The aim of this PhD dissertation is to develop analytical-numerical techniques for solving random initial value problems based on linear difference and differential equations and systems in the mean square sense.
Regarding the study made on difference equations, see Chapter 3, we extend to random context some of the main results that in the deterministic case are known for solving such equations and to study the asymptotic behavior of their solution stochastic processes. With respect to differential equations should be emphasized that the unifying element of the study in this piece is the extension to random scenario of the so-called Fröbenius method for computing the solution of differential equations as power series expansions. Chapters 4-7 deal with problems of both scalar and matrix type of first and second order, where the randomness enters into the model through the initial conditions and coefficients, and moreover uncertainty in the latter case, is considered in additive and multiplicative way. The problems based on random differential equations allow us to introduce relevant stochastic processes such as the exponential process (see Chapter 5) or the trigonometric sine and cosine processes as well as some of their basic algebraic properties (see Chapter 6). The final chapter is devoted to study the Hermite differential equation with random coefficients and, under certain conditions, its solutions are obtained as finite random series that allow us to define the random Hermite polynomials. Besides obtaining the solutions by means of mean square convergent random power series, for each of the considered problems, we provide approximations of the main statistical functions of the solution process, such as the mean and variance (or in the vector case, the so-called covariance matrix). These approximations are compared through illustrative examples with those obtained by other available methods.
The AMS (2010) thematic classification of this dissertation is: 65C20, 60H35, 65N12.