Large deviation techniques applied to three questions of when

Abstract: 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.