Betting on Monte Carlo: Stochastic Computational Methods for Finance

Speaker: 

Professor of Mathematics and Director of the Institute for Pure and Applied Mathematics (IPAM) Russel Caflisch

Institution: 

UCLA

Time: 

Thursday, April 28, 2011 - 4:00pm

Location: 

NS2 1201

Monte Carlo is a computational workhorse for valuation of financial securities and risk. It is directly applicable to almost all types of financial securities and is robust in that it is insensitive to the complexities of a security. On the other hand, Monte Carlo can be terribly slow and inaccurate. This talk will review the basics of Monte Carlo quadrature in the context of finance and methods for its acceleration, including variance reduction, quasi-Monte Carlo and ad hoc methods. American options, for which the exercise time is chosen by the option holder, are a class of securities to which Monte Carlo is not directly applicable. The talk will also describe the recently developed Least Square Monte Carlo (LSM) method for American options, some generalizations of LSM, and methods for estimating the accuracy of Monte Carlo for American options.

Geometric Flavored Arithmetic on Jacobians of Hyperelliptic Curves

Speaker: 

Craig Costello

Institution: 

UCI & Queensland University of Technology

Time: 

Monday, April 18, 2011 - 4:00pm

Location: 

RH 306

As an alternative to elliptic curve groups, Koblitz (1989) suggested Jacobians of hyperelliptic curves for use in public-key cryptography. Hyperelliptic curves can achieve the same level of discrete log based security as elliptic curves, whilst offering the potential advantage of being defined over much smaller fields. At present however, elliptic curves still outperform hyperelliptic curves in general, because of the significant difference in the complexity of computing group operations. Indeed, when deriving fast explicit formulas for elliptic curve computations, one is aided by the simple geometric "chord-and-tangent" description. In contrast, Cantor's algorithm for arithmetic in Jacobian groups suffers from more computationally heavy operations, such as Euclid's algorithm for finding the gcd of two polynomials, and the chinese remainder theorem. In this talk we discuss recent results which exploit a chord-and-tangent-like analogue for hyperelliptic curves. We give a simple description of higher genus Jacobian arithmetic and show that for genus 2 curves this gives rise to explicit formulas which are significantly faster than their Cantor-based counterparts. This is joint work with Kristin Lauter.

Strong Unique Continuation for the Navier-Stokes Equation with Non-analytic Forcing

Speaker: 

Mihaela Ignatova

Institution: 

University of Southern California

Time: 

Thursday, May 19, 2011 - 3:00pm

Location: 

RH 440R

The Navier-Stokes system is the classical model for the motion of a viscous incompressible homogeneous fluid. Physically, the equations
express Newton's second law of motion and the conservation of mass. The unknowns are the velocity vector field u and the scalar pressure field p, while the volume forces f and the kinematic viscosity are given. In this talk, we address the spatial complexity and the local behavior of solutions to the three dimensional (NSE) with general non-analytic forcing. Motivated by a result of Kukavica and Robinson in [4], we consider a system of elliptic-parabolic type for a diference of two solutions (u1; p1) and (u2; p2) of (NSE) with the same Gevrey forcing f. By proving delicate Carleman estimates with the same singular weights for the Laplacian and the heat operator (cf. [1, 2, 3]), we establish a quantitive estimate of unique continuation leading to the strong unique continuation property for solutions of the coupled elliptic-parabolic system. Namely, we obtain that if the velocity vector fields u1 and u2 are not identically equal, then their diference
u1-u2 has finite order of vanishing at any point. Moreover, we establish a polynomial estimate on the rate of vanishing, provided the forcing f lies in the Gevrey class for certain restricted range of the exponents. In particular, the necessary condition for the result in [4] is satisfied; thus a finite-dimensional family
of smooth solutions can be distinguished by comparing a finite number of their point values.
This is a joint work with Igor Kukavica.

References
[1] M. Ignatova and I. Kukavica, Unique continuation and complexity of solutions to parabolic partial diferential equations with Gevrey coeficients, Advances in Diferential Equations 15 (2010), 953-975.
[2] M. Ignatova and I. Kukavica, Strong unique continuation for higher order elliptic equations with Gevrey
coeficients, Journal of Diferential Equations (submitted in August, 2010).
[3] M. Ignatova and I. Kukavica, Strong unique continuation for the Navier-Stokes equation with non-analytic forcing, Journal of Dynamics and Diferential Equations (submitted in January, 2011).
[4] I. Kukavica and J.C. Robinson, Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem, Physica D 196 (2004), 45-66.

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