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Weak existence of a solution to a differential equation driven by a very rough fBm
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abstract
We prove that if $f:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous, then for every $H\in(0,1/4]$ 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 H\"older-continuous with H\"older exponent $\gamma$ for any $\gamma\in(0,H)$; 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.
Forward citations
Cited by 3 Pith papers
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Khinchin's and Chung's Laws of the Iterated Logarithm at Time Zero for the Linear Stochastic Fractional Diffusion Equation
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Temporal properties of the stochastic fractional heat equation with rough dependence in space
Temporal increments of the nonlinear stochastic fractional heat equation with rough spatial noise satisfy Khinchin's and Chung's laws of the iterated logarithm with explicit constants.
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