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Theses

Generalized Ornstein-Uhlenbeck Processes in Ruin Theory

Abstract : This thesis is concerned with the study of Generalized Ornstein-Uhlenbeck(GOU) processes and their application in ruin theory. The GOU processes,which are the solutions of certain linear stochastic differential equations, havebeen introduced in ruin theory by Paulsen in 1993 as models for the surpluscapital of insurance companies facing both insurance and market risks.In general, these processes were chosen as suitable models on ana prioribasis. The first and main contribution of this thesis is to show that GOUprocesses appear naturally as weak limits of random coefficient autoregres-sive processes which are used extensively in various domains of applied prob-ability. Using this result, the convergence in distribution of the ruin times,the convergence of the ultimate ruin probability and the moments are alsoshown.The rest of the thesis deals with the study of certain properties of GOU pro-cesses. In particular, the ruin problem for the GOU process is studied andnew bounds on the ruin probabilities are obtained. These results also general-ize some known upper bounds, asymptotic results and conditions for certainruin to the case when the market risk is modelled by a semimartingale.The final section of the thesis moves away from classical ruin theory and layssome first directions for the study of the law of GOU processes at fixed times.In particular, a partial integro-differential equation for the density, large andsmall-time asymptotics are obtained for these laws. This shift away fromthe ruin probability is explained by the fact that most risk measures used inpractice such as Value-at-Risk are based on these laws instead
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Jérôme Spielmann. Generalized Ornstein-Uhlenbeck Processes in Ruin Theory. Probability [math.PR]. Université d'Angers, 2019. English. ⟨NNT : 2019ANGE0066⟩. ⟨tel-03148887⟩

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