Solutions of Random Initial Value Problems

Abstract

We consider the random nonlinear initial value problem

$$ \frac{{dX}}{{dt}} = g\left( {t,X,A} \right);X\left( {{t_0}} \right) = {X_0} $$

(1)

where X = (X ₁ , X ₂ ,…, X _n )^T, g = (g ₁ , g ₂ ,…, g _n )^T, and T denotes transpose. The randomness in the problem arises from the vector of initial conditions X ₀ = (X ₁ , X ₂ ,…, X _n )^T, whose components are random variables with joint probability density function (jpdf) ${f_{{{x_0}}}}$ (x ₀ ), and the m random parameters A = (A ₁ , A ₂ ,…, A _m ), which appear on the right-hand side of eqn. (1) and which have the jpdf f_A (a). In general, A and X ₀ may be jointly distributed; however they are often independent in many applications.