A Priori Second-Moment Bounds for SDE Solutions

Why Bound the Running Maximum?

For an SDE, controlling the second moment at a single time does not by itself control the largest value reached over a whole time interval. A useful a priori estimate instead bounds the expected squared supremum of the solution. Such a bound quantifies how large the solution can become on a finite horizon and supplies an integrability guarantee for later arguments. “A priori” means that the estimate uses the equation’s coefficients and initial value, rather than an explicit formula for the solution.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Fix a finite horizon \(T>0\) and a deterministic initial value \(x_0\). Consider a continuous adapted solution of \(dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\), with \(X_0=x_0\). The drift integral is a Lebesgue integral. The stochastic integral is the Ito integral; its integrand is progressively measurable. For a process \(H\), write \(H\in\mathcal{L}^2([0,T])\) when it is progressively measurable and \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). Before localization, the solution’s stochastic integral may only be defined locally; the proof below constructs square-integrable stopped integrands.

The constants may be zero: for example, \(K_\sigma=0\) means the diffusion coefficient vanishes. These are squared-coefficient versions of linear growth. The estimate does not require the solution itself to be a martingale. Its stochastic-integral term will be a square-integrable martingale after stopping; the drift term generally prevents \(X\) from being a martingale.

The Finite-Horizon Bound

Proof. We first stop the solution before it can become large. For each integer \(n>|x_0|\), let \(\tau_n=\inf\{t\ge0:|X_t|\ge n\}\), with the infimum of the empty set defined as \(+\infty\), and let \(X^n_t=X_{t\wedge\tau_n}\). For \(s\le\tau_n\), continuity gives \(|X_s|\le n\). The process \(H^n_s=\mathbf{1}_{\{s\le\tau_n\}}\sigma(s,X_s)\) is progressively measurable. Its squared integral has finite expectation, since the growth bound gives \(|H^n_s|^2\le K_\sigma(1+n^2)\) on \([0,T]\). Thus \(H^n\in\mathcal{L}^2([0,T])\). The Ito integral \(M^n_t=\int_0^t H^n_s\,dB_s\) is a square-integrable martingale: in particular it is integrable and \(\mathbb{E}[M^n_t\mid\mathcal{F}_u]=M^n_u\) for \(u\le t\). By the Ito isometry, \(\mathbb{E}|M^n_t|^2=\mathbb{E}\int_0^t|H^n_s|^2\,ds\). These facts concern the stopped integral and do not assert that an unstopped local martingale is a true martingale.

Define \(A_n(t)=\mathbb{E}[\sup_{0\le u\le t}|X^n_u|^2]\). The stopped equation, the inequality \(|a+c+d|^2\le3(|a|^2+|c|^2+|d|^2)\), and Cauchy–Schwarz for the drift integral give \(\displaystyle A_n(t)\le3|x_0|^2+3t\int_0^t\mathbb{E}\big[\mathbf{1}_{\{s\le\tau_n\}}|\mu(s,X_s)|^2\big]\,ds+3\mathbb{E}\sup_{0\le u\le t}|M^n_u|^2.\) The standard Doob \(L^2\) maximal inequality for a square-integrable martingale states that \(\mathbb{E}\sup_{u\le t}|M^n_u|^2\le4\mathbb{E}|M^n_t|^2\); this is the standard maximal-inequality result used here. Substituting this inequality and the Ito isometry yields \(\displaystyle A_n(t)\le3|x_0|^2+3t\int_0^t\mathbb{E}\big[\mathbf{1}_{\{s\le\tau_n\}}|\mu(s,X_s)|^2\big]\,ds+12\int_0^t\mathbb{E}\big[\mathbf{1}_{\{s\le\tau_n\}}|\sigma(s,X_s)|^2\big]\,ds.\)

