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✓ 9 results · all verified · 6 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Stable sde schemes for superlinear drift

1 · Prerequisites

2 · Summary

This page develops three specific methods for SDEs outside the globally Lipschitz drift setting. The tamed Euler update replaces $b$ by $b_{h} = b / (1 + h ∣ b ∣)$ ; under a one-sided Lipschitz condition, a polynomial bound on $D b$ , globally Lipschitz diffusion, and all finite initial moments, its grid values have uniform finite-horizon moments and its continuous interpolation has strong path-supremum order one-half. The proof keeps the rare-event moment estimate separate from the error recurrence.

The drift-implicit Euler–Maruyama method applies the backward resolvent to $Y_{n} + σ (Y_{n}) Δ W_{n}$ . For globally dissipative drift with a unique resolvent and Lipschitz diffusion, the page proves finite-time mean-square stability and a grid-point strong order-one-half bound under the stated polynomial derivative and moment assumptions. A failed finite-precision solve is reported as a failure status and supplies no next scheme value.

For CIR, the page transforms $Z = \sqrt{X}$ and solves a quadratic implicit step with a unique strictly positive root under the Feller condition $2 κ θ \geq σ^{2}$ . The exact solution is shown to remain positive and nonexplosive; localization then proves convergence in probability on finite horizons. No $L^{p}$ error rate is asserted without further inverse moment control.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Locally Lipschitz Itô SDE on an open state domain

Definition

Assume AC. Fix a finite horizon $T > 0$ , finite dimensions $d, m \geq 1$ , and an open set $U \subseteq R^{d}$ . Use a complete usual filtered probability space and an adapted standard $m$ -dimensional Brownian motion whose increments after time $s$ are independent of $F_{s}$ , as in $d$ -dimensional Brownian motion and Continuous Brownian Ito processes. Let $b : [0, T] \times U \to R^{d}$ and $σ : [0, T] \times U \to R^{d \times m}$ be Borel maps. Require that they are bounded on every compact subset of $[0, T] \times U$ and locally Lipschitz in the state uniformly in time: for every compact $K \subset U$ there is $L_{K} < \infty$ such that, for every $t \in [0, T]$ and $x, y \in K$ , $∣ b (t, x) - b (t, y) ∣ + ∥ σ (t, x) - σ (t, y) ∥_{F} \leq L_{K} ∣ x - y ∣ .$ Let $ξ$ be $F_{0}$ -measurable, take values in $U$ almost surely, be independent of the future Brownian increments, and satisfy $E ∣ ξ ∣^{2} < \infty$ .

For $n \geq 1$ , set $K_{n} = {x \in U : ∣ x ∣ \leq n, dist (x, U^{c}) \geq n^{- 1}} .$ When $U = R^{d}$ , interpret $dist (x, U^{c}) = + \infty$ . If $U^{c} \neq \emptyset$ , the distance function is $1$ -Lipschitz, so each $K_{n}$ is closed; it is bounded, hence compact by Heine--Borel. For $U = R^{d}$ , each $K_{n}$ is a closed ball and is compact as well. As $n$ increases, the radius bound increases and the distance threshold decreases, so the sets are nested. Every $x \in U$ belongs to $int K_{n}$ for all sufficiently large $n$ : choose $n > ∣ x ∣$ and, when $U^{c} \neq \emptyset$ , $n^{- 1} < dist (x, U^{c})$ . Thus their interiors exhaust $U$ . $∣ d (x, A) - d (y, A) ∣ \leq d (x, y)$ , so the distance to a fixed nonempty set is $1$ -Lipschitz Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line

A local strong solution up to exit is the compatible family of stopped adapted continuous processes $X^{(n)}$ and stopping times $τ_{n}$ Continuous-time stopping times and stopped sigma-algebras such that $τ_{n} = \inf {t \in [0, T] : X_{t}^{(n)} \notin int K_{n}} \land T, X_{t \land τ_{n}}^{(n)} = ξ + \int_{0}^{t \land τ_{n}} b (s, X_{s}^{(n)}) d s + \int_{0}^{t \land τ_{n}} σ (s, X_{s}^{(n)}) d W_{s},$ with the equation holding indistinguishably and with $X^{(n)}$ constant after $τ_{n}$ . Empty early $K_{n}$ simply give $τ_{n} = 0$ . Compatibility means that for $m \geq n$ , $X^{(m)}$ stopped at $τ_{n}$ equals $X^{(n)}$ up to indistinguishability. The maximal lifetime is $ζ = \lim_{n} τ_{n}$ ; for $t < ζ$ , the common value of $X_{t}^{(n)}$ for sufficiently large $n$ defines the maximal local solution $X_{t}$ .

This definition records a local integral equation; it does not assert that a solution exists or that $ζ = T$ . Those are separate theorem obligations. The local coefficient conditions allow superlinear growth; global Lipschitz hypotheses will be imposed only on the coefficient truncations used in the existence proof.

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Lp maximal moment bound for Brownian Itô integrals

Statement

Assume AC and the usual filtered Brownian conventions of Continuous Brownian Ito processes. Let $G$ be a predictable $d \times m$ matrix process on $[0, T]$ with $E {(\int_{0}^{T} ∥ G_{s} ∥_{F}^{2} d s)}^{p / 2} < \infty$ for some real $p \geq 2$ , and let $M_{t} = \int_{0}^{t} G_{s} d W_{s}$ be its continuous vector Itô integral. There is a finite constant $C_{p, d}$ , independent of $G$ and $T$ , such that $E \sup_{0 \leq t \leq T} ∣ M_{t} ∣^{p} \leq C_{p, d} E {(\int_{0}^{T} ∥ G_{s} ∥_{F}^{2} d s)}^{p / 2} .$

Facts & Assumptions

Given: AC, finite $d, m$ , a usual filtered Brownian space, a predictable matrix integrand $G$ , and the finite $p / 2$ moment of its quadratic energy.

[F1]

The multidimensional Itô formula applies to each stopped real coordinate $M^{i}$ , driven by the $m$ -dimensional Brownian motion. For $f (x) = ∣ x ∣^{p}$ , $p \geq 2$ , one has $f \in C^{2} (R)$ and $f^{''} (x) = p (p - 1) ∣ x ∣^{p - 2}$ , with the continuous value $f^{''} (0) = 2$ when $p = 2$ and $f^{''} (0) = 0$ when $p > 2$ . Multidimensional Ito formula for Brownian-driven processes

[F2]

For a martingale $M$ , $∣ M ∣$ is a submartingale; for a nonnegative submartingale $Z$ with $Z_{N} \in L^{p}$ , Doob's finite time inequality gives $∥ \max_{k \leq N} Z_{k} ∥_{p} \leq \frac{p}{p - 1} ∥ Z_{N} ∥_{p}$ . Absolute value and powers of a martingale are submartingales Doob Lp maximal inequality

[F3]

Stopping a finite-energy Brownian Itô integral at a stopping time gives the integral of the stopped predictable integrand; a stopping time satisfies ${τ \leq t} \in F_{t}$ . Stopping an Ito integral Continuous-time stopping times and stopped sigma-algebras

[F4]

A continuous Brownian Itô process has an adapted continuous version and its finite-energy stochastic integral is a martingale. Continuous Brownian Ito processes

[F5]

Hölder's inequality with conjugate exponents $p / (p - 2)$ and $p / 2$ (for $p > 2$ ), and its $p = 2$ endpoint, gives $E [S^{p - 2} Q] \leq (E S^{p})^{(p - 2) / p} (E Q^{p / 2})^{2 / p}$ for nonnegative $S, Q$ . Holder's inequality for integrals, including the endpoint cases

[F6]

On nested dyadic grids that include $T$ , continuity makes the grid maxima increase to the path supremum; monotone convergence passes the expectations to that supremum. Monotone convergence for the integral

[F7]

Fatou's lemma passes a uniform bound on nonnegative stopped path suprema to their almost-sure limit. Fatou's lemma

[A1]

AC is inherited from the Brownian and Itô-integral interfaces; no choice is made in this estimate. The Axiom of Choice

Proof

Given: The assumptions in the Statement.

1.1F2F3F4F6given

For each coordinate $i$ , set $M_{t}^{i} = \sum_{j = 1}^{m} \int_{0}^{t} G_{s}^{i j} d W_{s}^{j}$ , $Q_{i} (t) = \int_{0}^{t} \sum_{j} ∣ G_{s}^{i j} ∣^{2} d s$ , and $M_{i}^{*} (t) = \sup_{u \leq t} ∣ M_{u}^{i} ∣$ . Stop at $ρ_{R} = \inf {t : Q_{i} (t) \lor M_{i}^{*} (t) \geq R} \land T$ , for $R > 0$ . These processes are continuous and adapted, and ${ρ_{R} \leq t} = {Q_{i} (t) \lor M_{i}^{*} (t) \geq R}$ for $t < T$ , so $ρ_{R}$ is a stopping time Continuous-time stopping times and stopped sigma-algebras. The stopped process is a continuous square-integrable martingale by [F3]–[F4], its quadratic energy is bounded by $R$ , and its path supremum is bounded by $R$ . Apply [F2] to $∣ M^{i} ∣$ sampled on nested dyadic grids including $T$ . By [F6], the grid maxima increase to the path supremum and monotone convergence passes the discrete bounds to $S_{i} (R) : = E \sup_{t \leq T} ∣ M_{t \land ρ_{R}}^{i} ∣^{p} \leq q^{p} E ∣ M_{T \land ρ_{R}}^{i} ∣^{p}$ , where $q = p / (p - 1)$ .

2.1F1F5step 1.1algebra

Apply [F1] to the stopped coordinate process. The stochastic term has mean zero because its integrand is bounded by the stopping level and has finite energy. Put $U_{i} (R) = \sup_{t \leq T} ∣ M_{t \land ρ_{R}}^{i} ∣$ . The Hessian term is bounded by $\frac{p (p - 1)}{2} U_{i} (R)^{p - 2} Q_{i} (R)$ , where $Q_{i} (R) = \int_{0}^{T \land ρ_{R}} \sum_{j} ∣ G_{s}^{i j} ∣^{2} d s$ . Therefore $S_{i} (R) \leq q^{p} \frac{p (p - 1)}{2} E [U_{i} (R)^{p - 2} Q_{i} (R)]$ . For $p = 2$ , interpret $U_{i} (R)^{0} = 1$ , giving $S_{i} (R) \leq C_{2} E Q_{i} (R)$ . For $p > 2$ , [F5] gives $S_{i} (R) \leq C_{p} S_{i} (R)^{(p - 2) / p} (E Q_{i} (R)^{p / 2})^{2 / p}$ .

3.1step 2.1givenalgebra

For $p = 2$ , step 2.1 gives the coordinate estimate. For $p > 2$ , if $S_{i} (R) = 0$ it is immediate; otherwise divide the bound in step 2.1 by $S_{i} (R)^{(p - 2) / p}$ and raise to the power $p / 2$ . In both cases $S_{i} (R) \leq C_{p} E Q_{i} (R)^{p / 2} \leq C_{p} E (\int_{0}^{T} ∥ G_{s} ∥_{F}^{2} d s)^{p / 2}$ . Since $∣ M_{t} ∣ \leq \sqrt{d} \max_{i} ∣ M_{t}^{i} ∣$ , summing the coordinate estimates gives the required bound for the stopped vector integral with constant $C_{p, d}$ .

