Optimal stopping for the exponential of a Brownian bridge
| Autor: | Tiziano De Angelis, Alessandro Milazzo |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
General Mathematics Structure (category theory) Expected value Bond/stock selling Free boundary problems 01 natural sciences FOS: Economics and business 010104 statistics & probability Mathematics::Probability Bellman equation Optimal stopping FOS: Mathematics Applied mathematics 0101 mathematics Mathematics - Optimization and Control Mathematics Brownian bridge Continuous boundary Regularity of value function 010102 general mathematics Probability (math.PR) Mathematical Finance (q-fin.MF) Exponential function Quantitative Finance - Mathematical Finance Optimization and Control (math.OC) Optimal stopping rule Statistics Probability and Uncertainty Martingale (probability theory) Mathematics - Probability |
| ISSN: | 0021-9002 |
| Popis: | In this paper we study the problem of stopping a Brownian bridge $X$ in order to maximise the expected value of an exponential gain function. In particular, we solve the stopping problem $$\sup_{0\le \tau\le 1}\mathsf{E}[\mathrm{e}^{X_\tau}]$$ which was posed by Ernst and Shepp in their paper [Commun. Stoch. Anal., 9 (3), 2015, pp. 419--423] and was motivated by bond selling with non-negative prices. Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we develop techniques that use pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory in order to find the optimal stopping rule and to show regularity of the value function. Comment: 22 pages, 6 figures |
| Databáze: | OpenAIRE |
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