Weak Error for Smooth Test Functions

Weak Error Measures Bias in Expectations

Numerical paths are often used to estimate quantities such as \(\mathbb{E}[f(X_T)]\), where \(X_T\) is the solution of an SDE and \(f\) is a chosen observable. A discretization can have noticeable pathwise error but a small error in this expectation: positive and negative deviations may cancel after averaging. Conversely, an observable that changes sharply can turn a small pathwise discrepancy into a large expectation error. Weak error describes the first kind of discrepancy, for a specified test function.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\). Consider the scalar SDE \(dX_t=\mu(X_t)\,dt+\sigma(X_t)\,dB_t\), with deterministic initial value \(X_0=x_0\). Here \(\mu\) is the drift and \(\sigma\) is the diffusion coefficient. Unless stated otherwise, assume they are globally Lipschitz and have at most linear growth: for some \(L,C<\infty\), \(|\mu(x)-\mu(y)|+|\sigma(x)-\sigma(y)|\le L|x-y|\) and \(|\mu(x)|+|\sigma(x)|\le C(1+|x|)\). These conditions ensure a unique strong solution with finite second moments on each bounded time interval. The stochastic integral in the integral form of the SDE is the Ito integral; its integrand \(\sigma(X_s)\) is adapted and progressively measurable, and satisfies \(\mathbb{E}\int_0^T \sigma(X_s)^2\,ds<\infty\) under these assumptions.

On a uniform grid \(t_n=nh\), \(h=T/N\), Euler–Maruyama is defined by \(X^h_{n+1}=X^h_n+\mu(X^h_n)h+\sigma(X^h_n)(B_{t_{n+1}}-B_{t_n})\), starting from \(X^h_0=x_0\). A test function \(f\) is a measurable function for which the expectations below exist.

In the usual rate statement, weak order one means that, for fixed \(T\), there is a constant \(C_{f,T}\) independent of \(h=T/N\) such that \(|e_{\mathrm{w}}(f,T,h)|\le C_{f,T}h\) for sufficiently small \(h\). The constant may depend on the coefficients, the time horizon, the initial value, and the test function. This is an upper bound, not a claim that the error is exactly proportional to \(h\): the signed error can vanish or decay faster for special choices of \(f\).

A Coupling Bound and Its Limits

Strong and weak errors are related when the test function is Lipschitz. Two random variables \(U,V\) are coupled if they are defined on the same probability space; no independence is required. A function \(f:\mathbb{R}\to\mathbb{R}\) is Lipschitz with constant \(L_f\) if \(|f(x)-f(y)|\le L_f|x-y|\) for all \(x,y\).

Proof. Linearity of expectation and the triangle inequality give \(|\mathbb{E}f(U)-\mathbb{E}f(V)|=|\mathbb{E}(f(U)-f(V))|\le\mathbb{E}|f(U)-f(V)|\). The Lipschitz condition bounds the last expression by \(L_f\mathbb{E}|U-V|\). Cauchy–Schwarz, applied to \(|U-V|\) and the constant random variable \(1\), gives \(\mathbb{E}|U-V|\le(\mathbb{E}|U-V|^2)^{1/2}\). The second-moment assumption makes these bounds finite. \(\square\)

In particular, a strong mean-square error bound of order \(h^{1/2}\) implies a weak error bound of order \(h^{1/2}\) for Lipschitz tests. It does not by itself give weak order one. A sharper weak rate can arise from cancellation in the expectation, but establishing that cancellation requires additional analysis of the scheme and typically greater smoothness of the coefficients and the test function.

Why Test-Function Regularity Matters

The coupling proposition explains one role of regularity: a Lipschitz function cannot magnify a small difference by more than its Lipschitz constant. Smoothness also matters when deriving higher weak rates. Taylor expansion averages increments rather than taking their absolute values, so odd centered Gaussian terms can cancel. To use this mechanism with controlled remainders, one needs derivatives of the test function and enough regularity of the coefficients.

