Second order linear differential equations with analytic uncertainties: stochastic analysis via the computation of the probability density function Abstract   This paper concerns the analysis of random second order linear differential equations. Usually, solving these equations consists of computing the first statistics of the response process, and that task has been an essential goal in the literature. A more ambitious objective is the computation of the solution probability density function. We present advances on these two aspects in the case of general random nonautonomous second order linear differential equations with analytic data processes. The Frobenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series. We demonstrate that the convergence requires the boundedness of the random input coefficients. Further, the mean square error of the Frobenius method is proved to decrease exponentially with the number of terms in the series, although not uniformly in time. Regarding the probability density function of the solution at a given time, which is the focus of the paper, we rely on the law of total probability to express it in closedform as an expectation. For the computation of this expectation, a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set. We prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation. The pointwise convergence turns out to be exponential under a Lipschitz hypothesis. As the density functions are expressed in terms of expectations, we propose a crude Monte Carlo sampling algorithm for their estimation. This algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper. After that, the efficiency of the algorithm is improved by employing the control variates method. Numerical examples corroborate the variance reduction of the Monte Carlo approach.

Explicit Milstein schemes with truncation for nonlinear stochastic differential equations: Convergence and its rate Abstract   Although some implicit numerical procedures have been developed to treat high nonlinearity, the question whether one can use explicit schemes to achieve convergence rate similar to that of Milsteinxe2x80x99s procedure remained open. This brings us to the current work that focuses on numerical solutions of stochastic differential equations using explicit schemes. Our main goals are to obtain order one convergence in the second moment in a finitetime interval. In contrast to the implicit schemes, explicit schemes are advantageous, easily implementable, and computationally less intensive. To overcome the difficulties due to superlinear growth of the coefficients, a truncation device is used in our algorithm. In addition to reaching aforementioned goals in the analysis part, numerical examples are provided to demonstrate our results.