---
tag: 5C3411G4C0
kind: definition
name: Stochastic Volterra equation
msc: [60H20, 45R05, 45D05]
defines: KA95J9V6AS
uses: [DND8CBEVXT, 22735FGJV6, JA95NH206E, YRNFE016FF, 130FCDWRT1]
sources:
  - url: https://en.wikipedia.org/wiki/Stochastic_differential_equation
---

## Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, P)$ be a probability space with a [[filtration|130FCDWRT1]], and let $W$ be a one-dimensional [[Wiener process|22735FGJV6]] with respect to this filtration.
Let $\tau > 0$ and $\Delta = \set{(t, s) : 0 \le s \le t \le \tau}$.
Let $a, b \colon \Delta \times \R \to \R$ be measurable functions, and let $X_0$ be a continuous [[adapted process|YRNFE016FF]] on $[0, \tau]$.
The *stochastic Volterra equation* with coefficients $a$, $b$ and free term $X_0$ is
$$X(t) = X_0(t) + \int_0^t a(t, s, X(s)) \, ds + \int_0^t b(t, s, X(s)) \, dW(s), \qquad 0 \le t \le \tau,$$
where the last integral is an Itô integral, see [[Itô calculus|JA95NH206E]].
A *solution* is a continuous adapted process $X$ on $[0, \tau]$ such that, for every $t \in [0, \tau]$:

1. almost surely $\int_0^t \abs{a(t, s, X(s))} \, ds < \infty$ and $\int_0^t b(t, s, X(s))^2 \, ds < \infty$;
2. the equation holds almost surely.

## 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|B219G9SHJA]] $dX = \tilde a(t, X) \, dt + \tilde b(t, X) \, dW$, $X(0) = \xi$.
If $b = 0$, the equation is a deterministic [[Volterra integral equation|DND8CBEVXT]] 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.

## Explanations

### L1 human

It is a Volterra equation in which part of the input arrives through random kicks $dW$, whose effect may depend on the time elapsed since they happened.