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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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