Feynman-Kac representation of fully nonlinear PDEs and applications

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance

Twistor Theory and Differential Equations

This is an elementary and self--contained review of twistor theory as a geometric tool for solving non-linear differential equations. Solutions to soliton equations like KdV, Tzitzeica, integrable chiral model, BPS monopole or Sine-Gordon arise from holomorphic vector bundles over T\CP^1. A different framework is provided for the dispersionless analogues of soliton equations, like dispersionless KP or $SU(\infty)$ Toda system in 2+1 dimensions. Their solutions correspond to deformations of (parts of) T\CP^1, and ultimately to Einstein--Weyl curved geometries generalising the flat Minkowski space. A number of exercises is included and the necessary facts about vector bundles over the Riemann sphere are summarised in the Appendix.Comment: 23 Pages, 9 Figure

Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations

We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian) ordinary differential equations by means of Newton-like iterations. We pay particular attention to symmetric symplectic IRK schemes (such as collocation methods with Gaussian nodes). For a $s$-stage IRK scheme used to integrate a $d$-dimensional system of ordinary differential equations, the application of simplified versions of Newton iterations requires solving at each step several linear systems (one per iteration) with the same $sd \times sd$ real coefficient matrix. We propose rewriting such $sd$-dimensional linear systems as an equivalent $(s+1)d$-dimensional systems that can be solved by performing the LU decompositions of $[s/2] +1$ real matrices of size $d \times d$. We present a C implementation (based on Newton-like iterations) of Runge-Kutta collocation methods with Gaussian nodes that make use of such a rewriting of the linear system and that takes special care in reducing the effect of round-off errors. We report some numerical experiments that demonstrate the reduced round-off error propagation of our implementation

Approximated Lax Pairs for the Reduced Order Integration of Nonlinear Evolution Equations

A reduced-order model algorithm, called ALP, is proposed to solve nonlinear evolution partial differential equations. It is based on approximations of generalized Lax pairs. Contrary to other reduced-order methods, like Proper Orthogonal Decomposition, the basis on which the solution is searched for evolves in time according to a dynamics specific to the problem. It is therefore well-suited to solving problems with progressive front or wave propagation. Another difference with other reduced-order methods is that it is not based on an off-line / on-line strategy. Numerical examples are shown for the linear advection, KdV and FKPP equations, in one and two dimensions

Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models

European options can be priced by solving parabolic partial(-integro) differential equations under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference discretization leads to a so-called full order model (FOM). Reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD). The early exercise constraint of American options is enforced by a penalty on subset of grid points. The presented numerical experiments demonstrate that pricing with ROMs can be orders of magnitude faster within a given model parameter variation range

Linear approach to the orbiting spacecraft thermal problem

We develop a linear method for solving the nonlinear differential equations of a lumped-parameter thermal model of a spacecraft moving in a closed orbit. Our method, based on perturbation theory, is compared with heuristic linearizations of the same equations. The essential feature of the linear approach is that it provides a decomposition in thermal modes, like the decomposition of mechanical vibrations in normal modes. The stationary periodic solution of the linear equations can be alternately expressed as an explicit integral or as a Fourier series. We apply our method to a minimal thermal model of a satellite with ten isothermal parts (nodes) and we compare the method with direct numerical integration of the nonlinear equations. We briefly study the computational complexity of our method for general thermal models of orbiting spacecraft and conclude that it is certainly useful for reduced models and conceptual design but it can also be more efficient than the direct integration of the equations for large models. The results of the Fourier series computations for the ten-node satellite model show that the periodic solution at the second perturbative order is sufficiently accurate.Comment: 20 pages, 11 figures, accepted in Journal of Thermophysics and Heat Transfe