For a one-parameter process of the form X, = X0 + & (b, d W, + & & ds, where W is a Wiener process and I+ d W is a stochastic integral, a twice continuously differentiable function f(X,) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a tw...

Simulation of beam cooling usually requires performing certain integral transformations every time step or so, which is a significant burden on the CPU. Examples are the dispersion integrals (Hilbert transforms) in the stochastic cooling, wake fields and IBS integrals. An original method is suggested for fast and sufficiently accurate computation of the integrals. This method is applied for the...

2 Young’s integrals and stochastic differential equations driven by fractional Brownian motions 4 2.1 Young’s integral and basic estimates . . . . . . . . . . . . . . . . . . 4 2.2 Stochastic differential equations driven by a Hölder path . . . . . . . 7 2.3 Multidimensional extension . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Fractional calculus . . . . . . . . . . . . . . . . . . . ...

Stochastic calculus is famous for providing the foundations for modern mathematical finance and is also used extensively in a large number of other areas of applied probability. The introductory text by Øksendal [3] strikes an excellent balance between theory and accessibility. Here we give a very brief review of the underlying concepts. A central notion for stochastic calculus is that of a (co...