4.1A1F3F7step 3.1given∎

As $R \to \infty$ , finite pathwise energy and continuity make $ρ_{R} = T$ eventually almost surely. By [F7], Fatou's lemma passes the uniform bound of step 3.1 to $E \sup_{t \leq T} ∣ M_{t} ∣^{p}$ . The constant depends only on $p, d$ . AC is exactly the inherited assumption [A1].

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Positive Lamperti drift-implicit scheme for the CIR process

Definition

Assume AC and the Brownian conventions of Locally Lipschitz Itô SDE on an open state domain on $[0, T]$ with $U = (0, \infty)$ . The input consists of $κ, θ, σ > 0$ satisfying the Feller condition $2 κ θ \geq σ^{2}$ , a step $h > 0$ , a current transformed value $z_{n} \geq 0$ , and a finite Brownian increment $Δ W_{n}$ . Set $a = \frac{4 κ θ - σ^{2}}{8}, A = 1 + \frac{κ h}{2}, c = z_{n} + \frac{σ}{2} Δ W_{n} .$ For valid parameters, $a > 0$ and $A > 0$ . Compute $z_{n + 1} = \frac{c + \sqrt{c^{2} + 4 A a h}}{2 A}, x_{n + 1} = z_{n + 1}^{2} .$ This is the positive-root solution of $A z_{n + 1}^{2} - c z_{n + 1} - a h = 0,$ which is the drift-implicit Euler equation for the Lamperti variable $Z = \sqrt{X}$ in the CIR model $d X_{t} = κ (θ - X_{t}) d t + σ \sqrt{X_{t}} d W_{t}$ .

Return invalid-input for parameters outside the stated domain or a non-finite input. Otherwise return success with $(z_{n + 1}, x_{n + 1})$ . In exact real arithmetic the discriminant is strictly positive and the displayed root is finite and positive for every $z_{n} \geq 0$ and $Δ W_{n} \in R$ ; the algorithm has no rejected root branch. A floating-point implementation that cannot verify a finite positive root returns solve-failure and does not advance the state. This is a specific implicit square-root update, not the ordinary Euler–Maruyama update for $X$ .

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Tamed Euler scheme for a superlinear drift

Definition

Assume AC and the Brownian conventions of Locally Lipschitz Itô SDE on an open state domain on $[0, T]$ with $U = R^{d}$ . The input is an autonomous drift $b : R^{d} \to R^{d}$ , diffusion $σ : R^{d} \to R^{d \times m}$ , initial state $ξ$ , and integer $N \geq 1$ with $h = T / N \leq 1$ . Define $t_{n} = n h$ and $b_{h} (x) = \frac{b (x)}{1 + h ∣ b (x) ∣} .$ Return the grid values, step process, continuous interpolation, and status as follows.

Set $Y_{0} = ξ$ .
For $n = 0, \dots, N - 1$ , set $Δ W_{n} = W_{t_{n + 1}} - W_{t_{n}}$ and $Y_{n + 1} = Y_{n} + h b_{h} (Y_{n}) + σ (Y_{n}) Δ W_{n} .$
Set ${\overset{ˉ}{Y}}_{t} = Y_{n}$ on $[t_{n}, t_{n + 1})$ , with ${\overset{ˉ}{Y}}_{T} = Y_{N}$ , and set $Y_{t} = Y_{n} + (t - t_{n}) b_{h} (Y_{n}) + σ (Y_{n}) (W_{t} - W_{t_{n}}), t \in [t_{n}, t_{n + 1}] .$ The shared endpoint formulas agree with the grid recursion.

If the dimensions, horizon, step count, or finite-valued coefficient inputs are invalid, return invalid-input without a trajectory. Otherwise return success and the sequence. In exact real arithmetic each update is a finite vector expression, and $∣ h b_{h} (x) ∣ \leq 1$ , so the finite recursion has no implicit solve or branch failure. The algorithm by itself asserts no moment, stability, or convergence property.

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Local strong existence and uniqueness up to exit for locally Lipschitz SDEs

Statement

Assume the data of Locally Lipschitz Itô SDE on an open state domain on $[0, T]$ , with $E ∣ ξ ∣^{2} < \infty$ . Then a local strong solution up to the exit time from $U$ exists on the given filtered space. It is unique up to indistinguishability before its maximal lifetime $ζ = \lim_{n} τ_{n}$ . Equivalently, its stopped solutions on the compact exhaustion $K_{n}$ form a compatible family as defined in that item. This theorem does not assert that $ζ = T$ ; a separate non-explosion argument is required.

Facts & Assumptions

Given: AC; the usual filtered Brownian space, open state domain, locally bounded and locally Lipschitz coefficients, and square-integrable initial state from the definition.

[F1]

A local solution is a compatible family of stopped continuous adapted processes on the compact exhaustion $K_{n}$ ; its stopped equation is the local Itô integral equation in the definition. Locally Lipschitz Itô SDE on an open state domain

[F2]

For globally Lipschitz coefficients of linear growth and an $L^{2}$ -initial state, the SDE has a unique global strong solution on the given filtered space. Global strong well-posedness for Lipschitz Itô SDEs

[F3]

For a nonempty set $A$ in a metric space, $∣ dist (x, A) - dist (y, A) ∣ \leq ∣ x - y ∣$ . $∣ d (x, A) - d (y, A) ∣ \leq d (x, y)$ , so the distance to a fixed nonempty set is $1$ -Lipschitz

[F4]

For a scalar predictable finite-energy integrand $H$ and one Brownian coordinate $W^{j}$ , $E \sup_{s \leq t} ∣ \int_{0}^{s} H_{u} d W_{u}^{j} ∣^{2} \leq 4 E \int_{0}^{t} H_{u}^{2} d u$ , and $E ∣ \int_{0}^{t} H_{u} d W_{u}^{j} ∣^{2} = E \int_{0}^{t} H_{u}^{2} d u$ . Doob maximal bound for the Ito integral Ito isometry and linearity in predictable L2

[F5]

Hölder's integral inequality at conjugate exponents $2, 2$ gives $∣ \int_{0}^{t} f (s) d s ∣^{2} \leq t \int_{0}^{t} ∣ f (s) ∣^{2} d s$ , applying the scalar estimate to each coordinate for vector $f$ . Holder's inequality for integrals, including the endpoint cases

[F6]

Euclidean Cauchy--Schwarz gives $∣ ⟨ x, y ⟩ ∣ \leq ∣ x ∣ ∣ y ∣$ . Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation

[F7]

If a nonnegative continuous $v$ satisfies $v (t) \leq C \int_{0}^{t} v (s) d s$ , then $v (t) = 0$ for all $t \leq T$ . Gronwall's integral inequality with variable and constant coefficients

[A1]

AC is inherited only through the Brownian and Itô-integral interfaces; the truncations, exhaustion and compatibility construction are indexed by $N$ . The Axiom of Choice

Proof

Given: The hypotheses and suppliers in the Statement and Facts.

1.1F1F3construct

For each $n$ , choose scalar piecewise-linear cutoffs $η_{n} (r)$ and $θ_{n} (r)$ with $η_{n} = 1$ on $[0, n]$ , $η_{n} = 0$ on $[n + 1, \infty)$ , $θ_{n} = 0$ on $[0, (2 n)^{- 1}]$ , and $θ_{n} = 1$ on $[n^{- 1}, \infty)$ . Put $χ_{n} (x) = η_{n} (∣ x ∣) θ_{n} (dist (x, U^{c}))$ on $U$ , extend it by zero outside $U$ , and use $θ_{n} \equiv 1$ when $U = R^{d}$ . The norm and distance functions are Lipschitz, so $χ_{n}$ is Lipschitz, equals one on $K_{n}$ , and has compact support inside $U$ .

2.1F1step 1.1given

Define $b_{n} = χ_{n} b$ and $σ_{n} = χ_{n} σ$ on $U$ , extended by zero outside $U$ . On a compact neighborhood of their support the original coefficients are bounded and locally Lipschitz; a finite subcover gives a uniform local Lipschitz constant there. Their products with $χ_{n}$ are therefore locally Lipschitz, including at the support boundary where the cutoff vanishes. A compactly supported locally Lipschitz map on $R^{d}$ is globally Lipschitz: use a finite cover and a Lebesgue number for nearby pairs, and boundedness divided by that Lebesgue number for pairs farther apart. Thus $b_{n}, σ_{n}$ are globally Lipschitz and have linear growth.

3.1F1F2step 2.1

By [F2], each truncated equation with coefficients $b_{n}, σ_{n}$ and initial value $ξ$ has a unique global strong solution $Z^{n}$ . If $K_{n} = \emptyset$ , take its exit time to be zero; otherwise stop at the first exit $τ_{n}$ from $int K_{n}$ . For $k \geq n$ , stop $Z^{n}, Z^{k}$ at the first time either leaves $int K_{n}$ . Before that stop both states lie in $K_{n}$ , where both truncations equal $b, σ$ .

4.1F1F2F4F5F6F7step 3.1algebra

Let $D = Z^{n} - Z^{k}$ up to the common stop $τ$ , and put $v (t) = E \sup_{s \leq t} ∣ D_{s \land τ} ∣^{2}$ , finite by [F2]. Write a finite-energy matrix integral coordinatewise as $M_{s}^{i} = \sum_{j = 1}^{m} \int_{0}^{s} G_{u}^{i j} d W_{u}^{j}$ . Since $∣ \sum_{j} a_{j} ∣^{2} \leq m \sum_{j} ∣ a_{j} ∣^{2}$ by [F6], summing the scalar bounds [F4] gives $E \sup_{s \leq t} ∣ M_{s} ∣^{2} \leq 4 m E \int_{0}^{t} ∥ G_{u} ∥_{F}^{2} d u$ . Apply this to $G_{u} = 1_{{u \leq τ}} (σ (u, Z_{u}^{n}) - σ (u, Z_{u}^{k}))$ ; boundedness on $K_{n}$ gives finite energy. The common coefficient Lipschitz constant $L_{n}$ on $K_{n}$ , [F5], and $D_{0} = 0$ now give $v (t) \leq (2 T L_{n}^{2} + 8 m L_{n}^{2}) \int_{0}^{t} v (s) d s .$ With $C_{n} = 2 T L_{n}^{2} + 8 m L_{n}^{2}$ , the continuous majorant $V (t) = C_{n} \int_{0}^{t} v (s) d s$ satisfies $v \leq V$ and $V (t) \leq C_{n} \int_{0}^{t} V (s) d s$ , so [F7] gives $V = v = 0$ . Hence the truncated solutions agree until either exits $K_{n}$ ; continuity makes those exit times equal, and $Z^{k}$ stopped at $τ_{n}$ equals $Z^{n}$ stopped there.

