Strong Convergence without Global Monotonicity

Definitions and mathematical statement

For an ${\mathbb{R}}^d$-valued random variable $Z$ on a probability space $(\Omega,\mathcal F,\mathbb P)$, define \[ \|Z\|_{L^2(\Omega;{\mathbb{R}}^d)}=\left({\mathbb{E}}[\|Z\|_{{\mathbb{R}}^d}^2]\right)^{1/2}, \] whenever this quantity is finite. Strong $L^2$ approximation compares an exact solution and its numerical approximation on a common probability space using this norm.

Problem. For multidimensional stochastic differential equations arising in applications whose coefficients do not satisfy global monotonicity assumptions, establish strong $L^2$ convergence rates for suitable numerical approximations. The task includes identifying the coefficient hypotheses and numerical schemes under which such rates hold; the supplied record does not state a universal assertion covering every equation that fails global monotonicity.

A possible precise finite-horizon target can be expressed as follows. Given an application-specific equation with an ${\mathbb{R}}^d$-valued solution $X=(X_t)_{0\leq t\leq T}$, where $T>0$, construct approximations $Y_t^h$, coupled with $X$, for admissible discretization parameters $h\in H\subset(0,\infty)$ with $0$ an accumulation point of $H$. Specify nonempty observation sets $I_h\subseteq[0,T]$ on which the approximations are defined. An algebraic strong $L^2$ rate would be an estimate \[ \sup_{t\in I_h}\|X_t-Y_t^h\|_{L^2(\Omega;{\mathbb{R}}^d)}\leq C h^r \qquad\text{for all }h\in H\text{ with }h\leq h_0, \] for some $r>0$, $C<\infty$, and $h_0>0$, with $C$ independent of $h$. Here $r$ refers to the root mean square error; the corresponding squared error bound has exponent $2r$.

This estimate is a formulation template, not an additional conjecture attributed to the source. The source does not specify an equation, a driving noise, coefficient domains or regularity, initial-data assumptions, existence and uniqueness hypotheses, moment bounds, the intended definition of global monotonicity, a numerical scheme, an error observation set, or a target rate. In particular, terminal-time error, maximum root mean square error over grid points, and the $L^2$ norm of a pathwise supremum are distinct targets; the supplied record does not choose among them. Any complete theorem requires these choices and hypotheses to be stated explicitly.

Short English statement

Establish quantitative strong $L^2$ error bounds for suitable numerical methods applied to multidimensional stochastic differential equations from applications that fail global monotonicity assumptions. The equation classes, replacement assumptions, schemes, and error norm must be specified before this broad research problem becomes a fully quantified theorem. The supplied dated curation labels the problem partially solved; that assessment is not independently verified here.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003391$, titled Strong Convergence without Global Monotonicity, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified. The supplied proposed year and year tag are $2017$.
• Supplied source citation: Mini-Workshop: Stochastic Differential Equations: Regularity and Numerical Analysis in Finite and Infinite Dimensions (2018), Oberwolfach Reports, pp. 453--498, DOI 10.4171/owr/2017/9. https://doi.org/10.4171/owr/2017/9. The record attributes the problem to workshop open-problem discussions. The citation's year $2018$ differs from the supplied proposed year $2017$; this discrepancy is retained without resolution.
• Supplied literature source, "Oberwolfach report": https://ems.press/content/serial-article-files/46671.
• Supplied literature source, "Hutzenthaler--Jentzen": https://doi.org/10.1214/19-AOP1345.
• Supplied literature source, "Dai--Wang": https://arxiv.org/abs/2401.00385.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem partially solved and reports strong $L^2$ convergence rates without global monotonicity for important classes of multidimensional stochastic differential equations, including stopped or tamed schemes and higher-order methods, while qualifying the broad problem as conditional on coefficient and scheme assumptions. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement corrected_verified and records a manual semantic, typographic, and LaTeX review claiming preservation of the mathematical target. This is supplied editorial metadata, not independent mathematical verification of this draft or of the literature assessment.