On \(\{s\le\tau_n\}\), \(X_s=X^n_s\). Hence, for either coefficient, the growth bound implies \(\mathbb{E}\big[\mathbf{1}_{\{s\le\tau_n\}}|\mu(s,X_s)|^2\big]\le K_\mu(1+A_n(s))\) and \(\mathbb{E}\big[\mathbf{1}_{\{s\le\tau_n\}}|\sigma(s,X_s)|^2\big]\le K_\sigma(1+A_n(s))\). Using \(t\le T\), we obtain \(\displaystyle A_n(t)\le3|x_0|^2+b\int_0^t(1+A_n(s))\,ds=3|x_0|^2+bt+b\int_0^t A_n(s)\,ds.\) In particular \(A_n(t)\le3|x_0|^2+bT+b\int_0^t A_n(s)\,ds\). Integral Gronwall gives \(A_n(t)\le(3|x_0|^2+bT)e^{bt}\). For completeness, if \(Y(t)=3|x_0|^2+bT+b\int_0^t A_n(s)\,ds\), then \(A_n\le Y\) and \(Y'(t)=bA_n(t)\le bY(t)\) almost everywhere; integrating the derivative of \(e^{-bt}Y(t)\) gives the stated inequality. This also covers \(b=0\).

Each continuous solution path is bounded on \([0,T]\). Therefore, for almost every outcome, \(\tau_n>T\) for all sufficiently large \(n\), and \(\sup_{u\le T}|X^n_u|^2\to\sup_{u\le T}|X_u|^2\). Fatou’s lemma now gives \(\displaystyle \mathbb{E}\sup_{u\le T}|X_u|^2\le\liminf_{n\to\infty}A_n(T)\le(3|x_0|^2+bT)e^{bT}.\) The right-hand side is finite, proving the theorem. \(\square\)

Proof. For each outcome and each fixed \(t\le T\), \(|X_t|^2\le\sup_{0\le u\le T}|X_u|^2\). Taking expectations and applying the theorem proves the claim. \(\square\)

Worked Applications

Why the Stopping and Maximal Inequality Matter

The localization is not cosmetic. Before the estimate is proved, one cannot simply assume that \(\mathbb{E}\int_0^T|\sigma(s,X_s)|^2\,ds\) is finite: linear growth bounds the integrand in terms of \(X\), and the desired moment bound is what will control \(X\). Stopping at \(\tau_n\) makes the integrand bounded on the finite horizon, so the Ito isometry and the square-integrable martingale estimates apply without circular reasoning. Fatou’s lemma then transfers the uniform bound back to the unstopped solution.

Common pitfall: using Ito isometry for a supremum. A wrong calculation replaces \(\mathbb{E}\sup_{u\le t}|M_u|^2\) by \(\mathbb{E}|M_t|^2=\mathbb{E}\int_0^tH_s^2\,ds\). Ito isometry evaluates the terminal-time second moment; it does not evaluate the maximum over earlier times. For the concrete choice \(H_s=1\) on \([0,1]\), the integral is \(M_t=B_t\). Its supremum includes both \(B_{1/2}^2\) and \(B_1^2\). The event \(\{|B_{1/2}|>|B_1|\}\) has positive probability, since the joint Gaussian vector \((B_{1/2},B_1)\) has a nonsingular covariance matrix. Consequently \(\mathbb{E}\sup_{0\le t\le1}|B_t|^2\ge\mathbb{E}\max(B_{1/2}^2,B_1^2)>\mathbb{E}B_1^2=1.\) The proposed equality with the Ito-isometry value \(1\) is false. The valid estimate used in the proof is Doob’s inequality, \(\mathbb{E}\sup_{u\le t}|M_u|^2\le4\mathbb{E}|M_t|^2\), which explains the factor \(12=3\cdot4\) in the theorem.

The bound is a finite-horizon estimate for an already specified solution. It does not by itself prove existence, uniqueness, or stability with respect to changes in initial conditions. Under the global Lipschitz setting, the existence-and-uniqueness theorem supplies a solution; this estimate gives a quantitative moment guarantee for it. Its main inputs are distinct: linear growth controls the coefficients, the Ito isometry and maximal inequality control the stochastic term, and Gronwall controls the feedback of the solution’s size into those coefficients.