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On the Law of the Minimum in a Class of Unidimensional SDEs

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Abstract

We prove that the law of the minimum \({m := {\rm min}_{t\in[0,1]}\xi(t)}\) of the solution \({\xi}\) to a one-dimensional stochastic differential equation with good nonlinearity has continuous density with respect to the Lebesgue measure. As a byproduct of the procedure, we show that the sets \({\{x \in C([0,1]) : {\rm inf} x \geq r\}}\) have finite perimeter with respect to the law \({\nu}\) of the solution \({\xi({\cdot})}\) in \({L^{2}(0,2)}\).

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Acknowledgements

The first author would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Scaling limits, rough paths, quantum field theory” when part of the work on this paper was undertaken. This work was supported by EPSRC Grant Number EP/R014604/1 and by the PRIN research project 2015233N54 “Deterministic and stochastic evolution equations”.

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Correspondence to Alessandra Lunardi.

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Lecture delivered at the Seminario Matematico e Fisico di Milano on February 23, 2018. Lectures delivered at RISM6 (Riemann International School of Mathematics) on July 25, 2018.

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Da Prato, G., Lunardi, A. & Tubaro, L. On the Law of the Minimum in a Class of Unidimensional SDEs. Milan J. Math. 87, 93–104 (2019). https://doi.org/10.1007/s00032-019-00295-2

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  • DOI: https://doi.org/10.1007/s00032-019-00295-2

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