Norm Estimates for Stochastic Integral Terms

Why Supremum-in-Time Estimates Are Needed

In estimates for stochastic differential equations, controlling a process at one fixed time is often not enough. A convergence argument for processes needs control over the entire interval, measured by the expected square of the largest absolute value. Drift terms and stochastic-integral terms require different estimates: Cauchy–Schwarz controls the time integral, while the Itô isometry together with Doob’s martingale maximal inequality controls the stochastic integral.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\), and fix \(T>0\). All stochastic integrals below are Itô integrals. Their integrands are progressively measurable and belong to \(\mathcal{L}^2\) on \([0,T]\), meaning that \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). For a continuous adapted process \(X\), define its supremum-in-time \(L^2\) norm by

when the expectation is finite. The supremum is taken over the whole interval, and the expectation is under \(\mathbb{P}\). In particular, this is not the same quantity as \(\sup_{0\le t\le T}\mathbb{E}[|X_t|^2]\).

The Drift and Stochastic-Integral Bounds

Proof. For every \(t\le T\), Cauchy–Schwarz for the ordinary time integral gives \(|A_t|^2\le t\int_0^t a_s^2\,ds\le T\int_0^T a_s^2\,ds\). The right-hand side does not depend on \(t\). Taking the supremum over \(t\) and then expectation proves the drift bound.

For the stochastic bound, the Itô isometry gives \(\mathbb{E}[M_T^2]=\mathbb{E}\int_0^T H_s^2\,ds<\infty\). The Itô integral is a square-integrable martingale: for \(0\le s\le t\le T\), its increment has conditional expectation \(\mathbb{E}[M_t-M_s\mid\mathcal{F}_s]=0\). This follows first for adapted step integrands from the zero conditional mean of Brownian increments, and then for all \(\mathcal{L}^2\) integrands by the \(L^2\) construction of the integral. Thus its martingale property and integrability hold on this interval.

We use Doob’s \(L^2\) maximal inequality, \(\mathbb{E}[\sup_{t\le T}|M_t|^2]\le4\mathbb{E}[|M_T|^2]\). For completeness, its continuous-time argument is as follows. Write \(M^*=\sup_{t\le T}|M_t|\). For \(\lambda>0\), stop at the first time \(|M_t|\) reaches \(\lambda\), capped at \(T\). On the event \(\{M^*\ge\lambda\}\), conditional Jensen’s inequality and the martingale property at this bounded stopping time give an expected terminal absolute value at least \(\lambda\). Hence \(\lambda\mathbb{P}(M^*\ge\lambda)\le\mathbb{E}[|M_T|\mathbf{1}_{\{M^*\ge\lambda\}}]\). Integrating the tail inequality up to \(R>0\), using the tail formula for a nonnegative random variable and Tonelli’s theorem, yields

If the last truncated expectation is zero the desired bound is immediate; otherwise divide by its square root and square. Letting \(R\) increase to infinity and using monotone convergence gives Doob’s inequality. Applying it to the Itô integral and then applying the Itô isometry proves the claimed stochastic bound. \(\square\)

The factor \(T\) in the drift estimate comes from integrating over time. The factor \(4\) in the stochastic estimate comes from the maximal inequality; the Itô isometry itself has constant \(1\). Keeping those two steps distinct helps avoid losing or misplacing constants.

Combining the Bounds for SDE Terms

Suppose the drift coefficient \(\mu(t,x)\) and diffusion coefficient \(\sigma(t,x)\) satisfy a common global Lipschitz bound: for all \(t\in[0,T]\) and \(x,y\in\mathbb{R}\), $$ |\mu(t,x)-\mu(t,y)|\le L|x-y|, \qquad |\sigma(t,x)-\sigma(t,y)|\le L|x-y|, $$ where \(L\ge0\). The drift is the coefficient of \(dt\), and the diffusion coefficient is the coefficient of \(dB_t\). Let \(X,Y\) be continuous adapted processes with finite \(S^2_T\) norm. Assume the coefficient processes evaluated along them are progressively measurable and square-integrable as needed for the time and Itô integrals below. Consider only the difference of their integral terms, $$ D_t=\int_0^t\big(\mu(s,X_s)-\mu(s,Y_s)\big)\,ds +\int_0^t\big(\sigma(s,X_s)-\sigma(s,Y_s)\big)\,dB_s. $$

Proof. Denote the drift integral in \(D_t\) by \(A_t\) and the stochastic integral by \(M_t\). For every \(t\), \(|A_t+M_t|^2\le2|A_t|^2+2|M_t|^2\). Taking the supremum and expectation preserves this inequality. The drift bound gives $$ \mathbb{E}\left[\sup_{t\le T}|A_t|^2\right] \le T\,\mathbb{E}\int_0^T|\mu(s,X_s)-\mu(s,Y_s)|^2\,ds \le L^2T^2\,\mathbb{E}\left[\sup_{t\le T}|X_t-Y_t|^2\right]. $$ Here the last inequality follows pointwise from the Lipschitz condition and \(\int_0^T|X_s-Y_s|^2ds\le T\sup_{t\le T}|X_t-Y_t|^2\). The stochastic bound and the diffusion Lipschitz condition give $$ \mathbb{E}\left[\sup_{t\le T}|M_t|^2\right] \le4\,\mathbb{E}\int_0^T|\sigma(s,X_s)-\sigma(s,Y_s)|^2\,ds \le4L^2T\,\mathbb{E}\left[\sup_{t\le T}|X_t-Y_t|^2\right]. $$ Multiplying these two bounds by \(2\), adding them, and collecting terms proves the corollary. \(\square\)

Common Mistake: Using the Isometry for a Supremum

A frequent error is to write \(\mathbb{E}[\sup_{t\le T}|M_t|^2]=\mathbb{E}[M_T^2]\) and then apply the Itô isometry. The isometry gives the terminal second moment, not the expected squared supremum. For example, with \(M_t=B_t\) on \([0,1]\), the isometry gives \(\mathbb{E}[M_1^2]=1\), while the maximal estimate supplied by Doob is \(\mathbb{E}[\sup_{t\le1}|B_t|^2]\le4\). The correct argument is to apply Doob’s inequality first and the Itô isometry second.

These bounds are useful in existence and uniqueness arguments because they turn coefficient differences into a multiple of the process difference in the same supremum norm. They do not, by themselves, show that this multiple is less than one, nor do they establish convergence of an iteration. Such conclusions require additional arguments. In particular, do not silently replace a supremum of expectations by an expectation of a supremum: the latter is the quantity controlled here.