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KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS
 

Summary: KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH
CONTINUOUS COEFFICIENTS
CLAUDIO ALBANESE
Abstract. Bidirectional valuation models are based on numerical methods to obtain kernels
of parabolic equations. Here we address the problem of robustness of kernel calculations vis
a vis floating point errors from a theoretical standpoint.
We are interested in kernels of one-dimensional diffusion equations with continuous co-
efficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in
the limit as h 0. We consider both semidiscrete triangulations with continuous time and
explicit Euler schemes with time step so small that the Courant condition is satisfied. We
find uniform bounds for the convergence rate as a function of the degree of smoothness. We
conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives
of the kernel and its first two space derivatives. The proof is constructive and is based on
a new technique of path conditioning for Markov chains and a renormalization group argu-
ment. We make the simplifying assumption of time-independence and use longitudinal Fourier
transforms in the time direction. Convergence rates depend on the degree of smoothness and
H¨older differentiability of the coefficients. We find that the fastest convergence rate is of order
O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit
schemes still converge for any degree of H¨older differentiability except that the convergence
rate is slower. H¨older continuity itself is not strictly necessary and can be relaxed by an

  

Source: Albanese, Claudio - Department of Mathematics, King's College London

 

Collections: Mathematics