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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.
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.

Proof

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

  • [RevuzYor1999]Revuz, Daniel, and Marc Yor. Continuous Martingales and Brownian Motion. Third edition. Springer, 1999.