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Global solutions to rough differential equations with unbounded vector fields. (English) Zbl 1254.60059

Donati-Martin, Catherine (ed.) et al., Séminaire de Probabilités XLIV. Papers based on the presentations at the 44th séminaire de probabilités, Dijon, France, June 2010. Berlin: Springer (ISBN 978-3-642-27460-2/pbk; 978-3-642-27461-9/ebook). Lecture Notes in Mathematics 2046, 215-246 (2012).
The aim of this study is to investigate the existence of global solutions to the rough differential equation (RDE) \[ y_t=y_0+\int_0^t f(y_s) dx_s \] in the case that the vector field \(f\) is unbounded and \(x_t\) is a rough path in \(T(\mathbb{R}^d)\) of finite \(p\)-variation, \(p\in[2,3)\). The author first thoroughly reviews available results for and approaches to the problem before introducing and justifying his approach of studying the Euler scheme following [A. M. Davie, AMRX, Appl. Math. Res. Express 2007, Article ID abm009 (2008; Zbl 1163.34005)]. In particular, solutions to the above RDE are understood in the sense of Davie. The main assumption in the paper is the following. \(f\) is a continuously differentiable mapping \(\mathbb{R}^d\) onto \(L(\mathbb{R}^d,\mathbb{R}^m)\) such that \(\nabla f\) is bounded and \(f\cdot\nabla f\) is \(\gamma\)-Hölder continuous with \((2+\gamma)/p>1\). The author proves the existence of a solution and under the additional assumption of its uniqueness. Furthermore, a rate of convergence for the Euler scheme is obtained, equations with geometric rough paths and connections to solutions in the sense of Lyons are discussed.
For the entire collection see [Zbl 1244.60005].

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations

Citations:

Zbl 1163.34005
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References:

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