Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proof of the Main Theorem

Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most of its time near a with σ small enough. In order to have more information about this behaviour, we introduce the following stopping times

\ That gives us:

\ Therefore, by induction in k, we get:

\ By Markov’s inequality, we get:

\ We remark that by Lemmas 3.1, 3.5 and derivations above, we have

\ This concludes the proof.

3.2.2 Exit-location

:::info This paper is available on arxiv under CC BY-SA 4.0 DEED license.

:::

:::info Authors:

(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;

(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;

(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

:::

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Table of Links

3.2 Proof of the main theorem

3.2.1 Kramers’ law

3.2.2 Exit-location

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