From One-Step Accuracy to a Whole-Path Error
A numerical method advances an approximation one time step at a time. Its local truncation error measures the discrepancy in a single step when the numerical method and the exact SDE start from the same state. Its stability measures how a difference in starting states affects the difference in the next numerical states when both use the same Brownian increment. These quantities address distinct issues: small local errors can accumulate, while an unstable update can amplify errors already present.
We work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Consider the scalar SDE \(dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\), with an initial value \(X_0\). Every Ito integral below is of the form \(\int_0^t H_s\,dB_s\), where \(H_s\) is adapted and progressively measurable and satisfies \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\) on the finite time interval in question. We assume the SDE has a solution with the second moments used below finite. For the general estimates, assume its exact one-step solution from a given starting state is well-defined.
Take a uniform grid \(t_n=nh\), \(n=0,\ldots,N\), with \(Nh=T\). Write the numerical update as \(Y_{n+1}=\Psi_n(Y_n)\). The update uses the Brownian increment on \([t_n,t_{n+1}]\); when comparing two starting states, we always use the same increment. Let \(\Phi_n(x)\) denote the exact solution at \(t_{n+1}\) when the SDE starts at \(x\) at time \(t_n\), using that same Brownian path over the step.
The global strong error at grid point \(t_n\) is \(e_n=X_{t_n}-Y_n\); “strong” refers to comparison on the same probability space and with the same Brownian motion, not just comparison of distributions. A bound on \(\mathbb{E}|e_n|^2\) is a mean-square, or \(L^2\), error bound. A weak error instead compares expectations of test functions of the two outputs, and is not what the definition above measures.
A Stability-and-Consistency Bound
The following result gives a useful sufficient condition for global control. Its local-error requirement is intentionally strong enough to combine with stability by the triangle inequality alone. Sharper stochastic estimates can be available when local errors have additional cancellation; they require more than this argument.
Proof. Insert the numerical update from the exact starting state between the exact and numerical next states. This gives \(e_{n+1}=\Phi_n(X_{t_n})-\Psi_n(Y_n)=\delta_{n+1}+\Psi_n(X_{t_n})-\Psi_n(Y_n)\). The stability assumption, followed by expectation, gives \(\|\Psi_n(X_{t_n})-\Psi_n(Y_n)\|_2\le(1+Lh)^{1/2}\|e_n\|_2\). The triangle inequality in \(L^2\) therefore yields \(\|e_{n+1}\|_2\le C h^{p+1}+(1+Lh)^{1/2}\|e_n\|_2\). Since \(e_0=0\), repeated substitution gives the stated sum. For the uniform bound, \(1+Lh\le e^{Lh}\), so each factor \((1+Lh)^{j/2}\le e^{Ljh/2}\le e^{LT/2}\). There are \(n\le N=T/h\) terms, hence the sum is at most \((T/h)e^{LT/2}\). Multiplication by \(C h^{p+1}\) gives \(C T e^{LT/2}h^p\). At \(n=0\), the sum is empty and the initial error is zero, so the result includes that case. \(\square\)
This theorem illustrates a general pattern: stability controls propagation, while local accuracy controls what is added at each step. The resulting \(h^p\) global rate follows from adding \(N\), or order \(1/h\), local \(L^2\) bounds of size \(h^{p+1}\). The proof does not use independence or cancellation between successive defects. Consequently, it can be conservative for stochastic schemes.
Worked Example: An Additive-Noise Scheme with Zero Local Defect
Consider \(dX_t=2\,dt+3\,dB_t\), with \(X_0=x\), and use the Euler–Maruyama update. Its exact one-step solution is \(X_{t_{n+1}}=X_{t_n}+2h+3(B_{t_{n+1}}-B_{t_n})\), which is exactly the Euler–Maruyama update. The stochastic integral here has integrand \(H_s=3\), which is progressively measurable and satisfies \(\mathbb{E}\int_0^T H_s^2\,ds=9T<\infty\).
