A stochastic particle method for some one-dimensional nonlinear p.d.e.

✍ Scribed by Mireille Bossy; Denis Talay

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 317 KB
• Volume: 38
• Category: Article
• ISSN: 0378-4754
• DOI: 10.1016/0378-4754(93)e0065-d

✦ Synopsis

We consider the one-dimensional nonlinear ED.E. in the weak sense:

R

When the initial condition is a probability on ~, the solution Ut is the distribution of the random variable X t where (Xt) is a nonlinear stochastic process in the sense of McKean.

Our purpose is to study a stochastic particle algorithm for the computation of the cumulative distribution function of Ut. This method is based upon the moving of particles according to the law of a Markov chain approximating (Xt), and the approximation of (Eb(x, Xt), t <~ T) by means of empirical distributions.

For a bounded Lipschitz function b, we prove the convergence of the method with a rate of convergence of order O( 1/x/~+ x/~) where N is the number of particles and ,4 is the time step.