public 01:34:42

Jacob Bedrossian : Positive Lyapunov exponents for 2d Galerkin-Navier-Stokes with stochastic forcing

  -   Applied Math and Analysis ( 436 Views )

In this talk we discuss our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander's condition. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry.

public 01:14:21

Valery A. Kholodnyi : Foreign Exchange Option Symmetry and a Coordinate-Free Description of a Foreign Exchange Option Market

  -   Applied Math and Analysis ( 28 Views )

In spite of the fact that symmetries play one of the major roles in physics, the ir usage in finance is relatively new and, to the best of our knowledge, can be traced to 1995 when Kholodnyi introduced the beliefs-preferences gauge symmetry. In this talk we present another symmetry, foreign exchange option symmetry, int roduced by Kholodnyi and Price in 1996. Foreign exchange option symmetry associa tes financially equivalent options on opposite sides of the foreign exchange mar ket. In a two-currency market, the foreign exchange option symmetry is formalized in terms of the one-dimensional Kelvin transform. In a multiple-currency market the foreign exchange option symmetry is formalized in terms of differential geometr y on graphs, that is, in terms of vector lattice bundles on graphs and connectio ns on these bundles. Foreign exchange option symmetry requires no assumptions on the nature of a prob ability distribution for exchange rates. In fact, it does not even require the a ssumptions of the existence of such a distribution. Furthermore, the symmetry is applicable not only to a foreign exchange market but to any financial market as well. The practical applications of the foreign exchange option symmetry range from th e detection of a new type of true arbitrage to the detection of inconsistent mod els of foreign exchange option markets and the development of algorithms and sof tware to value and analyze portfolios of foreign exchange options.

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