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Let W = (W i ) i∈N be an infinite dimensional Brownian motion and (X t ) t≥0 a continuous adapted n-dimensional process. Set τ R = inf{t : |X t − x t | ≥ R t }, where x t , t ≥ 0 is a R n -valued deterministic differentiable curve and R t > 0, t ≥ 0 a time-dependent radius. We assume that, up to τ R , the process X solves the following (not necessarily Markov) S D E :Under local conditions on the coefficients, we obtain lower bounds for P(τ R ≥ T ) as well as estimates for distribution functions and expectations. These results are discussed in the elliptic and log-normal frameworks. An example of a diffusion process that satisfies the weak Hörmander condition is also given.

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In this paper we give upper bounds for both the Value at Risk VaR α , 0 < α < 1, and for ruin probabilities associated with the supremum of a process driven by a Brownian motion and a compound Poisson process. We obtain lower bounds for the same Value at Risk, and for different cases we discuss the behavior of the bounds for small α. We prove our bounds are "asymptotically" optimal, as α tends to zero. The ruin probabilities obtained are related to other bounds found in recent literature.

In this paper we examine the large deviations principle (LDP) for sequences of classic Cramér-Lundberg risk processes under suitable time and scale modifications, and also for a wide class of claim distributions including (the non-superexponential) exponential claims. We prove two large deviations principles: first, we obtain the LDP for risk processes onwith the Skorohod topology. In this case, we provide an explicit form for the rate function, in which the safety loading condition appears naturally. The second theorem allows us to obtain the LDP for Aggregate Claims processes on with a different time-scale modification. As an application of the first result we estimate the ruin probability, and for the second result we work explicit calculations for the case of exponential claims.