*Tid:* ** 17 december 2013 kl 13.15-15.00. (observera tiden)**
**Seminarierummet 3721**, Institutionen för
matematik, KTH, Lindstedts vägen 25, plan 7.
Karta!
*Föredragshållare:*
**
Thorbjörn Gudmundsson, KTH
**
**Titel:**
Licentiat seminarium: Markov Chain Monte Carlo for rare-event simulation in heavy-tailed settings.
**Diskutant:**
Ola Hössjer, Stockholms Universitet
**Abstract:**
In this thesis a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalising constant. Using the MCMC methodology a Markov chain is simulated, with that conditional distribution as its invariant distribution, and information about the normalising constant is extracted from its trajectory.
The algorithm is described in full generality and applied to four different problems of computing rare-event probability. The first problem considers a random walk exceeding a high threshold, where the increments are independent, identically distributed and heavy-tailed. The second problem is an extension of the first one to a heavy-tailed random sum exceeding a high threshold. The third problem considers a stochastic recurrence equation exceeding a high threshold, where the innovations are independent and identically distributed and heavy-tailed. The final problem considers the ruin probability for an insurance company with risky investments.
An unbiased estimator of the reciprocal probability for each corresponding problem is constructed whose normalised variance vanishes asymptotically. The algorithm is illustrated numerically and compared to existing importance sampling algorithms.
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