# Download Diffusion, Markov processes and martingales. Ito calculus by L. C. G. Rogers, David Williams PDF

By L. C. G. Rogers, David Williams

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Example text

Buy Do Carmo’s (1976, 1992) two books, and start reading the one on curves and surfaces. If you are as bad learner as I am, do what I did, that is, all the exercises. Once you read about two dimensional surfaces and understand that curvature is a geometric name for a second order Taylor formula, you will have enough intuition to digest the abstract Riemannian manifolds — which really copy the classical theory of surfaces in R3 . After reading Do Carmo’s books, I found Chavel (1996) and some parts of Spivak (1970) most valuable.

From its definition, we infer that expp (ΩA,p) coincide with the set ωA,p = { q ∈ ΛI(A) : there exists a unique minimizing geodesic through q which meets DA orthogonally at p }. Set ωA = ωA,p . p∈DA On ωA , we can define a projection πA onto DA as follows. Any point q in ωA can be written in a unique way as q = expp (u) for (p, u) ∈ DA × Np DA . We set πA (q) = p. In other words, q is on a unique geodesic starting from DA and orthogonal to DA ; the projection of q on DA is the starting point of this geodesic in DA .

There are by now a few books on large deviations. Dembo and Zeitouni (1993) and Dupuis and Ellis (1997) are good starting points to the huge literature. From a different perspective, and restricted essentially to the univariate case, Jensen (1995) may be closer to what we are looking for here. Introducing the set Λc suggests that the variations of − log f are important. It has to do with the following essential remark. The negative exponential function is the only one — up to an asymptotic equivalence — for which integrating on an interval of length of order 1 produces a relative variation of order 1 on the integral.