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Gronwall's Lemma

Gronwall's lemma is a basic estimate for integral inequalities. It is often used when a quantity is bounded by an initial term plus the accumulated size of the same quantity over time.
The terms in Gronwall's integral inequality
GRONWALL'S LEMMA
Let ϕ\phi be a nonnegative locally bounded Borel function on R+\mathbb{R}_{+}. Suppose that, for every t≥0t \geq 0,
ϕ(t)≤a+b∫0tϕ(s) ds,\begin{equation*}\phi(t) \leq a + b \int_{0}^{t} \phi(s) \, ds,\end{equation*}
where a≥0a \geq 0 and b≥0b \geq 0 are constants. Then for t≥0t \geq 0
ϕ(t)≤aebt.\begin{equation*}\phi(t) \leq a e^{bt}.\end{equation*}
In particular, if a=0a = 0, then ϕ≡0\phi \equiv 0.
The point of the lemma is that the integral term cannot amplify the function faster than an exponential factor. Thus, once the initial bound aa is known, the whole function is controlled on every finite interval.
Fix t≥0t \geq 0. From the assumed inequality, for every s∈[0,t]s \in [0,t] we have
ϕ(s)≤a+b∫0sϕ(u) du.\begin{equation*}\phi(s) \leq a + b \int_{0}^{s} \phi(u) \, du.\end{equation*}
Substituting this bound into the integral term gives
ϕ(t)≤a+b∫0t(a+b∫0sϕ(u) du)ds=a+abt+b2∫0t(t−u)ϕ(u) du≤a+abt+b2t∫0tϕ(u) du.\begin{align*}\phi(t)&\leq a + b \int_{0}^{t} \left(a + b \int_{0}^{s} \phi(u) \, du \right) ds \\&= a + abt + b^{2} \int_{0}^{t} (t-u)\phi(u) \, du \\&\leq a + abt + b^{2}t \int_{0}^{t} \phi(u) \, du.\end{align*}
Repeating the same substitution inductively yields, for every n≥0n \geq 0,
ϕ(t)≤a(1+bt+⋯+(bt)nn!)+bn+1tnn!∫0tϕ(u) du.\begin{equation*}\phi(t) \leq a \left(1 + bt + \cdots + \frac{(bt)^n}{n!}\right)+\frac{b^{n+1} t^n}{n!} \int_{0}^{t} \phi(u) \, du.\end{equation*}
Since ϕ\phi is locally bounded, the integral ∫0tϕ(u) du\int_{0}^{t}\phi(u)\,du is finite. Therefore
bn+1tnn!∫0tϕ(u) du⟶0,n→∞.\begin{equation*}\frac{b^{n+1} t^n}{n!} \int_{0}^{t} \phi(u) \, du \longrightarrow 0,\qquad n \to \infty.\end{equation*}
Letting n→∞n \to \infty in the previous estimate gives
ϕ(t)≤a∑k=0∞(bt)kk!=aebt.\begin{equation*}\phi(t) \leq a \sum_{k=0}^{\infty} \frac{(bt)^k}{k!} = a e^{bt}.\end{equation*}
If a=0a=0, then ϕ(t)≤0\phi(t) \leq 0 for every t≥0t \geq 0. Since ϕ\phi is nonnegative, this implies ϕ(t)=0\phi(t)=0 for every t≥0t \geq 0.

References

  • [RevuzYor2005]Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer, 2005.