OPTIMAL PREDICTION PROBLEMS IN INSURANCE

Abstract

We consider the optimal prediction problem of stopping a L\'evy process as close as possible to a given distance $b\geq0$ from its ultimate supremum, under a squared error penalty function. We work with spectrally negative L\'evy processes and double exponential jump diffusion processes. For both cases first with solve an optimal prediction problem defined in terms of a positive penalty function. We prove that we can express this problem as an equivalent optimal stopping problem driven by the process reflected in its running supremum. After showing that for any non-decreasing penalty function the problem is trivial, we focus on the case of the quadratic penalty function mention above. Under some mild conditions, the solution was fully characterised. For a spectrally negative L\'evy process the result is written in terms of the well-studied scale functions. The proof consists in a number of lemmas, where we use results from fluctuation theory. For a double exponential jump diffusion process, writing two verification lemmas and solving a free-boundary problem, we find an explicit solution. We find that in both cases the solution has an interesting non-trivial structure: if $b$ is large than a certain threshold then it is optimal to stop as soon as the difference between the running supremum and the position of the process excess a certain level, typically strictly smaller than $b$, while if $b$ is smaller than this threshold then it is optimal to stop immediately (independent of the running supremum and position of the process). Moreover for processes of unbounded variation our solutions satisfy the Smooth Pasting Condition, while in the bounded variation case we have Continuous Pasting. We also present some examples to illustrate our results.

Date of Award	1 Aug 2020
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Kees Van Schaik (Supervisor) & Neil Walton (Supervisor)