5.1A1F1step 4.1∎

The stopped family $X^{(n)} : = Z_{\cdot \land τ_{n}}^{n}$ is compatible by step 4.1 and satisfies the stopped integral equation because each cutoff equals the original coefficients on $K_{n}$ . The exit times increase, since the compact sets are nested and the paths agree up to each earlier exit. Since $⋃_{n} int K_{n} = U$ , its lifetime $ζ = \lim_{n} τ_{n}$ and common values before $ζ$ give a local strong solution in the sense of [F1]. For any two local solutions, their $K_{n}$ -stopped path suprema are bounded by $∣ ξ ∣ + n$ and hence are square-integrable; the same difference estimate therefore proves equality up to the minimum of their lifetimes. AC is exactly [A1].

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CIR Lamperti equation and unique positive implicit root

Statement

Let $κ, θ, σ > 0$ satisfy the Feller condition $2 κ θ \geq σ^{2}$ , and put $a = \frac{4 κ θ - σ^{2}}{8} .$ For a positive local strong solution of the CIR SDE $d X_{t} = κ (θ - X_{t}) d t + σ \sqrt{X_{t}} d W_{t}$ on $U = (0, \infty)$ , the Lamperti variable $Z_{t} = \sqrt{X_{t}}$ satisfies $d Z_{t} = (\frac{a}{Z_{t}} - \frac{κ}{2} Z_{t}) d t + \frac{σ}{2} d W_{t}$ up to every compact exit time. Moreover, for every $h > 0$ , $z \geq 0$ , and $Δ W \in R$ , the drift-implicit step $z_{+} = z + h (\frac{a}{z_{+}} - \frac{κ}{2} z_{+}) + \frac{σ}{2} Δ W$ has exactly one positive root. It is the value defined by Positive Lamperti drift-implicit scheme for the CIR process. The boundary input $z = 0$ is included for the numerical root statement; the CIR solution theorem below uses $X_{0} > 0$ .

Facts & Assumptions

Given: AC; positive parameters satisfying the Feller condition; a positive local CIR solution; and a positive step $h$ .

[F1]

For a continuous Brownian Itô process $X$ and $f \in C^{1, 2}$ , Itô's formula has drift $f^{'} b + \frac{1}{2} f^{''} σ^{2}$ and diffusion $f^{'} σ$ . Multidimensional Ito formula for Brownian-driven processes

[F2]

The numerical root procedure sets $A = 1 + κ h / 2$ , $c = z + σ Δ W / 2$ , and returns $(c + \sqrt{c^{2} + 4 A a h}) / (2 A)$ . Positive Lamperti drift-implicit scheme for the CIR process

[F3]

Every nonnegative real number has a unique nonnegative square root, and squaring is strictly increasing on nonnegative reals. Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$

[F4]

On $(0, \infty)$ , $(x^{1 / 2})^{'} = \frac{1}{2} x^{- 1 / 2}$ and $(x^{1 / 2})^{''} = - \frac{1}{4} x^{- 3 / 2}$ . Continuity and derivatives of positive-base real powers

[F5]

For $0 < ε < R$ , a $C^{2}$ function $g$ on $(0, \infty)$ has a global $C^{2}$ extension agreeing with it on $[ε, R]$ : choose a smooth cutoff $χ$ with support in $[ε / 2, 2 R] \subset (0, \infty)$ and equal to $1$ on $[ε, R]$ , and set $\tilde{g} = χ g$ on $(0, \infty)$ and $0$ on $(- \infty, 0]$ . Such a cutoff is given by $χ (x) = η ((x - ε / 2) / (ε / 2)) η ((2 R - x) / R)$ , where $ρ (u) = e^{- 1 / u}$ for $u > 0$ , $ρ (u) = 0$ for $u \leq 0$ , and $η (u) = ρ (u) / (ρ (u) + ρ (1 - u))$ . The stopped state remains in $[ε, R]$ , so the global Itô formula for $\tilde{g}$ uses exactly $g, g^{'}, g^{''}$ there.

[A1]

AC is inherited from the local Brownian and Itô interfaces; no additional choice is used. The Axiom of Choice

Proof

Given: The hypotheses and facts in the Statement and Facts.

1.1F1F4F5givenalgebra

For each $n$ let $D_{n} = {1 / n < X_{0} < n} \in F_{0}$ , and let $τ_{n}$ be the first exit from $[1 / n, n]$ . The events $D_{n}$ increase to a probability-one event because $X_{0} > 0$ is finite. Use [F5] to extend $f (x) = \sqrt{x}$ from that interval to a global $C^{2}$ function equal to zero off its positive support, then apply [F1] to the stopped process $X^{τ_{n}}$ . On $D_{n}$ its state stays in $[1 / n, n]$ , so the extended function and its derivatives agree with $f$ , $f^{'} (x) = 1 / (2 \sqrt{x})$ and $f^{''} (x) = - 1 / (4 x^{3 / 2})$ by [F4]. The resulting drift is $\frac{κ (θ - X_{t})}{2 \sqrt{X_{t}}} - \frac{σ^{2} X_{t}}{8 X_{t}^{3 / 2}} = \frac{4 κ θ - σ^{2}}{8 Z_{t}} - \frac{κ}{2} Z_{t} = \frac{a}{Z_{t}} - \frac{κ}{2} Z_{t}$ and the diffusion is $σ / 2$ . The countable compact exhaustion proves the identity up to every compact exit and hence on the local solution's lifetime.

1.2givenalgebra

The Feller inequality implies $4 κ θ - σ^{2} \geq σ^{2} > 0$ , so $a > 0$ . Thus for the numerical step $A = 1 + κ h / 2 > 0$ and $4 A a h > 0$ .

2.1F2F3step 1.2algebra

Multiplying the drift-implicit equation by $z_{+}$ and collecting terms gives $A z_{+}^{2} - c z_{+} - a h = 0$ , with $c = z + σ Δ W / 2$ . Its discriminant is $c^{2} + 4 A a h > c^{2}$ , so [F3] gives a square root strictly larger than $∣ c ∣$ . The quadratic formula gives the positive root $z_{+} = (c + \sqrt{c^{2} + 4 A a h}) / (2 A)$ ; the other root has negative numerator $c - \sqrt{c^{2} + 4 A a h} < 0$ and positive denominator, hence is negative.

3.1A1step 2.1givenalgebra∎

Every positive solution must be one of those two quadratic roots, and only the first is positive. Substitution verifies the displayed positive root in the original equation, so it is the unique admissible step, including when $z = 0$ . AC is exactly [A1].

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Uniform finite-horizon moments for tamed Euler approximations

Statement

Assume AC. Let $d, m \geq 1$ be finite, $T > 0$ , and let $b, σ, ξ, W$ satisfy the coefficient, Brownian, and initial-moment hypotheses of Global existence and finite-horizon moments for one-sided Lipschitz SDEs. For each integer $N \geq \max (1, ⌈ T ⌉)$ , put $h = T / N$ and define $Y_{0}, \dots, Y_{N}$ by Tamed Euler scheme for a superlinear drift. Then for every finite $p \geq 1$ , $\sup_{N \geq \max (1, ⌈ T ⌉)} \max_{0 \leq n \leq N} E ∣ Y_{n} ∣^{p} < \infty .$ The bound may depend on $p, T$ , the coefficient constants, and the initial moments, but not on $N$ or $n$ .

Facts & Assumptions

Given: AC; the autonomous coefficient and initial-moment hypotheses in the Statement; and the deterministic grid $t_{n} = n h$ .

[F1]

The tamed update is $Y_{n + 1} = Y_{n} + h b (Y_{n}) / (1 + h ∣ b (Y_{n}) ∣) + σ (Y_{n}) Δ W_{n}$ ; in particular $∣ h b (Y_{n}) / (1 + h ∣ b (Y_{n}) ∣) ∣ \leq 1$ . Tamed Euler scheme for a superlinear drift

[F2]

The sampled filtration $(F_{t_{n}})_{n = 0}^{N}$ is increasing. Filtration and filtered probability space

[F3]

Each increment $Δ W_{n}$ is independent of $F_{t_{n}}$ and has law $N_{m} (0, h I_{m})$ . $d$ -dimensional Brownian motion

[F4]

For a nonnegative submartingale $Z$ and $r > 1$ , $∥ \max_{k \leq N} Z_{k} ∥_{r} \leq r / (r - 1) ∥ Z_{N} ∥_{r}$ . Doob Lp maximal inequality

[F5]

The coordinates of $Δ W_{n}$ are independent $N (0, h)$ variables by Standard normal and normal laws. Their density gives, for $2 a h < 1$ , $E e^{a ∣ Δ W_{n} ∣^{2}} = (1 - 2 a h)^{- m / 2}$ : in one coordinate, integrate $(2 π h)^{- 1 / 2} e^{- (1 / (2 h) - a) x^{2}}$ and rescale to the normalized Gaussian density. The same density and $x = \sqrt{h} u$ show $E ∣ Δ W_{n} ∣^{r} = h^{r / 2} E ∣ G ∣^{r} < \infty$ for every finite $r$ , where $G \sim N_{m} (0, I_{m})$ .

[F6]

For predictable $G$ and $p \geq 2$ , $∥ \sup_{t \leq T} ∣ \int_{0}^{t} G_{s} d W_{s} ∣ ∥_{p} \leq C_{p, d} ∥ (\int_{0}^{T} ∥ G_{s} ∥_{F}^{2} d s)^{1 / 2} ∥_{p}$ . Lp maximal moment bound for Brownian Itô integrals

[F7]

Hölder's inequality yields the $L^{r}$ triangle inequality for $r \geq 1$ , the three-factor bound $E ∣ U V W ∣ \leq (E ∣ U ∣^{3} E ∣ V ∣^{3} E ∣ W ∣^{3})^{1 / 3}$ , and $E [1_{A} ∣ Z ∣^{p}] \leq P (A)^{1 / 2} (E ∣ Z ∣^{2 p})^{1 / 2}$ . Holder's inequality for integrals, including the endpoint cases Cauchy-Schwarz for random variables

[F8]

For a nonnegative random variable $Z$ and $a > 0$ , $P (Z \geq a) \leq E Z / a$ . Markov's inequality for random variables

[F9]

For a finite family of events, the probability of their union is at most the sum of their probabilities. The finite union bound

[A1]

AC is inherited through the Brownian, normal, filtration, and Itô interfaces; the estimates below use only countable grid constructions. The Axiom of Choice

Proof

Given: The hypotheses and suppliers in the Statement and Facts.

1.1F1givenalgebra

The derivative bound, integrated on the segment from $0$ to $x$ , gives $∣ b (x) ∣ \leq B (1 + ∣ x ∣^{q + 1})$ for a finite $B$ . One-sided Lipschitzness with $y = 0$ gives $⟨ x, b (x) ⟩ \leq B (1 + ∣ x ∣^{2})$ , and global Lipschitzness of $σ$ gives $∥ σ (x) ∥_{F} \leq B (1 + ∣ x ∣)$ . Increase $B$ so these bounds hold with one constant. For $∣ x ∣ \geq 1$ they imply $∥ σ (x) ∥_{F}^{2} \leq B_{1} ∣ x ∣^{2}$ , $⟨ x, b (x) ⟩ \leq B_{1} ∣ x ∣^{2}$ , and $∣ b (x) ∣^{2} \leq B_{1} ∣ x ∣^{2 q + 2}$ . Thus, whenever $1 \leq ∣ x ∣ \leq N^{1 / (2 q)}$ and $h = T / N$ , $h^{2} ∣ b (x) ∣^{2} \leq B_{1} T h ∣ x ∣^{2}$ . Fix once and for all $Λ \geq \max {1, 2 B_{1} (1 + T), 1 + \sup_{∣ x ∣ \leq 1} (∣ b (x) ∣ + ∥ σ (x) ∥_{F})}$ ; continuity and compactness make this choice finite.

