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Young and rough differential inclusions. (English) Zbl 1481.34024

The article contains an interesting result on Young und rough differential inclusions. Firstly the authors introduce the concept of Young differential inclusion. Let \(x\) be an element of \(C^\alpha ([0,T ], \mathbb{R}^l)\) with \(\alpha \in (1/2, 1]\), and let \(F : \mathbb{R}^l \rightarrow \dot{2}^{L(R^l,R^d)}\) be a set-valued map. A solution to the Young differential inclusion \[dz_t\in F (z_t)dx_t, z_0 = \xi \in \mathbb{R}^d\] is a pair of paths \((z, v)\), defined on the time interval \([0,T ]\), with \(v\) a \(L(\mathbb{R}^l, \mathbb{R}^d)\)-valued path of finite \(p\)-variation such that \(\alpha + 1/p > 1\), and for all \(0\leq t\leq T\), \(v_t \in F (z_t)\) and \[z_t = \xi + \int_0^t v_s dx_s.\]
Then a theorem on the existence of solutions for such systems is proved.
Secondly, the notion of solution to the rough differential inclusion \[dz_t \in F (z_t) dt + G(z_t) dX_t\] is given for an \(\alpha\)-Hölder rough path \(X\) with \(\alpha \in (1/3, 1/2]\), a set-valued map \(F\) and a single-valued one form \(G\).
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 is proven.
All the results obtained in the article are rigorously substantiated and strictly proven. Also, these results provide a deeper understanding of stochastic analysis and disentangles in that setting probabilistic and dynamical matters.

MSC:

34A60 Ordinary differential inclusions
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References:

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