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Register a new userTransportation cost-information inequalities and applications to random dynamical systems and diffusions
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We first give a characterization of the L^1-transportation cost-information inequality on a metric space and next find some appropriate sufficient condition to transportation cost-information inequalities for dependent sequences. Applications to random dynamical systems and diffusions are studied.
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We provide a rather general perfection result for crude local semi-flows taking values in a Polish space showing that a crude semi-flow has a modification which is a (perfect) local semi-flow which is invariant under a suitable metric dynamical system. Such a (local) semi-flow induces a (local) random dynamical system. Then we show that this result can be applied to several classes of stochastic differential equations driven by semimartingales with stationary increments such as equations with locally monotone coefficients and equations with singular drift. For these examples it was previously unknown whether they generate a (local) random dynamical system or not.
This work examines a class of switching jump diffusion processes. The main effort is devoted to proving the maximum principle and obtaining the Harnack inequalities. Compared with the diffusions and switching diffusions, the associated operators for switching jump diffusions are non-local, resulting in more difficulty in treating such systems. Our study is carried out by taking into consideration of the interplay of stochastic processes and the associated systems of integro-differential equations.
Let $T$ be a random ergodic pseudometric over $mathbb R^d$. This setting generalizes the classical emph{first passage percolation} (FPP) over $mathbb Z^d$. We provide simple conditions on $T$, the decay of instant one-arms and exponential quasi-independence, that ensure the positivity of its time constants, that is almost surely, the pseudo-distance given by $T$ from the origin is asymptotically a norm. Combining this general result with previously known ones, we prove that The known phase transition for Gaussian percolation in the case of fields with positive correlations with exponentially fast decayholds for Gaussian FPP, including the natural Bargmann-Fock model; The known phase transition for Voronoi percolation also extends to the associated FPP; The same happens for Boolean percolation for radii with exponential tails, a result which was known without this condition. We prove the positivity of the constant for random continuous Riemannian metrics, including cases with infinite correlations in dimension $d=2$. Finally, we show that the critical exponent for the one-arm, if exists, is bounded above by $d-1$. This holds forbond Bernoulli percolation, planar Gaussian fields, planar Voronoi percolation, and Boolean percolation with exponential small tails.
We extend the Bismut-Elworthy-Li formula to non-degenerate jump diffusions and payoff functions depending on the process at multiple future times. In the spirit of Fournie et al [13] and Davis and Johansson [9] this can improve Monte Carlo numerics for stochastic volatility models with jumps. To this end one needs so-called Malliavin weights and we give explicit formulae valid in presence of jumps: (a) In a non-degenerate situation, the extended BEL formula represents possible Malliavin weights as Ito integrals with explicit integrands; (b) in a hypoelliptic setting we review work of Arnaudon and Thalmaier [1] and also find explicit weights, now involving the Malliavin covariance matrix, but still straight-forward to implement. (This is in contrast to recent work by Forster, Lutkebohmert and Teichmann where weights are constructed as anticipating Skorohod integrals.) We give some financial examples covered by (b) but note that most practical cases of poor Monte Carlo performance, Digital Cliquet contracts for instance, can be dealt with by the extended BEL formula and hence without any reliance on Malliavin calculus at all. We then discuss some of the approximations, often ignored in the literature, needed to justify the use of the Malliavin weights in the context of standard jump diffusion models. Finally, as all this is meant to improve numerics, we give some numerical results with focus on Cliquets under the Heston model with jumps.
We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on R d , with d $ge$ 1. We provide new results on the uniqueness and stability of the associated optimal transportation potentials , namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.
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