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A random Euler scheme for Carathéodory differential equations. (English) Zbl 1160.65003
The Caratheodory differential equation has the form of
\[ \dot{x} =f(t,x), x(0)=x_0 \] with a vector field \( f\) measurable in \( t\) instead of continuous. The Euler type method of computing the solution is based on an equidistant discretisation of \([0,T]\):
\[ \tilde{x}(t_{i+1})=\tilde{x}(t_i)+\int_{t_i}^{t_{i+1}}f(\tau,\tilde{x{t_i}})d\tau, i=0,\dots,n-1 \] A Monte Carlo approximation \(\frac{1}{m}\frac{T}{n}\sum_{j=1}^mf(\tau_j,\tilde{x{t_i}})\) with \(\tau_j\) i.i.d uniformly distributed on \([t_i,t_{i+1}]\) is performed to compute the above integrals to construct a so-called random Euler scheme, of which the solution is denoted by \(x_{n,m}(t)\). Under mild conditions, the \(L_p\) maximum error of the solution is proven bounded by \(C(\frac{1}{n}+\frac{1}{\sqrt{nm}})\). In the case of \(m=n\), the maximum error is also shown to have an order of \(\frac{1}{n^{1-\epsilon}}\) for any \(\epsilon>0\). A numerical example is shown for a comparison with the exact solution.

65C05 Monte Carlo methods
65L70 Error bounds for numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
Full Text: DOI
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