Abstract. We prove Freidlin-Wentzell Large Deviation estimates under rather minimal assumptions. This allows to derive Wentzell-Freidlin Large Deviation estimates for diffusions on the positive half line with coefficients that are neither bounded nor Lipschitz continuous. This applies to models of interest in Finance, i.e. the CIR and the CEV models, which are positive diffusion processes whose diffusion coefficient is only Hölder continuous.
We use integration by parts formulas to give estimates for the L p norm of the Riesz transform. This is motivated by the representation formula for conditional expectations of functionals on the Wiener space already given in Malliavin and Thalmaier (2006). As a consequence, we obtain regularity and estimates\ud
for the density of non-degenerated functionals on the Wiener space. We also give a semi-distance which characterizes the convergence to the boundary of the set of the strict positivity points for the density
In this paper we develop simulation techniques in order to evaluate single and double barrier options with general features. Our method is based on Sharp Large Deviation estimates, which allow one to improve the usual Monte Carlo procedure. Numerical results are provided and show the validity of the proposed simulation algorithm. Copyright Blackwell Publishers Inc 1999.
We give estimates of the distance between the densities of the laws of two functionals F and G on the Wiener space in terms of the Malliavin-Sobolev norm of F − G. We actually consider a more general framework which allows one to treat with similar (Malliavin type) methods functionals of a Poisson point measure (solutions of jump type stochastic equations). We use the above estimates in order to obtain a criterion which ensures that convergence in distribution implies convergence in total variation distance; in particular, if the functionals at hand are absolutely continuous, this implies convergence in L 1 of the densities.We define now the functional spaces and the differential operators.Simple functionals. A random variable F is called a simple functional if there exists f ∈ C ∞ p (R J ) such that F = f (V ). We denote through S the set of simple functionals. Simple processes. A simple process is a random variable U = (U 1 , . . . , U J ) in R J such that U i ∈ S for each i ∈ {1, . . . , J}. We denote by P the space of the simple processes. On P we define the scalar product ·, · : P × P → S, (U, V ) → U,
We consider a general one-dimensional diffusion process and we study the probability of crossing a boundary for the associated pinned diffusion as the time at which the conditioning takes place goes to zero. We provide asymptotics for this probability as well as a first order development. We consider also the cases of two boundaries possibly depending on the time. We give applications to simulation.
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