpha>1/2$, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, $gamma$-Holder continuous set-valued map on the interval $[0,1]$ has a selection with finite $p$-variation, for $p>1/gamma$. We also give a notion of solution to the rough differential inclusion $$ dz_t in F(z_t)dt + G(z_t)d{bf X}_t, $$ for an $alpha$-Holder rough path $bf X$ with $alphain left(frac{1}{3},frac{1}{2}right]$, a set-valued map $F$ and a single-valued one form $G$. Then, we prove the existence of a solution to the inclusion when $F$ is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.
The purpose of this note is to provide an alternative proof of two transformation formulas contiguous to that of Kummers second transformation for the confluent hypergeometric function ${}_1F_1$ using a differential equation approach.
We introduce and study the properties of a new family of fractional differential and integral operators which are based directly on an iteration process and therefore satisfy a semigroup property. We also solve some ODEs in this new model and discuss applications of our results.
Within the rough path framework we prove the continuity of the solution to random differential equations driven by fractional Brownian motion with respect to the Hurst parameter $H$ when $H in (1/3, 1/2]$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
