Abstract
Let X a be a Markov process with generator ∑ i,j ∂ i ( a ij∂ j · ) where a is a uniformly elliptic symmetric matrix. Thanks to the fundamental works of T. Lyons, stochastic differential equations driven by X a can be solved in the “rough path sense”; that is, pathwise by using a suitable stochastic area process. Our construction of the area, which generalizes previous works of Lyons–Stoica and then Lejay, is based on Dirichlet forms associated to subellitpic operators. This enables us in particular to discuss large deviations and support descriptions in suitable rough path topologies. As typical rough path corollary, Freidlin–Wentzell theory and the Stroock–Varadhan support theorem remain valid for stochastic differential equations driven by X a.
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Peter Friz is a Leverhulme Fellow.
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Friz, P., Victoir, N. On uniformly subelliptic operators and stochastic area. Probab. Theory Relat. Fields 142, 475–523 (2008). https://doi.org/10.1007/s00440-007-0113-y
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DOI: https://doi.org/10.1007/s00440-007-0113-y