Thus \(\delta_{n+1}=0\) at every step. For any two starting values \(x,y\), the coupled updates are \(x+2h+3\Delta B_n\) and \(y+2h+3\Delta B_n\). Their difference is \(x-y\), so the stability inequality holds with \(L=0\), with equality. Starting from the same \(X_0\), every numerical value equals the exact grid value, and the global error is zero.
Check: on a step of length \(h=\tfrac14\), both updates add the same drift \(2\cdot\tfrac14=\tfrac12\) and the same noise \(3\Delta B_n\). Their difference remains zero; no approximation of the drift or diffusion contributes a local defect in this constant-coefficient equation.
Local Error and Stability Need Separate Checks
Worked Example: Drift Error and Stability for Euler–Maruyama
Consider the degenerate SDE \(dX_t=-X_t\,dt\), with \(X_0=2\) and no diffusion. The exact one-step solution from a starting value \(x\) is \(e^{-h}x\); Euler–Maruyama gives \((1-h)x\). For \(h=\tfrac12\) and \(x=2\), the exact value is \(2e^{-1/2}\), whereas the numerical value is \(1\). The local defect is \(2(e^{-1/2}-\tfrac12)\), which is negative because \(e^{-1/2}<\tfrac12\) is false; in fact \(e^{-1/2}>\tfrac12\), so the defect is positive. Its magnitude is \(2(e^{-1/2}-\tfrac12)\).
For stability, updates from \(x\) and \(y\) differ by \((1-h)(x-y)\). At \(h=\tfrac12\), their squared difference is \(\tfrac14|x-y|^2\), so the stability bound holds with \(L=0\), since \(\tfrac14\le1\). Over two steps, the numerical value is \(2(1-\tfrac12)^2=\tfrac12\), while the exact value at time \(1\) is \(2e^{-1}\). This comparison shows how the local discrepancy is carried through repeated updates.
Check: for a general small step, Taylor's formula gives \(e^{-h}=1-h+\tfrac12h^2+O(h^3)\). Thus the local defect from a starting value \(x\) is \(\tfrac12xh^2+O(h^3)\). The update difference is multiplied by \(1-h\), so for \(0<h<2\) its absolute value is less than the starting difference. Neither statement alone is a global error calculation; the theorem combines local bounds with stability over all steps.
The stability definition concerns two numerical updates, not an exact update and a numerical one. Comparing those two objects is the role of the local defect. Keeping the comparisons separate avoids a common ambiguity: a method can have a small one-step defect yet amplify perturbations, or be stable while making a substantial local approximation error.
Worked Example: A Stochastic Local Error for Geometric Brownian Motion
Consider \(dX_t=X_t\,dB_t\), with starting value \(x\) on a single interval of length \(h\). Euler–Maruyama gives \(x(1+\Delta B)\), where \(\Delta B=B_{t+h}-B_t\sim N(0,h)\). The exact solution is \(x\exp(\Delta B-\tfrac12h)\). To verify this candidate, apply Ito's formula to \(f(t,z)=z\exp(B_t-\tfrac12t)\) as a function of \(B_t\) and \(t\), or equivalently to \(g(t,b)=x\exp(b-\tfrac12t)\). Its derivatives are \(g_t=-\tfrac12g\), \(g_b=g\), and \(g_{bb}=g\); Ito's formula gives \(dg=g_b\,dB_t+(g_t+\tfrac12g_{bb})\,dt=g\,dB_t\), with initial value \(x\). The integrand \(X_s\) is progressively measurable, and \(\mathbb{E}\int_0^T X_s^2\,ds=x^2\int_0^T e^s\,ds<\infty\).
The local defect is \(x(e^{\Delta B-h/2}-1-\Delta B)\). For \(Z\sim N(0,h)\), the normal moment-generating function gives \(\mathbb{E}e^{Z-h/2}=1\), \(\mathbb{E}[Ze^{Z-h/2}]=h\), and \(\mathbb{E}e^{2Z-h}=e^h\). Also \(\mathbb{E}Z=0\) and \(\mathbb{E}Z^2=h\). Expanding the square therefore gives \(\mathbb{E}|\delta|^2=x^2\{e^h+1+h-2(1+h)\}=x^2(e^h-1-h)\). For \(x=1\), this is \(e^h-1-h\), whose Taylor expansion is \(\tfrac12h^2+O(h^3)\). Thus the local root-mean-square error is of order \(h\), not \(h^2\).
