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Department of Mathematics, University of California San Diego

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Tail estimates and applications for rough differential equations

Abstract:

Consider a solution to an ordinary differential equation driven along smooth vector fields $V=\left( V^{1},...,V^{d}\right)$ of linear growth. $$dy_{t}=V\left( y_{t}\right) dx_{t},\text{ started from }y_{0}.\label{ode}%$$ If $x$ has finite $1-$variation then Gronwall's inequality gives a bound on $y$ of the type% $\left\vert y\right\vert _{1-var}\leq C\exp\left( C\left\vert x\right\vert _{1-var}\right) .$ If $x$ has finite $p-$variation for $p>2$ then rough path theory needs to be used to understand (\ref{ode}), and a corresponding growth estimate of the form $\left\vert y\right\vert _{p-var}\leq C\exp\left( C\left\vert \mathbf{x}% \right\vert _{p-var}^{p}\right)$ can be derived in some cases. For a large class of random rough paths $\mathbf{x=x}\left( \omega\right)$, e.g. the Brownian rough path, the right hand side of this inequality is not integrable (Fernique's theorem). This has implications for some applications of interest, such as showing the existence and smoothness of densities of RDEs via Malliavin calculus. In this talk we show how this obstacle can be bypassed by consideration of the so-called \textit{accumulated p-variation }$M\left( \mathbf{x}\right)$ of a $p$-rough path $\mathbf{x}$ over $\left[ 0,t\right]$ which is given by% \[ M\left( \mathbf{x}\right) =\sup_{\overset{D=\left\{ 0=t_{0}

Host: Bruce Driver

AP&M 6402

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