1.2F2F3F4F5F7given

The recursion and Brownian adaptedness show by induction that $Y_{k}$ is $F_{t_{k}}$ -measurable for every $k$ . For $0 \leq k < N$ , define $α_{k} = 1_{{∣ Y_{k} ∣ \geq 1}} \frac{⟨ Y_{k}, σ (Y_{k}) Δ W_{k} ⟩}{∣ Y_{k} ∣^{2}}, S_{0} = 0, S_{j} = \sum_{k < j} α_{k} (1 \leq j \leq N) .$ Since $α_{k}$ is $F_{t_{k + 1}}$ -measurable, $(S_{j})$ is adapted to the sampled filtration. For $0 \leq n \leq N$ , also define $D_{n} = (Λ + ∣ ξ ∣) \exp ⁣ (Λ T + \max_{0 \leq u \leq n} \sum_{k = u}^{n - 1} (Λ h + Λ ∣ Δ W_{k} ∣^{2} + α_{k})) .$ On $∣ Y_{k} ∣ \geq 1$ , write $α_{k} = v_{k} \cdot Δ W_{k}$ with $v_{k} = σ (Y_{k})^{T} Y_{k} / ∣ Y_{k} ∣^{2}$ ; global Lipschitzness of $σ$ gives $∣ v_{k} ∣ \leq ∥ σ (Y_{k}) ∥_{F} / ∣ Y_{k} ∣ \leq C$ . Conditionally on $F_{t_{k}}$ its variance is $h ∣ v_{k} ∣^{2} \leq C^{2} h$ ; on $∣ Y_{k} ∣ < 1$ , $α_{k} = 0$ . Brownian independence and the Gaussian density therefore give $E [α_{k} ∣ F_{t_{k}}] = 0$ and, by completing the square in each coordinate, $E [e^{r α_{k}} ∣ F_{t_{k}}] \leq e^{C^{2} r^{2} h / 2} (r \in R) .$ Conditional Jensen shows that $e^{Q S_{j}}$ and $e^{- Q S_{j}}$ are nonnegative submartingales for each $Q > 0$ . Iterating the preceding bound with $r = 2 Q$ gives terminal second moments at most $e^{2 C^{2} Q^{2} T}$ for both processes. Doob's $L^{2}$ inequality [F4] then gives $\sup_{N} E e^{Q \max_{j \leq N} ∣ S_{j} ∣} < \infty$ : the exponential of the maximum is bounded by the sum of the maxima of $e^{Q S_{j}}$ and $e^{- Q S_{j}}$ , whose expectations are bounded by their $L^{2}$ maxima. For any $r > 0$ , choose $Q \geq \max (1, r)$ ; since $x^{r / Q} \leq 1 + x$ for $x \geq 0$ , this also bounds $\sup_{N} E e^{r \max_{j \leq N} ∣ S_{j} ∣}$ . Finally, for $N \geq 4 r Λ T$ , [F5] and $- \log (1 - u) \leq 2 u$ on $0 \leq u \leq 1 / 2$ give $E \exp ⁣ (r Λ \sum_{k < N} ∣ Δ W_{k} ∣^{2}) = \prod_{k < N} (1 - 2 r Λ h)^{- m / 2} \leq e^{2 r m Λ T} .$ Every suffix sum $\sum_{k = u}^{j - 1} α_{k} = S_{j} - S_{u}$ is bounded in absolute value by $2 \max_{i \leq N} ∣ S_{i} ∣$ . Since $\max_{n \leq N} D_{n}^{p} \leq (Λ + ∣ ξ ∣)^{p} e^{2 p Λ T} \exp ⁣ (p Λ \sum_{k < N} ∣ Δ W_{k} ∣^{2} + 2 p \max_{j \leq N} ∣ S_{j} ∣),$ the three-factor Hölder bound [F7], the all-moment assumption on $ξ$ , the square-exponential estimate with $r = 3 p$ , and the exponential-martingale bound with $r = 6 p$ give a uniform moment for $\max_{n \leq N} D_{n}^{p}$ when $N \geq N_{0} (p) : = \max {1, ⌈ T ⌉, ⌈ 12 p Λ T ⌉}$ . The finitely many smaller $N$ will be handled separately.

1.3F1F3F6F7givenalgebra

Set $a_{n} = (E ∣ Y_{n} ∣^{p})^{1 / p}$ for $p \geq 2$ . Decompose the recursion as $Y_{n} = ξ + \sum_{k < n} h b_{h} (Y_{k}) + σ (0) W_{t_{n}} + \sum_{k < n} (σ (Y_{k}) - σ (0)) Δ W_{k} .$ The drift sum has norm at most $N$ by [F1]. The last sum is a continuous Brownian Itô integral with a predictable step integrand. By [F6], global Lipschitzness of $σ$ , and the $L^{p / 2}$ triangle inequality from [F7], its $L^{p}$ norm is at most $C_{p} {(\sum_{k < n} h ∥ σ (Y_{k}) - σ (0) ∥_{L^{p}}^{2})}^{1 / 2} \leq C_{p} {(h \sum_{k < n} a_{k}^{2})}^{1 / 2} .$ The pointwise inequality $∣ u_{1} + \dots + u_{4} ∣^{p} \leq 4^{p - 1} \sum_{i} ∣ u_{i} ∣^{p}$ and the bounded $p$ th moments of $W_{t}$ on $[0, T]$ therefore give $a_{n}^{2} \leq C_{p} (1 + ∥ ξ ∥_{L^{p}}^{2} + N^{2}) + C_{p} h \sum_{k < n} a_{k}^{2} .$ Iterating this finite recurrence yields $\max_{n \leq N} a_{n}^{2} \leq C_{p} (1 + ∥ ξ ∥_{L^{p}}^{2} + N^{2}) e^{C_{p} T}$ , hence $\max_{n \leq N} ∥ Y_{n} ∥_{L^{p}} \leq C_{p} (1 + N)$ .

2.1F1step 1.1step 1.2givenalgebraconstruct

Write $Δ W_{k} = W_{t_{k + 1}} - W_{t_{k}}$ and use $α_{k}, D_{n}$ from step 1.2. Define $Ω_{n} = {\max_{k < n} D_{k} \leq N^{1 / (2 q)}, \max_{k < n} ∣ Δ W_{k} ∣ \leq 1} .$ On $∣ Y_{k} ∣ \leq 1$ and $∣ Δ W_{k} ∣ \leq 1$ , [F1] gives $∣ Y_{k + 1} ∣ \leq 1 + h ∣ b (Y_{k}) ∣ + ∥ σ (Y_{k}) ∥_{F} \leq Λ$ . On $1 \leq ∣ Y_{k} ∣ \leq N^{1 / (2 q)}$ , expand the squared update. The drift cross term is at most $2 B_{1} h ∣ Y_{k} ∣^{2}$ ; the squared taming increment is at most $B_{1} T h ∣ Y_{k} ∣^{2}$ by step 1.1; the diffusion square is at most $B_{1} ∣ Y_{k} ∣^{2} ∣ Δ W_{k} ∣^{2}$ ; and the drift-noise cross term is at most the sum of those two square terms. The remaining cross term is exactly $2 ∣ Y_{k} ∣^{2} α_{k}$ . With the fixed $Λ$ chosen in step 1.1 large enough for these coefficients, $1 + u \leq e^{u}$ therefore gives $∣ Y_{k + 1} ∣ \leq ∣ Y_{k} ∣ \exp (Λ h + Λ ∣ Δ W_{k} ∣^{2} + α_{k}) .$ Induction on $n$ , starting either from $Y_{0} = ξ$ or from the last index with $∣ Y_{k} ∣ \leq 1$ , now gives $1_{Ω_{n}} ∣ Y_{n} ∣ \leq D_{n}$ : the first step after such an index is at most $Λ$ , and the later factors are bounded by the suffix sum in $D_{n}$ .

2.2F3F5F8F9step 1.2givenalgebra

Markov's inequality [F8] and step 1.2 imply, for any $r > 0$ , $P ⁣ (\max_{n \leq N} D_{n} > N^{1 / (2 q)}) \leq N^{- r / (2 q)} E \max_{n \leq N} D_{n}^{r} .$ Also, by the finite union bound [F9] and the $r$ th Gaussian moment, $P (\max_{k < N} ∣ Δ W_{k} ∣ > 1) \leq C_{r} N h^{r / 2} = C_{r, T} N^{1 - r / 2}$ . Choosing $r$ arbitrarily large in these bounds shows that for every $s > 0$ there is $C_{s}$ with $P (Ω_{N}^{c}) \leq C_{s} N^{- s}$ for all $N$ .

3.1A1F1F7step 2.1step 1.2step 2.2step 1.3algebra∎

For $p \geq 1$ , on $Ω_{N}$ step 2.1 bounds $E [1_{Ω_{N}} ∣ Y_{n} ∣^{p}] \leq E D_{n}^{p} \leq C_{p}$ ; here $Ω_{N} \subseteq Ω_{n}$ . On the complement, [F7] and step 1.3 give $E [1_{Ω_{N}^{c}} ∣ Y_{n} ∣^{p}] \leq P (Ω_{N}^{c})^{1 / 2} ∥ Y_{n} ∥_{L^{2 p}}^{p} \leq C_{s} N^{- s / 2} (1 + N)^{p} .$ Choose $s > 2 p$ . This is uniformly bounded. For the finitely many small $N$ excluded in step 1.2, the finite recursion and linear growth of $σ$ give finite moments by induction; taking their finite maximum completes the uniform bound. For $1 \leq p < 2$ , apply Hölder on the probability space to the $p = 2$ result. AC is exactly [A1].

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Global existence and finite-horizon moments for one-sided Lipschitz SDEs

Statement

Assume AC. Let $d, m \geq 1$ be finite, $T > 0$ , and let $W$ and $ξ$ satisfy the filtered Brownian hypotheses in Locally Lipschitz Itô SDE on an open state domain with $U = R^{d}$ . Consider the autonomous SDE $d X_{t} = b (X_{t}) d t + σ (X_{t}) d W_{t}, X_{0} = ξ,$ where $b \in C^{1} (R^{d}; R^{d})$ , $σ : R^{d} \to R^{d \times m}$ is globally Lipschitz, and for constants $L, C < \infty$ and an integer $q \geq 1$ , $⟨ x - y, b (x) - b (y) ⟩ \leq L ∣ x - y ∣^{2}, ∥ D b (x) ∥ \leq C (1 + ∣ x ∣^{q}) .$ Suppose $E ∣ ξ ∣^{p} < \infty$ for every finite $p \geq 1$ . Then the maximal local strong solution is global on $[0, T]$ , is unique up to indistinguishability, and for every finite $p \geq 1$ , $E \sup_{0 \leq t \leq T} ∣ X_{t} ∣^{p} < \infty .$ The bound may depend on $p, T$ , the coefficient constants, and the initial moments. No global Lipschitz condition on the drift is assumed.

