Stochastic Volterra equation

Remarks

If $a(t, s, x) = \tilde a(s, x)$ and $b(t, s, x) = \tilde b(s, x)$ do not depend on $t$ and $X_0(t) = \xi$ is constant in time, the equation is the integral form of the stochastic differential equation $dX = \tilde a(t, X) \, dt + \tilde b(t, X) \, dW$, $X(0) = \xi$. If $b = 0$, the equation is a deterministic Volterra integral equation for each path. Systems with values in $\R^n$ and several Wiener processes are defined in the same way.

Example: $a = 0$, $b(t, s, x) = x$ and $X_0 = 1$. The equation is $X(t) = 1 + \int_0^t X(s) \, dW(s)$, and $X(t) = \exp(W(t) - t/2)$ is a solution. Indeed, Itô's formula applied to $f(t, w) = e^{w - t/2}$ gives $dX = X \, dW + \big(-\tfrac{1}{2} + \tfrac{1}{2}\big) X \, dt = X \, dW$.

Example with memory: $a = 0$, $X_0 = 0$ and $b(t, s, x) = (t - s)^{H - 1/2}$ with $H > 0$. Clause 1 holds since $\int_0^t (t - s)^{2H - 1} \, ds = t^{2H} / (2H) < \infty$, and the solution is the process $X(t) = \int_0^t (t - s)^{H - 1/2} \, dW(s)$, up to a choice of a continuous version.

Non-example: for $b(t, s, x) = (t - s)^{-1/2}$, the case $H = 0$, the integral $\int_0^t (t - s)^{-1} \, ds$ diverges, so clause 1 fails for every $t > 0$ and the equation has no solution.