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Large deviation techniques applied to three questions of when ...
Stockholm University, Faculty of Science, Department of Mathematics.
2005 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Large deviation techniques are used to solve three problems; when is a distant convex barrier passed, when to accept a sequence of gambles and when is the time of ruin. This work is the collection of four papers.

Paper I: When is a convex barrier passed for the first time by a random walk? With a certain scaling, the limit distribution of the first passage time to a convex barrier decays exponentially in the scale parameter. Under the Esscher transform, which changes the drift, four properties are proved. First, the limit distribution of the overshoot is distributed as an overshoot over a linear barrier. Secondly, the stopping time is asymptotically normally distributed when it is properly normalized. Thirdly, the overshoot and the asymptotic normal part are asymptotically independent. Fourthly, the overshoot over a linear barrier is bounded by an exponentially distributed random variable. The combination of these building blocks gives the function that multiplies the exponential part in the asymptotic distribution.

Paper II: This paper continues the work on first passage times to a convex barrier. The rate of exponential decay changes for times before the most probable passage time. The function multiplying the exponential part also alters to a constant divided by the square root of the scale parameter. An auxiliary result used in the proof of this is the characteristic function of the stopping time which decays exponentially in its non-central parts.

Paper III: A sequence of gambles which initially is rejected, is eventually accepted when the gambles follow a large deviation principle and the utility function is non-satiated and bounded from below in a certain way. The number of gambles required for acceptance is computed.

Paper IV: In classical risk theory, the surplus can increase to infinity. This is however not realistic in real life. To get a more applicable model, a variable premium that depends on the level of the surplus is introduced. The variable premium is assumed to be linear such that the buffer is a generalized Ornstein-Uhlenbeck process. Asymptotically, the time to ruin with a certain scaling is shown to be exponentially distributed.

Place, publisher, year, edition, pages
Stockholm: Matematiska institutionen , 2005. , 110 p.
Keyword [en]
Large deviations, first passage time, stopping time, rate function, convex barrier, fallacy of large numbers, utility function, insurance, ruin, variable premium, generalized Ornstein-Uhlenbeck process
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:su:diva-697ISBN: 91-7155-132-8 (print)OAI: oai:DiVA.org:su-697DiVA: diva2:197382
Public defence
2005-11-11, sal 14, hus 5, Kräftriket, Stockholm, 10:15
Opponent
Supervisors
Available from: 2005-10-20 Created: 2005-10-20Bibliographically approved
List of papers
1. Tools to estiamate the first passage time to a convex barrier
Open this publication in new window or tab >>Tools to estiamate the first passage time to a convex barrier
2005 In: Journal of applied probability, ISSN 0021-9002, Vol. 42, no 1, 61-81 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-24363 (URN)
Note
Part of urn:nbn:se:su:diva-697Available from: 2005-10-20 Created: 2005-10-20Bibliographically approved
2. Tools to estimate the first passage time to a convex barrier
Open this publication in new window or tab >>Tools to estimate the first passage time to a convex barrier
2005 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 42, no 1, 61-81 p.Article in journal (Refereed) Published
Abstract [en]

he first passage time of a random walk to a barrier (constant or concave) is of great importance in many areas, such as insurance, finance, and sequential analysis. Here, we consider a sum of independent, identically distributed random variables and the convex barrier cb(n/c), where c is a scale parameter and n is time. It is shown, using large-deviation techniques, that the limit distribution of the first passage time decays exponentially in c. Under a tilt of measure, which changes the drift, four properties are proved: the limit distribution of the overshoot is distributed as an overshoot over a linear barrier; the stopping time is asymptotically normally distributed when it is properly normalized; the overshoot and the asymptotic normal part are asymptotically independent; and the overshoot over a linear bather is bounded by an exponentially distributed random variable. The determination of the function that multiplies the exponential part is guided by consideration of these properties.

Identifiers
urn:nbn:se:su:diva-24364 (URN)
Note
Part of urn:nbn:se:su:diva-697Available from: 2005-10-20 Created: 2005-10-20 Last updated: 2017-12-13Bibliographically approved
3. When to accept a sequence of gambles
Open this publication in new window or tab >>When to accept a sequence of gambles
2005 (English)In: Journal of Mathematical Economics, ISSN 0304-4068, E-ISSN 1873-1538, Vol. 41, no 8, 974-982 p.Article in journal (Refereed) Published
Abstract [en]

When is a sequence of gambles, which is initially rejected eventually accepted? The eventual acceptance is defined as a pair property between the utility function and the sequences of gambles. A sequence of gambles is accepted when the gambles follow a large deviation principle and the utility function is non-satiated and bounded from below in a certain way. The number of gambles required for acceptance is computed.

Place, publisher, year, edition, pages
Elsevier B.V., 2005
Keyword
Fallacy of large numbers; Eventual acceptance property; Large deviation; Rate function; Expected utility
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-24365 (URN)10.1016/j.jmateco.2004.12.007 (DOI)
Note
Part of urn:nbn:se:su:diva-697Available from: 2005-10-20 Created: 2005-10-20 Last updated: 2017-12-13Bibliographically approved
4. Ruin probabilities for a risk-process with variable premium
Open this publication in new window or tab >>Ruin probabilities for a risk-process with variable premium
2005 Article in journal (Refereed) Submitted
Identifiers
urn:nbn:se:su:diva-24366 (URN)
Note
Part of urn:nbn:se:su:diva-697Available from: 2005-10-20 Created: 2005-10-20Bibliographically approved

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