Facts & Assumptions

Given: AC; the locally Lipschitz SDE data, one-sided Lipschitz drift, polynomial derivative bound, globally Lipschitz diffusion, and all finite initial moments in the Statement.

[F1]

The stopped local strong solution exists uniquely up to exit from compact subsets of the open state domain; if its lifetime is finite, the exhaustion $K_{n}$ records exit from every compact subset. Local strong existence and uniqueness up to exit for locally Lipschitz SDEs

[F2]

For $V \in C^{1, 2}$ and a continuous Brownian Itô process, Itô's formula has drift $\partial_{t} V + \nabla V \cdot b + \frac{1}{2} (σ σ^{T}) : D^{2} V$ . Multidimensional Ito formula for Brownian-driven processes

[F3]

For a predictable matrix integrand $G$ and $p \geq 2$ , $E \sup_{t \leq T} ∣ \int_{0}^{t} G_{s} d W_{s} ∣^{p} \leq C_{p, d} E (\int_{0}^{T} ∥ G_{s} ∥_{F}^{2} d s)^{p / 2}$ . Lp maximal moment bound for Brownian Itô integrals

[F4]

A finite-energy Itô integral has mean zero and its second moment equals the expected coefficient energy. Ito isometry and linearity in predictable L2

[F5]

[F6]

Hölder's integral inequality gives ${(\int_{0}^{T} f (s) d s)}^{p} \leq T^{p - 1} \int_{0}^{T} f (s)^{p} d s$ for nonnegative measurable $f$ and $p \geq 1$ . Holder's inequality for integrals, including the endpoint cases

[F7]

Fatou's lemma passes uniform stopped nonnegative moment bounds to an almost sure limit. Fatou's lemma

[F8]

If $u (t) \leq A + \int_{0}^{t} B u (s) d s$ with nonnegative continuous $u$ and constants $A, B \geq 0$ , then $u (t) \leq A e^{B t}$ . Gronwall's integral inequality with variable and constant coefficients

[F9]

Dominated convergence gives continuity of $u_{R} (t) = E V_{p} (X_{t \land τ_{R}})$ : the stopped paths are continuous and $V_{p} (X_{t \land τ_{R}}) \leq V_{p} (ξ) + (1 + R^{2})^{p / 2}$ . Dominated convergence

[F10]

Tonelli's theorem interchanges expectation and time integration for the nonnegative measurable function $V_{p} (X_{s \land τ_{R}})$ . Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[A1]

AC is inherited through the local Brownian and Itô interfaces; no choice is used to stop, localize, or pass to limits. The Axiom of Choice

Proof

Given: The assumptions and suppliers in the Statement and Facts.

1.1F2F5givenalgebra

For $p \geq 2$ define $V_{p} (x) = (1 + ∣ x ∣^{2})^{p / 2}$ . Direct differentiation gives $\nabla V_{p} = p (1 + ∣ x ∣^{2})^{p / 2 - 1} x$ and $D^{2} V_{p} = p (1 + ∣ x ∣^{2})^{p / 2 - 1} I + p (p - 2) (1 + ∣ x ∣^{2})^{p / 2 - 2} x x^{T} .$ Thus $∥ D^{2} V_{p} (x) ∥ \leq C_{p} (1 + ∣ x ∣^{2})^{p / 2 - 1}$ . The one-sided Lipschitz condition with $y = 0$ and [F5] imply $⟨ x, b (x) ⟩ \leq L ∣ x ∣^{2} + ∣ b (0) ∣ ∣ x ∣ \leq C_{0} (1 + ∣ x ∣^{2})$ . Global Lipschitz continuity gives $∥ σ (x) ∥_{F}^{2} \leq C_{1} (1 + ∣ x ∣^{2})$ . Substitution into the generator formula [F2] yields $\nabla V_{p} (x) \cdot b (x) + \frac{1}{2} (σ σ^{T}) : D^{2} V_{p} (x) \leq C_{p} V_{p} (x) .$

2.1F2F4F8F9F10step 1.1given

For integer $R \geq 1$ , stop the maximal solution at its first exit $τ_{R}$ from ${∣ x ∣ < R}$ , or at its lifetime if earlier; for $U = R^{d}$ these are the compact-exhaustion stops. If $∣ ξ ∣ \geq R$ , then $τ_{R} = 0$ and the stopped integrands vanish. Otherwise all active states lie in ${\overline{B}}_{R}$ , so the coefficients, derivatives of $V_{p}$ , and Itô integrand are bounded and the stochastic integral in [F2] has mean zero by the Itô isometry. The integrable bound in [F9] makes $u_{R} (t) = E V_{p} (X_{t \land τ_{R}})$ continuous, as required by [F8]. Taking expectations in Itô's formula, using step 1.1 and Tonelli [F10], gives $u_{R} (t) \leq E V_{p} (ξ) + C_{p} \int_{0}^{t} u_{R} (s) d s .$ Applying [F8] gives $\sup_{R} \sup_{t \leq T} E V_{p} (X_{t \land τ_{R}}) \leq E V_{p} (ξ) e^{C_{p} T} < \infty$ .

3.1F1F7step 2.1given

Set $E_{R} = {\sup_{t \leq T} ∣ X_{t \land τ_{R}} ∣ \geq R}$ . On $E_{R}$ , either $∣ ξ ∣ \geq R$ and $τ_{R} = 0$ , or the continuous path reaches radius $R$ and is stopped there; hence $∣ X_{T \land τ_{R}} ∣ \geq R$ . Step 2.1 therefore gives $R^{p} P (E_{R}) \leq E V_{p} (X_{T \land τ_{R}}) \leq E V_{p} (ξ) e^{C_{p} T} .$ If a maximal path fails to extend as a solution through $T$ , it must reach every radius by $T$ : otherwise it agrees with a globally defined truncation whose cutoff is one on a larger ball containing that path, giving an extension by local uniqueness. Thus the failure event is contained in every $E_{R}$ and has probability zero as $R \to \infty$ . The solution extends globally through $T$ , and Fatou's lemma [F7] applied at each time to the stopped bounds gives $\sup_{t \leq T} E V_{p} (X_{t}) < \infty$ .

4.1A1F1F3F6F7step 3.1givenalgebra∎

The derivative bound and the fundamental theorem of calculus on the segment from $0$ to $x$ give $∣ b (x) ∣ \leq C_{2} (1 + ∣ x ∣^{q + 1})$ ; global Lipschitz continuity gives $∥ σ (x) ∥_{F} \leq C_{3} (1 + ∣ x ∣)$ . For each $p \geq 2$ , applying step 3.1 at moments $p (q + 1)$ and $p$ therefore implies $E {(\int_{0}^{T} ∣ b (X_{s}) ∣ d s)}^{p} < \infty, E {(\int_{0}^{T} ∥ σ (X_{s}) ∥_{F}^{2} d s)}^{p / 2} < \infty,$ by Hölder's integral inequality [F6]. The integral equation gives $\sup_{t \leq T} ∣ X_{t} ∣^{p} \leq 3^{p - 1} (∣ ξ ∣^{p} + {(\int_{0}^{T} ∣ b (X_{s}) ∣ d s)}^{p} + \sup_{t \leq T} {∣ \int_{0}^{t} σ (X_{s}) d W_{s} ∣}^{p}) .$ The first two expectations are finite by the hypothesis and the preceding bound; [F3] bounds the last by the diffusion-energy moment already shown finite. For $1 \leq p < 2$ , Hölder's inequality [F6] on the probability space gives $E \sup_{t \leq T} ∣ X_{t} ∣^{p} \leq (E \sup_{t \leq T} ∣ X_{t} ∣^{2})^{p / 2}$ . Uniqueness is the local uniqueness in [F1], and AC is exactly [A1].

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Drift-implicit Euler–Maruyama with explicit solve-failure status

Definition

Assume AC and the Brownian conventions of Locally Lipschitz Itô SDE on an open state domain on $[0, T]$ with $U = R^{d}$ . An admissible mathematical input has finite dimensions $d, m \geq 1$ , $T > 0$ , $N \geq 1$ with $h = T / N \leq 1$ , an autonomous continuous drift $b : R^{d} \to R^{d}$ , a finite-valued Borel diffusion $σ : R^{d} \to R^{d \times m}$ , and a finite $F_{0}$ -measurable initial state $ξ$ . It also has a constant $μ \geq 0$ such that $⟨ x - y, b (x) - b (y) ⟩ \leq - μ ∣ x - y ∣^{2} (x, y \in R^{d}) .$ This global condition is a precondition on the mathematical input, not a runtime test over all $x, y$ . For $h > 0$ , set $H_{h} (x) = x - h b (x)$ and let $R_{h} = H_{h}^{- 1}$ , which is defined uniquely on all of $R^{d}$ by Backward Euler has a globally defined nonexpansive resolvent (contractive when strongly dissipative).

First check that $d, m$ are positive integers, $T$ is finite and positive, $N$ is a positive integer, $0 < h = T / N \leq 1$ , and $ξ$ is finite. If any of these checks fails, return invalid-input. An exact execution then sets $Y_{0} = ξ$ and, for $n = 0, \dots, N - 1$ , computes $u_{n} = Y_{n} + σ (Y_{n}) (W_{t_{n + 1}} - W_{t_{n}}), Y_{n + 1} = R_{h} (u_{n}) .$ It returns the grid values, the left-endpoint step process ${\overset{ˉ}{Y}}_{t} = Y_{n} (t_{n} \leq t < t_{n + 1}), {\overset{ˉ}{Y}}_{T} = Y_{N},$ and the continuous interpolation $Y_{t} = Y_{n} + (t - t_{n}) b (Y_{n + 1}) + σ (Y_{n}) (W_{t} - W_{t_{n}}), t_{n} \leq t \leq t_{n + 1} .$ This is a pathwise continuous extension of the grid values. At each shared endpoint the interpolation agrees with the grid update.

A finite implementation may instead receive a root-solver interface $S$ and an integer work cap $M \geq 0$ . Each call $S (H_{h}, u_{n}, M)$ must terminate after at most $M$ declared solver iterations and return either certified-root(z, proof) or no-certificate(last-iterate, residual) (with last-iterate = null if the cap permits no iterate). The supplied exact verifier must terminate and accepts a certificate precisely when it proves that $z \in R^{d}$ is finite and $H_{h} (z) = u_{n}$ . At the first call returning no-certificate, or at the first certificate rejected by the verifier, stop and return solve-failure with zero-based failed index $n$ , the valid grid prefix through $Y_{n}$ , the step process on $[0, t_{n}]$ , the interpolation on $[0, t_{n}]$ , and the solver diagnostic. For no-certificate, retain its last iterate and residual; for a rejected certificate with candidate $z$ , report $z$ and residual $H_{h} (z) - u_{n}$ . If the diagnostic has no finite candidate, its iterate and residual are null. Do not assign $Y_{n + 1}$ or continue. A verified root equals $R_{h} (u_{n})$ by uniqueness. If every call is verified, return success and the exact grid, step process, and interpolation above. The global dissipativity condition remains an admissibility precondition and is not tested by these finite-value checks. Thus a failed finite solve records failure of certification, not absence of a mathematical root. The algorithm makes no convergence or mean-square stability claim.

