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Carr–Nadtochiy’s weak reflection principle for Markov chains on \(\mathbf {Z}^d\)

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A Correction to this article was published on 08 October 2020

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Abstract

The reflection principle for Brownian motion gives a way to calculate the joint distribution of a hitting time and a one dimensional marginal. The Carr–Nadtochiy transform is a formulation that generalizes the reflection principle in this respect. The transform originated from a way to hedge so-called barrier options in the literature of financial mathematics. The existence of the transform has been established only for one dimensional diffusion processes. In the present paper, the existence is proved for a fairly general class of Markov chains in the multi dimensional lattice \(\mathbf {Z}^d\). The difficulty is that the reflection boundary is not a one-point set, contrasting the one dimensional cases. It is solved in this paper by looking at the problem in an algebraic way.

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Funding

This work was supported by JSPS KAKENHI Grant Numbe 20K03731.

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  1. Faculty of Mathematics and Physics, Institute of Science Engineering, Kanazawa University, Kanazawa, Japan

    Yuri Imamura

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  1. Yuri Imamura
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Correspondence to Yuri Imamura.

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The original version of this article was revised due to equation renumbering.

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Imamura, Y. Carr–Nadtochiy’s weak reflection principle for Markov chains on \(\mathbf {Z}^d\). Japan J. Indust. Appl. Math. 38, 257–267 (2021). https://doi.org/10.1007/s13160-020-00436-w

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  • DOI: https://doi.org/10.1007/s13160-020-00436-w

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