Itô and Stratonovich Stochastic Calculus

We provide a detailed hands-on tutorial for the R Development Core Team [2014] add-on package Sim.DiffProc [Guidoum and Boukhetala, 2014], for symbolic and floating point computations in stochastic calculus and stochastic differential equations (SDEs). The package implement is introduced and it is explains how to use the snssde1d, snssde2d and snssde3d main functions in this package, for simulate uniand multidimensional SDEs, notice that, in this version of the package, multidimensional SDEs need to have diagonal noise. 1 Background and motivation Differential equations are used to describe the evolution of a system. SDEs arise when a random noise is introduced into ordinary differential equations (ODEs). Let us consider first an example to illustrate the need for simulated and to analyze the properties of solution of SDEs. Many (or even most) processes in nature and technology are driven by (temperature, energy, velocity, concentration,. . . ) changes. Such processes are called diffusion (or dispersion) processes because the quantity considered (e.g., temperature) is distributed to an equilibrium state is established (i.e., until the differences that drive the process are minimized). There are many examples of diffusion processes. Diffusion is responsible for the distribution of sugar throughout a cup of coffee. Diffusion is the mechanism by which oxygen moves into our cells. Diffusion is of fundamental importance in many disciplines of physics, economics, mathematical finance, chemistry, and biology: diffusion is relevant to the sintering process (powder metallurgy, production of ceramics), the chemical reactor design, catalyst design in the chemical industry, doping during the production of semiconductors, and the transport of necessary materials such as amino acids within biological cells. The diffusion processes {Xt, t ≥ 0} solutions to SDEs, with slight notational variations, are standard in many books with applications in different fields, see, e.g., Soong [1973], Rolski et all [1998], Øksendal [2000], Klebaner [2005], Henderson and Plaschko [2006], Racicot and Théoret [2006], Allen [2007], Jedrzejewski [2009], Platen and Bruti-Liberati [2010], Stefano [2011], Heinz [2011],. . . . If Xt is a differentiable function defined for t ≥ 0, f(x, t) is a function of x and t, and the following relation is satisfied for all t, 0 ≤ t ≤ T , dXt dt = X ′ t = f(Xt, t), and X0 = x0, (1) then Xt is a solution of the ODE with the initial condition x0. The above equation can be written in other forms (by continuity of X ′ t): Xt = X0 + ∫ t