Random integral equation

Remarks

Random Volterra equations, with $\int_a^t$ in place of $\int_a^b$, and nonlinear random equations such as $x(t, \omega) = h(t, \omega) + \int_a^b k(t, s, \omega) f(s, x(s, \omega)) \, ds$ are defined in the same way. A random solution is a stochastic process indexed by $t \in [a, b]$. The integral is an ordinary Lebesgue integral, computed separately for each $\omega$.

Example: $[a, b] = [0, 1]$, $Y$ a random variable with values in $[0, 1]$, $k(t, s, \omega) = Y(\omega)/2$ and $h = 1$. A constant $x(t, \omega) = c(\omega)$ solves the equation when $c = 1 + Y c / 2$, that is, $c = 2/(2 - Y)$. This $c$ is measurable, so $x(t, \omega) = 2/(2 - Y(\omega))$ is a random solution.

Non-example: the equation $x(t) = 1 + \int_0^t x(s) \, dW(s)$, with $W$ a Wiener process, is not a random integral equation in this sense. Almost every path of $W$ has unbounded variation, so the integral cannot be computed path by path as a Lebesgue–Stieltjes integral. It is an Itô integral, and the equation is a stochastic Volterra equation.