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CIR paths stay positive and do not explode under the Feller condition

Statement

Assume AC and the Brownian conventions of Locally Lipschitz Itô SDE on an open state domain. Let $κ, θ, σ > 0$ , let $x_{0} > 0$ be deterministic, and suppose $2 κ θ \geq σ^{2}$ . Then the CIR SDE $d X_{t} = κ (θ - X_{t}) d t + σ \sqrt{X_{t}} d W_{t}, X_{0} = x_{0},$ has a unique global strong solution on every finite horizon. Its paths remain strictly positive and finite, and for every finite $p \geq 1$ , $\sup_{0 \leq t \leq T} E X_{t}^{p} < \infty .$ No negative-moment bound or strong numerical convergence rate is asserted.

Facts & Assumptions

Given: AC; the local SDE definition on $U = (0, \infty)$ ; positive $κ, θ, σ, x_{0}$ with $2 κ θ \geq σ^{2}$ ; and the Lamperti equation supplier.

[F1]

For $Z = \sqrt{X}$ , the local Lamperti equation is $d Z = (a / Z - κ Z / 2) d t + σ d W / 2$ , where $a = (4 κ θ - σ^{2}) / 8 > 0$ . CIR Lamperti equation and unique positive implicit root

[F2]

Locally Lipschitz coefficients on an open domain have a unique local strong solution up to exit from compact subsets. Local strong existence and uniqueness up to exit for locally Lipschitz SDEs

[F3]

Itô's formula gives the generator $L f = q f^{'} + \frac{1}{2} c^{2} f^{''}$ for $d Z = q (Z) d t + c d W$ . Multidimensional Ito formula for Brownian-driven processes

[F4]

Stopping an Itô integral gives the integral of the stopped predictable integrand; for finite energy this integral is square-integrable, has mean zero, and satisfies the Itô isometry. Stopping an Ito integral Ito isometry and linearity in predictable L2

[F5]

For $0 < ε < R$ , choose an exhaustion set $K_{n}$ with $[ε, R] \subset int K_{n}$ . The local stopped solution $Z^{(n)}$ is continuous and adapted on $[0, T]$ . Its first exit from $(ε, R)$ is a stopping time: the event of exit by $t$ is determined by the minimum and maximum on $[0, t]$ , measurable by rational-time approximation and the stopping-time definition. Because $[ε, R] \subset int K_{n}$ , this exit agrees with $τ_{ε} \land τ_{R}$ before the exhaustion exit. Local strong existence and uniqueness up to exit for locally Lipschitz SDEs The rationals embed densely in the reals Continuous-time stopping times and stopped sigma-algebras

[F6]

If $u (t) \leq A + \int_{0}^{t} B u (s) d s$ with nonnegative continuous $u$ , then $u (t) \leq A e^{B t}$ . Gronwall's integral inequality with variable and constant coefficients

[F7]

For $0 < ε < R$ and $g \in C^{2} ((0, \infty))$ , a smooth cutoff with support in $[ε / 2, 2 R] \subset (0, \infty)$ and equal to $1$ on $[ε, R]$ gives a global $C^{2}$ extension $\tilde{g}$ by setting $\tilde{g} = χ g$ on $(0, \infty)$ and $0$ elsewhere. Explicitly, use $χ (z) = η ((z - ε / 2) / (ε / 2)) η ((2 R - z) / R)$ , where $ρ (u) = e^{- 1 / u}$ for $u > 0$ , $ρ (u) = 0$ for $u \leq 0$ , and $η (u) = ρ (u) / (ρ (u) + ρ (1 - u))$ . On a stopped path in $[ε, R]$ , $\tilde{g}$ and its first two derivatives agree with those of $g$ .

[F8]

Dominated convergence gives continuity of $t \mapsto E V_{p} (Z_{t \land τ_{ε, R}})$ because the stopped path is continuous and $V_{p}$ is bounded on $[ε, R]$ . Dominated convergence

[F9]

Tonelli's theorem interchanges expectation and time integration for nonnegative measurable functions. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F10]

Fatou's lemma passes uniform bounds for stopped nonnegative moments to the almost-sure limit. Fatou's lemma

[A1]

AC is inherited from the local Brownian and Itô interfaces. The Axiom of Choice

Proof

Given: The hypotheses and suppliers in the Statement and Facts.

1.1F1F2givenalgebra

Put $z_{0} = \sqrt{x_{0}}$ and $q (z) = a / z - κ z / 2$ on $U = (0, \infty)$ , with constant diffusion $c = σ / 2$ . The function $q$ is $C^{1}$ , hence bounded and Lipschitz on every compact subset of $U$ ; $c$ is constant, and $z_{0} > 0$ is deterministic with finite second moment. Apply [F2] directly to these coefficients and initial value to obtain the unique maximal local solution $Z$ of $d Z = q (Z) d t + c d W$ . This construction uses the local theorem for the displayed transformed coefficients; it does not reverse [F1], which only transforms a pre-existing CIR solution. Put $δ = \frac{8 a}{σ^{2}} = \frac{4 κ θ}{σ^{2}} - 1.$ The Feller inequality gives $δ \geq 1$ .

1.2F3F4F5F7givenalgebra

Define $s (z) = \int_{1}^{z} u^{- δ} \exp ⁣ (\frac{2 κ u^{2}}{σ^{2}}) d u, z > 0.$ Then $s^{''} (z) / s^{'} (z) = - δ / z + 4 κ z / σ^{2}$ , and direct substitution gives $L s (z) = s^{'} (z) (\frac{a}{z} - \frac{κ}{2} z + \frac{σ^{2}}{8} (- \frac{δ}{z} + \frac{4 κ z}{σ^{2}})) = 0.$ Fix $0 < ε < z_{0} < R$ , define $τ_{ε} = \inf {t < ζ : Z_{t} \leq ε}$ and $τ_{R} = \inf {t < ζ : Z_{t} \geq R}$ (with $\inf \emptyset = \infty$ ), and set $τ_{ε, R} = τ_{ε} \land τ_{R} \land T$ . By [F5], this interval-exit time is a stopping time. Extend $s$ globally by [F7] and apply [F3] to the stopped process. The stopped stochastic integrand is bounded and has finite energy, so [F4] gives $E s (Z_{T \land τ_{ε, R}}) = s (z_{0})$ . On $A_{ε, R} = {τ_{ε} \leq T, τ_{ε} < τ_{R}}$ the stopped value is $s (ε)$ ; on its complement it is at most $s (R)$ . Therefore $P (A_{ε, R}) \leq \frac{s (R) - s (z_{0})}{s (R) - s (ε)} .$ Since $s^{'} (z) = z^{- δ} \exp (2 κ z^{2} / σ^{2})$ and $δ \geq 1$ , $s (ε) \to - \infty$ as $ε ↓ 0$ . Thus, for each fixed upper level $R$ , the probability of reaching arbitrarily small positive levels before $R$ by time $T$ is zero. Taking a countable union over integer $R > z_{0}$ excludes lower-boundary termination on paths bounded above; lower-boundary termination can therefore remain only on paths with unbounded upper excursions by time $T$ .

2.1F2F3F4step 1.2givenalgebra

For $V (z) = z^{2}$ , the global Itô formula gives $L V (z) = 2 z q (z) + c^{2} = κ θ - κ z^{2} \leq κ θ .$ For $0 < ε < z_{0} < R$ , stop at $τ_{ε, R}$ as above. The stopped stochastic integral has mean zero by [F4], so the pathwise generator bound gives $E Z_{T \land τ_{ε, R}}^{2} \leq z_{0}^{2} + κ θ T .$ On the event of hitting $R$ before $ε$ by time $T$ , the stopped value is $R$ , hence $R^{2} P (τ_{R} \leq T, τ_{R} < τ_{ε}) \leq z_{0}^{2} + κ θ T .$ For fixed $R$ , as $ε ↓ 0$ these upper-exit events increase to ${τ_{R} \leq T}$ : on every path that reaches $R$ by $T$ , the positive continuous path up to its first hit has a positive minimum. Thus $P (τ_{R} \leq T) \leq (z_{0}^{2} + κ θ T) / R^{2}$ , and letting $R \to \infty$ rules out upper escape by time $T$ . A local solution cannot terminate while remaining in a compact subset of $(0, \infty)$ , by [F2]. Step 1.2 excludes lower-boundary termination on bounded paths, and this estimate excludes paths with unbounded upper excursions; hence $Z$ is global and strictly positive on $[0, T]$ .

3.1F1F2F3step 2.1givenalgebra

Set $X = Z^{2}$ . Applying [F3] to the global $C^{2}$ function $z \mapsto z^{2}$ gives $d X_{t} = (2 Z_{t} q (Z_{t}) + c^{2}) d t + 2 c Z_{t} d W_{t} = κ (θ - X_{t}) d t + σ \sqrt{X_{t}} d W_{t},$ since $2 a + c^{2} = κ θ$ and $2 c = σ$ . Thus the CIR equation has a global positive strong solution. If $\tilde{X}$ is any other positive local strong CIR solution with initial value $x_{0}$ , [F1] makes $\sqrt{\tilde{X}}$ a local solution of the same $q, c$ equation; local uniqueness in [F2] makes it equal to $Z$ up to every compact exit, so $\tilde{X} = X$ .

4.1A1F3F4F6F7F8F9F10step 2.1step 3.1givenalgebra∎

For any $p \geq 1$ , let $V_{p} (z) = z^{2 p}$ on $(0, \infty)$ . For each $0 < ε < z_{0} < R$ , use [F7] to extend $V_{p}$ to a global $C^{2}$ function agreeing on $[ε, R]$ , and apply [F3] to the stopped process. Its generator on the stopped path is $L V_{p} (z) = (2 p a + \frac{p (2 p - 1) σ^{2}}{4}) z^{2 p - 2} - p κ z^{2 p} \leq C_{p} (1 + z^{2 p}) .$ The stopped stochastic integral has mean zero by [F4]. For fixed $ε, R$ , dominated convergence [F8] makes $u (t) = E V_{p} (Z_{t \land τ_{ε, R}})$ continuous, and Tonelli [F9] justifies the expectation of its time-integrated generator. Gronwall [F6] gives $\sup_{0 < ε < z_{0} < R} \sup_{t \leq T} E V_{p} (Z_{t \land τ_{ε, R}}) \leq C_{p, T} (1 + z_{0}^{2 p}) .$ Since $Z$ is global, continuous and strictly positive on $[0, T]$ , these stopping times equal $T$ eventually along any sequence $ε ↓ 0$ , $R ↑ \infty$ on almost every path. Fatou's lemma [F10] then yields $\sup_{t \leq T} E Z_{t}^{2 p} < \infty$ , and $X = Z^{2}$ gives the claimed CIR moment bound. AC is exactly [A1].

TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passOpen item page →

Strong order one-half for tamed Euler under polynomial drift growth

Statement

Assume AC. Let $d, m \geq 1$ , $T > 0$ , $b \in C^{1} (R^{d}; R^{d})$ , and $σ : R^{d} \to R^{d \times m}$ be globally Lipschitz. Suppose constants $L, C < \infty$ and an integer $q \geq 1$ satisfy $⟨ x - y, b (x) - b (y) ⟩ \leq L ∣ x - y ∣^{2}, ∥ D b (x) ∥ \leq C (1 + ∣ x ∣^{q}) .$ Let $X$ be the unique strong solution of the corresponding autonomous SDE from Global existence and finite-horizon moments for one-sided Lipschitz SDEs, with initial state $ξ$ having all finite moments. For $N \geq \max (1, ⌈ T ⌉)$ put $h = T / N$ , and let $\tilde{Y}$ be the continuous tamed Euler interpolation from Tamed Euler scheme for a superlinear drift. There is a constant $C_{T}$ , independent of $N$ , such that $E \sup_{0 \leq t \leq T} ∣ X_{t} - {\tilde{Y}}_{t} ∣^{2} \leq C_{T} h .$ Thus the tamed Euler method has strong order one-half in the path-supremum $L^{2}$ norm under the displayed hypotheses.

Facts & Assumptions

Given: AC; the autonomous SDE and coefficient hypotheses in the Statement; $h = T / N \leq 1$ ; and the tamed update and interpolation.

[F1]

The local Itô SDE definition specifies the filtered Brownian conventions and the integral equation that a local strong solution must satisfy; it does not assert existence. The global adapted continuous solution used here is supplied by [F2]. Locally Lipschitz Itô SDE on an open state domain

[F2]

Under the stated one-sided Lipschitz, polynomial derivative, and diffusion Lipschitz assumptions, $X$ is global and has finite path-supremum moments of every finite order. Global existence and finite-horizon moments for one-sided Lipschitz SDEs

[F3]

The tamed grid values satisfy $\sup_{N} \max_{n} E ∣ Y_{n} ∣^{p} < \infty$ for every finite $p \geq 1$ . Uniform finite-horizon moments for tamed Euler approximations

[F4]

The algorithm gives ${\tilde{Y}}_{t} = Y_{n} + (t - t_{n}) b (Y_{n}) / (1 + h ∣ b (Y_{n}) ∣) + σ (Y_{n}) (W_{t} - W_{t_{n}})$ on $[t_{n}, t_{n + 1}]$ . Tamed Euler scheme for a superlinear drift

[F5]

For a continuous Brownian Itô process $Z$ and $f \in C^{1, 2}$ , Itô's formula for $∣ Z ∣^{2}$ has drift $2 ⟨ Z, b_{Z} ⟩ + ∥ σ_{Z} ∥_{F}^{2}$ and martingale integrand $2 Z^{T} σ_{Z}$ . Multidimensional Ito formula for Brownian-driven processes

[F6]

For a finite-energy row-vector integrand $H$ against an $m$ -dimensional Brownian motion, $E \sup_{s \leq t} ∣ \int_{0}^{s} H_{u} d W_{u} ∣ \leq 16 \sqrt{m} E (\int_{0}^{t} ∥ H_{u} ∥_{2}^{2} d u)^{1 / 2}$ . An L1 maximal bound for finite-energy Brownian Itô integrals

[F7]

Cauchy--Schwarz for random variables bounds $E ∣ U V ∣ \leq (E ∣ U ∣^{2})^{1 / 2} (E ∣ V ∣^{2})^{1 / 2}$ . Cauchy-Schwarz for random variables

[F8]

If $v (t) \leq A + \int_{0}^{t} B v (s) d s$ with nonnegative continuous $v$ and constants $A, B \geq 0$ , then $v (t) \leq A e^{B t}$ . Gronwall's integral inequality with variable and constant coefficients

[F9]

A standard normal variable has finite absolute moments of every finite order, since $\int_{0}^{\infty} x^{r} e^{- x^{2} / 2} d x < \infty$ ; scaling gives $E ∣ W_{s} - W_{t} ∣^{r} = C_{r, m} ∣ s - t ∣^{r / 2}$ . Standard normal and normal laws

[F10]

Tonelli's theorem interchanges expectation and time integration for the nonnegative error majorant used below. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F11]

For fixed finite $N$ , the interpolation has finite fourth path-supremum moment: the grid values and coefficients have finite eighth moments by [F3] and polynomial drift growth, while the Brownian path supremum has finite eighth moment by the $p = 8$ maximal Itô-integral bound with constant integrand. Apply Cauchy--Schwarz to each product of a grid diffusion coefficient and a Brownian interval supremum, then sum over the finitely many intervals in [F4]. In particular the second path-supremum moment is finite as well. Lp maximal moment bound for Brownian Itô integrals

[F12]

Since $e$ has continuous paths and $E \sup_{s \leq T} ∣ e_{s} ∣^{2} < \infty$ by [F2] and [F11], dominated convergence makes $v (t) = E \sup_{s \leq t} ∣ e_{s} ∣^{2}$ continuous. Dominated convergence

[A1]

AC is inherited from the SDE and Brownian Itô interfaces; the estimates below use only the specified Brownian increments and stopping conventions. The Axiom of Choice

Proof

Given: The assumptions and facts in the Statement and Facts.

1.1F1F2F3F4F5given

Let ${\underline{Y}}_{s} = Y_{n}$ on $[t_{n}, t_{n + 1})$ and $δ_{s} = {\tilde{Y}}_{s} - {\underline{Y}}_{s}$ . Induction in the recursion shows $Y_{n}$ is $F_{t_{n}}$ -measurable, so the step coefficient processes have predictable representatives, using $Y_{n}$ on $(t_{n}, t_{n + 1}]$ for stochastic integrands; changing values at grid times does not change the integrals. The interpolation is adapted and continuous and satisfies ${\tilde{Y}}_{t} = ξ + \int_{0}^{t} \frac{b ({\underline{Y}}_{s})}{1 + h ∣ b ({\underline{Y}}_{s}) ∣} d s + \int_{0}^{t} σ ({\underline{Y}}_{s}) d W_{s} .$ Its coefficient energies are finite by [F2]–[F3] and the polynomial growth of $b$ . Hence $e_{t} = X_{t} - {\tilde{Y}}_{t}$ is a continuous Brownian Itô process and [F5] applies.

1.2F3F4F9givenalgebra

On an interval $[t_{n}, t_{n + 1}]$ , $δ_{s} = (s - t_{n}) b_{h} (Y_{n}) + σ (Y_{n}) (W_{s} - W_{t_{n}})$ . For $r = 2, 4$ , $∣ (s - t_{n}) b_{h} (Y_{n}) ∣ \leq h ∣ b (Y_{n}) ∣$ , and the Brownian increment is independent of $F_{t_{n}}$ with $r$ th moment $C_{r, m} (s - t_{n})^{r / 2}$ by [F9]. The moment bounds [F3], polynomial growth $∣ b (x) ∣ \leq C_{1} (1 + ∣ x ∣^{q + 1})$ , and linear growth of $σ$ give $\sup_{n} E ∣ b (Y_{n}) ∣^{r} < \infty$ and $\sup_{n} E ∥ σ (Y_{n}) ∥_{F}^{r} < \infty$ . Since $h^{r} \leq h^{r / 2}$ for $h \leq 1$ , these estimates imply $\sup_{n < N} \sup_{s \in [t_{n}, t_{n + 1}]} E ∣ δ_{s} ∣^{r} \leq C_{r} h^{r / 2}, r = 2, 4.$ Also, with $b_{h} (y) = b (y) / (1 + h ∣ b (y) ∣)$ , $∣ b (y) - b_{h} (y) ∣ \leq h ∣ b (y) ∣^{2}$ , so $\sup_{n} E ∣ b (Y_{n}) - b_{h} (Y_{n}) ∣^{2} \leq C h^{2}$ by [F3].

2.1F2F3F7F10step 1.2givenalgebra

Put $G_{s} = σ (X_{s}) - σ ({\underline{Y}}_{s})$ and $A_{s} = 1 + ∣ X_{s} ∣^{q} + ∣ {\underline{Y}}_{s} ∣^{q}$ . The derivative bound and integration along the segment from $X_{s}$ to ${\underline{Y}}_{s}$ give $∣ b (X_{s}) - b ({\underline{Y}}_{s}) ∣ \leq C A_{s} ∣ X_{s} - {\underline{Y}}_{s} ∣$ . Since $X_{s} - {\underline{Y}}_{s} = e_{s} + δ_{s}$ , one-sided Lipschitzness gives $2 ⟨ e_{s}, b (X_{s}) - b ({\underline{Y}}_{s}) ⟩ = 2 ⟨ X_{s} - {\underline{Y}}_{s}, b (X_{s}) - b ({\underline{Y}}_{s}) ⟩ - 2 ⟨ δ_{s}, b (X_{s}) - b ({\underline{Y}}_{s}) ⟩ \leq C ∣ e_{s} ∣^{2} + C (1 + A_{s}^{2}) ∣ δ_{s} ∣^{2} .$ Here the second term uses $2 A_{s} ∣ δ_{s} ∣ ∣ e_{s} ∣ \leq ∣ e_{s} ∣^{2} + A_{s}^{2} ∣ δ_{s} ∣^{2}$ . The taming correction contributes at most $∣ e_{s} ∣^{2} + ∣ b ({\underline{Y}}_{s}) - b_{h} ({\underline{Y}}_{s}) ∣^{2}$ , and $∥ G_{s} ∥_{F}^{2} \leq C (∣ e_{s} ∣^{2} + ∣ δ_{s} ∣^{2})$ by global Lipschitzness of $σ$ . Thus a nonnegative $H_{s} = C (∣ e_{s} ∣^{2} + (1 + A_{s}^{2}) ∣ δ_{s} ∣^{2} + ∣ b ({\underline{Y}}_{s}) - b_{h} ({\underline{Y}}_{s}) ∣^{2})$ satisfies $2 ⟨ e_{s}, b (X_{s}) - b_{h} ({\underline{Y}}_{s}) ⟩ + ∥ G_{s} ∥_{F}^{2} \leq H_{s} .$ By [F2]–[F3], Cauchy--Schwarz [F7], and the $r = 2, 4$ increment estimates in step 1.2, Tonelli [F10] gives $E \int_{0}^{t} H_{s} d s \leq C \int_{0}^{t} v (s) d s + C h$ , where $v (t) = E \sup_{s \leq t} ∣ e_{s} ∣^{2}$ .

