Euler–Maruyama on a predictable grid with bridge-refinement status

Assume AC and use the coefficient, Brownian, and initial-value conventions of Euler–Maruyama recursion, on a filtered space satisfying the usual conditions. The inputs are $T>0$, the coefficients $b,\sigma$, the Brownian path or increment source, an initial value $X_0$, a deterministic maximum step $h_{\max }>0$, and a deterministic work budget $B\in {\mathbb{N}}_0$. Stopped Brownian values are measurable at their stopped sigma-algebras by the upper-grid approximation and null-completion argument of Continuous adapted processes are measurable at bounded stopping times, which applies to any complete filtration, and each stopping time ${\tau }_n$ is ${\mathcal{F}}_{{\tau }_n}$-measurable by the event tests of Continuous-time stopping times and stopped sigma-algebras. Since $h_n$ is ${\mathcal{F}}_{{\tau }_n}$-measurable, Borel coefficient evaluation and the predictable-grid update keep $Y_n$ measurable at ${\mathcal{F}}_{{\tau }_n}$ by induction. The continuous interpolation is adapted as well: at each fixed $t$, the active-interval events are in ${\mathcal{F}}_t$ by the stopping-time tests, and each term uses stopped values restricted to $\{{\tau }_n\le t\}$. Choose one of the following modes.

1. Predictable-grid mode. Supply a finite predictable grid Predictable stopping-time grid with $h_n\le h_{\max }$ on every active interval. Initialize $Y_0=X_0$ and process its intervals in order until the grid reaches $T$ or the budget is exhausted. Each interval uses the actual Brownian increment and update $Y_{n+1}=Y_n+b({\tau }_n,Y_n)h_n+\sigma ({\tau }_n,Y_n)(W_{{\tau }_{n+1}}-W_{{\tau }_n})\mathrm{.}$ The continuous interpolation on the interval is $Y_t=Y_n+b({\tau }_n,Y_n)(t-{\tau }_n)+\sigma ({\tau }_n,Y_n)(W_t-W_{{\tau }_n})\mathrm{.}$ This is the Euler–Maruyama update of Euler–Maruyama recursion with the supplied interval length; the fixed-grid supplier itself makes no variable-grid convergence assertion.

2. Endpoint-refinement mode. Supply a total Borel-measurable Boolean-valued accept/reject rule on the finite-dimensional current trial data. Initialize a queue with a finite partition $0=u_0<\cdots <u_J=T$ whose root intervals have lengths at most $h_{\max }$, together with their sampled Brownian increments. Pop the earliest interval $[u,u+\mathrm{\ell }]$, form the Euler trial $Y^{\ast }=Y+b(u,Y)\mathrm{\ell }+\sigma (u,Y)\mathrm{\Delta }W$, and apply the supplied Borel accept/reject rule (a Borel function of the finite trial history and current trial data). If accepted, set $Y\leftarrow Y^{\ast }$, append the right-endpoint/state pair to the accepted path record, and advance to the interval's right endpoint. If rejected, retain its parent increment, split it with Dyadic Brownian bridge increment tree, and put its left then right children at the front of the queue; do not update $Y$ on the rejected parent and do not redraw the parent or either child. This branch preserves the sampled path tree, but an endpoint-dependent acceptance rule does not by itself establish the stopped-measurability condition of a predictable-grid controller; the strong-error theorem on this page makes no error claim for this branch.

For either mode, initialize current_time=0, current_state=X_0, and trial_count=0. Maintain current_time and current_state as the endpoint and Euler state of the accepted path. In endpoint-refinement mode, also initialize the accepted path record with $(0,X_0)$; append each right-endpoint/state pair only when its interval is accepted. Count every trial node, accepted or rejected, as one work unit; after each trial, increment trial_count by one. Before the first trial and after every trial, first check whether the mode's completion condition holds: the supplied grid has reached $T$ in predictable-grid mode, or the queue is empty and the accepted path reaches $T$ in endpoint-refinement mode. If so, return with status: complete. Otherwise, if the count has reached $B$, return with status: inconclusive; if not, perform the next trial. Thus a budget $B=0$ returns inconclusive, and a last trial that reaches $T$ returns complete.

Both statuses use the output record (mode, status, current_time, current_state, approximation, pending_work, trial_count). On completion, current_time=T, current_state=Y_T, and pending_work is empty. The approximation is the full continuous Euler interpolation in predictable-grid mode; in endpoint-refinement mode it is the accepted ordered endpoint/state record together with the retained increment tree. On an inconclusive return, trial_count=B, current_state is the state at current_time, and approximation is the partial output through that time: the continuous interpolation restricted to $[0,\mathtt{\text{current\_time}}]$ in predictable-grid mode, and the accepted endpoint/state record and retained tree in endpoint-refinement mode. pending_work is the remaining ordered grid suffix or the current ordered tree queue, respectively. The tree assigns no values at non-dyadic times, so endpoint-refinement mode returns its accepted node record rather than a continuous interpolation. A finite budget never changes an inconclusive result into a successful certificate.