For Euler–Maruyama, a standard sufficient setting for a weak order-one estimate is that \(\mu\) and \(\sigma\) are globally Lipschitz with bounded continuous derivatives through order four, and that \(f\) has bounded continuous derivatives through order four. In that setting, for fixed \(T\), a standard weak order-one result gives \(|\mathbb{E}f(X_T)-\mathbb{E}f(X^h_N)|\le C_{f,T}h\). The constant is not asserted to be uniform over all such test functions. The proof of this rate is the subject of the subsequent weak-order analysis; the statement here records the regularity it uses, rather than deriving the rate.

These bounded smoothness assumptions are convenient sufficient conditions, not necessary conditions in every individual problem. Variants allow derivatives and coefficients of polynomial growth, provided appropriate moment bounds are available and the test function has matching growth and integrability. A rate statement must specify which version is being used. For instance, a polynomially growing \(f\) requires enough moments of both the exact solution and the approximation for \(\mathbb{E}|f(X_T)|\) and \(\mathbb{E}|f(X^h_N)|\) to be finite.

Global Error as a Sum of One-Step Defects

A useful way to organize a weak-error proof is to separate the one-step discrepancy from the way later steps act on it. Assume the SDE is time-homogeneous and Markov, and define its exact transition operator \(P_h\) and the Euler transition operator \(Q_h\) by \(P_hg(x)=\mathbb{E}[g(X_h^x)]\) and \(Q_hg(x)=\mathbb{E}[g(x+\mu(x)h+\sigma(x)\sqrt{h}\,Z)]\), where \(X_0^x=x\) and \(Z\sim N(0,1)\). The exact transition expectation is well-defined for functions with suitable integrability. The Euler expectation uses one fresh normal increment at each step.

Proof. The Markov property gives \(\mathbb{E}f(X_T)=(P_h^Nf)(x_0)\); the independent Brownian increments in Euler–Maruyama give \(\mathbb{E}f(X^h_N)=(Q_h^Nf)(x_0)\). For any two operators \(A,B\) whose displayed compositions are defined, \(A^N-B^N=\sum_{k=0}^{N-1}B^k(A-B)A^{N-1-k}\). Indeed, the \(k\)-th summand is \(B^kA^{N-k}-B^{k+1}A^{N-1-k}\); summing makes consecutive terms cancel, leaving \(A^N-B^N\). Taking \(A=P_h\), \(B=Q_h\), and evaluating at \(x_0\) gives the formula. If \(N=0\), the sum is empty and both sides are zero, so the identity includes that degenerate case as well. \(\square\)

This identity does not by itself give a rate: it isolates the one-step defect \((P_h-Q_h)g\), but that defect is applied to functions \(g=P_h^{N-1-k}f\), whose derivatives and growth must also be controlled. A weak-rate proof therefore needs both a local comparison and bounds that remain valid as the test function is propagated through the transition operators. This is one reason coefficient regularity and test-function regularity appear together in sufficient hypotheses.

Common Pitfalls and Practical Interpretation

A weak error statement is incomplete if it does not name the test function, the time horizon, and the limiting regime. The expressions \(\mathbb{E}f(X_T)\) and \(\mathbb{E}f(X^h_N)\) must both be finite. When reporting a rate, keep the test function fixed (or specify a norm over a stated family of tests) and let \(h=T/N\) tend to zero with \(T\) fixed.

A common mistake is to replace weak error by the mean-square path error. For example, concluding that an \(O(h^{1/2})\) strong error automatically proves an \(O(h)\) weak error is invalid: the coupling bound only transfers the available strong rate. Conversely, a small weak error for one \(f\) does not imply that the simulated paths are close; the identity test in the worked example had zero weak error while the mean-square discrepancy was positive.

Finally, do not assume that all test functions qualify for the same rate. Bounded smooth tests are compatible with standard Taylor-based estimates; indicators and other discontinuous functions may require separate distributional or smoothing arguments. The one-step-defect decomposition clarifies the strategy: specify the observable, control the local weak defect for the functions that occur, and ensure coefficient and moment conditions support those bounds.