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Centre of Financial Mathematics Seminar Series
November 29, 2017 @ 1:30 pm - 2:30 pm
Dr. Budhi Surya (School of Mathematics and Statistics, Victoria University of Wellington, New Zealand)
Some identities concerning excursion below zero with fixed duration of spectrally negative Levy process.
Motivated by some financial problems, e.g., discussed in Broadie et al. (J. Finance, Vol. LXII, p.1341-1377, 2007) and Chesney et al (Adv. Appl. Probab. Vol. 29, p. 165-184, 1997), we present some identities concerning excursion below zero with fixed duration of spectrally negative Levy process. Our main object of interest is the first time the excursion exceeds the specified duration. It is a stopping time and is usually referred to as Parisian ruin-time. We first derive the Laplace transform of this stopping time. Secondly, taking account of the result, we derive using Esscher transform of measure the joint Laplace transform of the stopping time and the corresponding position of the Levy process. Thirdly, we derive the q-resolvent kernel of the Levy process, which corresponds to the total discounted occupation time of the Levy process within an interval prior to the stopping time. The results have semi-explicit expressions in terms of the q-scale function and the law of the Levy process.
This talk is based on recent joint work with Ronnie Loeffen (University of Manchester) and Zbigniew Palmowski (University of Wroclaw).
Please let me know if you want to join the speaker for lunch at Expresso Warriors before the seminar at 12pm.