The introduction of the anticipating (or Skorohod) integral in [8] and of the anticipating stochastic calculus in [7] has opened the question of solving anticipating stochastic differential equations. In general, the existence and uniqueness of the solution for these equations is not known. The difficulty of solving such equations is due to the fact that the classical method of Picard’s iterati...

We present in these lectures, in an informal manner, the very basic ideas and results of stochastic calculus, including its chain rule, the fundamental theorems on the representation of martingales as stochastic integrals and on the equivalent change of probability measure, as well as elements of stochastic differential equations. These results suffice for a rigorous treatment of important appl...

monte carlo methods designed base on stochastic reiterations and can be useful in uncertainty conditions. the purpose of this research was investigation of uncertainty to rank desertification indicators. in order to decrease uncertainty have been applied monte carlo reiterations for each value of indicator, if rank of xi is higher than rank of xk constantly. using multi dimension integrals the ...

We describe a method of random generation of the integrals A 1;2 (t; t + h) = Z t+h t Z s t dw 1 (r)dw 2 (s) ? Z t+h t Z s t dw 2 (r)dw 1 (s) together with the increments w 1 (t+h)?w 1 (t) and w 2 (t+h)?w 2 (t) of a two-dimensional Brownian path (w 1 (t);w 2 (t)). The method chosen is based on Marsaglia's `rectangle-wedge-tail' method, gen-eralised to higher dimensions. The motivation is the ne...

Abstract. We show that an anticipating stochastic forward integral introduced in [8] by means of fractional calculus is an extension of other forward integrals known from the literature. The latter provide important classes of integrable processes. In particular, we investigate the deterministic case for integrands and integrators from optimal Besov spaces. Here the forward integral agrees with...

Abstract. This paper derives several identities for the iterated integrals of a general semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Their form and derivations are combinatorial. The formulae simplify for continuous or finite-variation semimartingales, especially for counting processes. The results are motivated by chaotic representation of martinga...

We give a simple criterion for a stochastic process Z := H+K ·W , where H and K are respectively continuous and left-continuous processes independent of the driving Brownian motion W , which ensures that Z has the conditional full support property, introduced by Guasoni, Rásonyi, and Schachermayer, in connection to pricing models with transaction costs. As an application of this result, we show...

In Burrage and Burrage (1996) it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In ...