Check: the defect has mean zero, since \(\mathbb{E}e^{\Delta B-h/2}=1\) and \(\mathbb{E}\Delta B=0\). Its variance is therefore the computed mean square. Also \(e^h-1-h\ge0\) for \(h\ge0\), as follows from the exponential series, and it is zero at \(h=0\).
When Stochastic Errors Cancel
A local defect with conditional mean zero can accumulate differently from a sequence of errors whose absolute sizes are simply added. The next result isolates a setting where that cancellation can be used. Its assumptions are stated directly; they are not automatic consequences of mean-square stability for a general nonlinear scheme.
Proof. Since \(e_n\) is \(\mathcal{F}_{t_n}\)-measurable and \(\eta_{n+1}\) has conditional mean zero, \(\mathbb{E}[e_n\eta_{n+1}]=\mathbb{E}[e_n\mathbb{E}[\eta_{n+1}\mid\mathcal{F}_{t_n}]]=0\). Expanding the square and taking expectations yields \(\mathbb{E}|e_{n+1}|^2=a^2\mathbb{E}|e_n|^2+\mathbb{E}\eta_{n+1}^2\le(1+Lh)\mathbb{E}|e_n|^2+C h^{2p+1}\). Starting from \(e_0=0\), iteration gives the stated sum. For each \(j<n\le N\), \((1+Lh)^j\le e^{Ljh}\le e^{LT}\); there are at most \(N=T/h\) terms. Hence the sum is bounded by \((T/h)e^{LT}\), giving \(C T e^{LT}h^{2p}\). The case \(n=0\) has zero error and an empty sum. \(\square\)
This proposition shows why one-step mean-square errors of size \(h^{2p+1}\) can support a global mean-square bound of order \(h^{2p}\), when conditional orthogonality applies. In more general SDE schemes, the local defect may interact with the propagated error, and one must verify the needed conditional estimates rather than assume cancellation. The stability-and-triangle-inequality theorem remains a valid, more conservative alternative when its stronger local bound is available.
Common Pitfall: Local Order Is Not Automatically Global Order
A tempting but incorrect calculation is: “The local root-mean-square error is \(O(h)\), so after \(N=T/h\) steps the global root-mean-square error is \(O(h)\).” This assumes that local errors combine in a way that does not follow from their individual sizes. The triangle inequality only gives a sum of root-mean-square sizes: \(N\cdot O(h)=O(1)\) when \(T\) is fixed. To obtain a smaller global bound, one needs stronger local estimates, or additional structure such as conditional mean-zero errors and suitable orthogonality. Conversely, merely observing that a scheme is stable does not control the local defects it adds.
In practice, check the two ingredients separately. First compare a numerical step and an exact step started from the same state, using the same Brownian path. Then compare numerical steps started from different states, again using the same Brownian increment. Finally, use a global argument whose hypotheses match the local-error information available. The distinction matters because drift error and diffusion error can have different sizes, and stochastic cancellation is a property to establish, not a replacement for calculation.
Check Your Understanding
For each question, keep the local defect, stability estimate, and global error argument distinct.
- For \(dX_t=3\,dt+2\,dB_t\), write the exact one-step update and determine the Euler–Maruyama local defect.
- For the deterministic update \(\Psi(x)=(1-2h)x\), calculate the stability multiplier and state when its absolute value is at most one.
- If \(\|\delta_{n+1}\|_2\le C h^3\) and the hypotheses of the stability-and-consistency theorem hold, what global \(L^2\) order does that theorem give over fixed \(T\)?
- In the conditional-mean-zero proposition, identify the conditional expectation that makes the cross term vanish.
- Why does a local root-mean-square error of order \(h\), by itself, not imply a global root-mean-square error of order \(h\)?