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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.

MSC:
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
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