The paradigm of complex probability and Monte Carlo methods

In 1933, Andrey Nikolaevich Kolmogorov established the system of five axioms that define the concept of mathematical probability. This system can be developed to include the set of imaginary numbers and this by adding a supplementary three original axioms. Therefore, any experiment can be performed in the set of complex probabilities which is the summation of the set of real probabilities and the set of imaginary probabilities. The purpose here is to include additional imaginary dimensions to the experiment taking place in the ‘real’ laboratory in and hence to evaluate all the probabilities. Consequently, the probability in the entire set  =  +  is permanently equal to one no matter what the stochastic distribution of the input random variable in is, therefore the outcome of the probabilistic experiment in can be determined perfectly. This is due to the fact that the probability in is calculated after subtracting from the degree of our knowledge the chaotic factor of the random experiment. This novel complex probability paradigm will be applied to the classical probabilistic Monte Carlo numerical methods and to prove as well the convergence of these stochastic procedures in an original way.

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