2.2F2F3F6F7F11step 1.2algebra

The martingale in Itô's identity is $M_{t} = 2 \int_{0}^{t} e_{s}^{T} G_{s} d W_{s}$ . For fixed $N$ , [F2], [F3], [F11], and Cauchy--Schwarz [F7] give $E \int_{0}^{T} ∣ e_{s} ∣^{2} ∥ G_{s} ∥_{F}^{2} d s < \infty$ , since $e$ has a finite fourth path-supremum moment and $G_{s}$ has uniformly finite fourth moments. Thus [F6] applies and gives $E \sup_{s \leq t} ∣ M_{s} ∣ \leq 32 \sqrt{m} E [\sup_{s \leq t} ∣ e_{s} ∣ {(\int_{0}^{t} ∥ G_{s} ∥_{F}^{2} d s)}^{1 / 2}] .$ Using Cauchy--Schwarz [F7] and $u v \leq ε u^{2} + (4 ε)^{- 1} v^{2}$ (which follows by expanding $(\sqrt{ε} u - v / (2 \sqrt{ε}))^{2} \geq 0$ ), the Lipschitz bound on $σ$ , and step 1.2, this is at most $\frac{1}{2} v (t) + C \int_{0}^{t} v (s) d s + C h$ .

3.1A1F2F3F5F8F10F11F12step 2.1step 2.2algebra∎

For each finite $N$ , [F2] and [F11] give $E \sup_{s \leq T} ∣ e_{s} ∣^{2} < \infty$ , and [F12] makes $v (t)$ continuous. Taking the supremum over times up to $t$ in Itô's identity [F5], the pointwise majorant in step 2.1 gives $\sup_{u \leq t} \int_{0}^{u} H_{s} d s \leq \int_{0}^{t} H_{s} d s$ ; now take expectations and insert steps 2.1–2.2. Moving $v (t) / 2$ to the left gives $v (t) \leq C \int_{0}^{t} v (s) d s + C h$ . Gronwall [F8] yields $v (T) \leq C_{T} h$ . AC is inherited exactly through [A1], so the stated strong order follows.

TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Lamperti drift-implicit CIR scheme converges on finite horizons in probability

Statement

Assume AC, $κ, θ, σ > 0$ , $2 κ θ \geq σ^{2}$ , and deterministic $X_{0} = x_{0} > 0$ . Let $X$ be the positive CIR solution from CIR paths stay positive and do not explode under the Feller condition. For $h = T / N$ , initialize $Z_{0} = \sqrt{x_{0}}$ and apply Positive Lamperti drift-implicit scheme for the CIR process to obtain $Z_{n}$ and $X_{n}^{h} = Z_{n}^{2}$ . Then for every $T > 0$ , $\max_{0 \leq n \leq N} ∣ X_{t_{n}} - X_{n}^{h} ∣ ⟶ 0 in probability as N \to \infty .$ This is convergence in probability on each finite horizon; no $L^{p}$ error rate is asserted.

Facts & Assumptions

Given: AC; positive CIR parameters and initial state; the exact positive solution and the transformed drift-implicit algorithm.

[F1]

The positive transformed solution $Z = \sqrt{X}$ satisfies $d Z = q (Z) d t + c d W$ , where $q (z) = a / z - κ z / 2$ , $a > 0$ , $c = σ / 2$ ; the algorithm returns the unique positive root of $z_{n + 1} = z_{n} + h q (z_{n + 1}) + c Δ W_{n}$ . CIR Lamperti equation and unique positive implicit root Positive Lamperti drift-implicit scheme for the CIR process

[F2]

The exact CIR solution is global, continuous, and strictly positive on each finite horizon. CIR paths stay positive and do not explode under the Feller condition

[F3]

Brownian increments over deterministic intervals have centered normal law with variance equal to interval length. $d$ -dimensional Brownian motion Standard normal and normal laws

[F4]

For a nonnegative random variable $U$ and $r > 0$ , $P (U > r) \leq E U / r$ . Markov's inequality for random variables

[A1]

AC is inherited through the Brownian and numerical inputs; no additional choice is used. The Axiom of Choice

Proof

Given: The hypotheses and suppliers in the Statement and Facts.

1.1F2given

The exact path $Z = \sqrt{X}$ is continuous and strictly positive on the compact interval $[0, T]$ by [F2]. Hence its pathwise minimum is positive and its maximum is finite. For every $η > 0$ choose $0 < ε < R$ so that the event $E_{ε, R} = {ε \leq Z_{t} \leq R for every t \leq T}$ has probability at least $1 - η$ ; these events increase to a probability-one event as $ε ↓ 0$ and $R ↑ \infty$ .

1.2F1givenalgebra

Write $c = σ / 2$ and $r_{n + 1} = \int_{t_{n}}^{t_{n + 1}} (q (Z_{s}) - q (Z_{t_{n + 1}})) d s .$ The exact SDE and the implicit recursion give $Z_{t_{n + 1}} = Z_{t_{n}} + h q (Z_{t_{n + 1}}) + c Δ W_{n} + r_{n + 1}, Z_{n + 1} = Z_{n} + h q (Z_{n + 1}) + c Δ W_{n} .$ Let $e_{n} = Z_{n} - Z_{t_{n}}$ . Since $q^{'} (z) = - a / z^{2} - κ / 2 \leq - κ / 2$ , the mean-value theorem gives $(u - v) (q (u) - q (v)) \leq - κ ∣ u - v ∣^{2} / 2$ for positive $u, v$ . Subtract the two updates and take the inner product with $e_{n + 1}$ : $(1 + h κ / 2) ∣ e_{n + 1} ∣^{2} \leq ∣ e_{n + 1} ∣ ∣ e_{n} - r_{n + 1} ∣ .$ If $e_{n + 1} = 0$ the next bound is immediate; otherwise divide by $(1 + h κ / 2) ∣ e_{n + 1} ∣$ to get $∣ e_{n + 1} ∣ \leq ∣ e_{n} ∣ + ∣ r_{n + 1} ∣$ . As $e_{0} = 0$ , $\max_{n \leq N} ∣ e_{n} ∣ \leq \sum_{n < N} ∣ r_{n + 1} ∣$ .

2.1F1F3step 1.2givenalgebra

On $E_{ε, R}$ , the drift $q$ is bounded by $B$ and is Lipschitz with a constant $L$ . For $s \in [t_{n}, t_{n + 1}]$ , the exact transformed equation gives $∣ Z_{s} - Z_{t_{n + 1}} ∣ \leq B h + c ∣ W_{s} - W_{t_{n + 1}} ∣ .$ A centered normal variable of variance $v > 0$ has absolute first moment $\frac{2}{\sqrt{2 π v}} \int_{0}^{\infty} x e^{- x^{2} / (2 v)} d x = \sqrt{2 v / π}$ by direct integration of its density; at $v = 0$ both sides are zero. Thus, using $1_{E_{ε, R}} \leq 1$ (without assuming independence of this path event from the increment) and [F3], $E [1_{E_{ε, R}} ∣ r_{n + 1} ∣] \leq L \int_{t_{n}}^{t_{n + 1}} (B h + c \sqrt{2 (t_{n + 1} - s) / π}) d s \leq C_{ε, R} (h^{2} + h^{3 / 2}) .$ Summing over $n < N$ and using $N h = T$ gives $E [1_{E_{ε, R}} \sum_{n < N} ∣ r_{n + 1} ∣] \leq C_{ε, R, T} \sqrt{h}$ .

3.1A1F2F4step 1.1step 1.2step 2.1givenalgebra∎

For any $ρ > 0$ , steps 1.2–2.1 and [F4] give $P (\max_{n \leq N} ∣ e_{n} ∣ > ρ) \leq P (E_{ε, R}^{c}) + ρ^{- 1} C_{ε, R, T} \sqrt{h} .$ First let $N \to \infty$ with $ε, R$ fixed, then let $η \to 0$ using step 1.1. Thus $\max_{n} ∣ Z_{n} - Z_{t_{n}} ∣ \to 0$ in probability. On $E_{ε, R} \cap {\max_{n} ∣ e_{n} ∣ \leq 1}$ , $∣ Z_{n} + Z_{t_{n}} ∣ \leq 2 R + 1$ , and hence $\max_{n} ∣ X_{n}^{h} - X_{t_{n}} ∣ \leq (2 R + 1) \max_{n} ∣ Z_{n} - Z_{t_{n}} ∣ .$ The probability of the complementary event is arbitrarily small by the event construction in step 1.1 and the convergence just proved. This gives the stated convergence in probability for $X_{n}^{h} = Z_{n}^{2}$ . No $L^{p}$ rate follows from this localization argument. AC is exactly [A1].

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Strong order one-half for drift-implicit Euler–Maruyama

Statement

Assume AC. Let $d, m \geq 1$ be finite, $T > 0$ , and consider the autonomous SDE with $b \in C^{1} (R^{d}; R^{d})$ and globally Lipschitz $σ : R^{d} \to R^{d \times m}$ . Suppose that for constants $μ \geq 0$ , $C < \infty$ , and an integer $q \geq 1$ , $⟨ x - y, b (x) - b (y) ⟩ \leq - μ ∣ x - y ∣^{2}, ∥ D b (x) ∥ \leq C (1 + ∣ x ∣^{q}) .$ Let $ξ$ have all finite moments and let $X$ be the corresponding global strong solution. For $N \geq \max (1, ⌈ T ⌉)$ , put $h = T / N$ and let $Y_{n}$ be the exact-resolvent drift-implicit Euler–Maruyama values from Drift-implicit Euler–Maruyama with explicit solve-failure status. There is a constant $C_{T}$ independent of $N$ such that $E \max_{0 \leq n \leq N} ∣ X_{t_{n}} - Y_{n} ∣^{2} \leq C_{T} h .$ Moreover, for two such numerical sequences driven by the same Brownian increments but with initial values $ξ, \tilde{ξ}$ , $E ∣ Y_{n} - {\tilde{Y}}_{n} ∣^{2} \leq e^{L_{σ}^{2} t_{n}} E ∣ ξ - \tilde{ξ} ∣^{2},$ where $L_{σ}$ is a global Lipschitz constant of $σ$ . The second estimate is finite-time mean-square stability; it does not claim strict contraction when diffusion is present.

Facts & Assumptions

Given: AC; the dissipative SDE and moment hypotheses in the Statement; $h = T / N \leq 1$ ; and the drift-implicit scheme.

[F1]

The backward Euler resolvent exists on all of $R^{d}$ and obeys $∣ R_{h} (u) - R_{h} (v) ∣ \leq (1 + h μ)^{- 1} ∣ u - v ∣$ . Backward Euler has a globally defined nonexpansive resolvent (contractive when strongly dissipative)

[F2]

The exact SDE solution is global and has finite path-supremum moments of every finite order. Global existence and finite-horizon moments for one-sided Lipschitz SDEs

[F3]

The exact SDE satisfies the local Itô integral equation from the definition. For a scalar predictable finite-energy integrand $H$ and each Brownian coordinate $W^{j}$ , $E ∣ \int_{a}^{b} H_{s} d W_{s}^{j} ∣^{2} = E \int_{a}^{b} H_{s}^{2} d s$ ; the matrix bound needed below is derived coordinatewise in step 1.3. Locally Lipschitz Itô SDE on an open state domain Ito isometry and linearity in predictable L2

[F4]

The Brownian Itô integral satisfies the $L^{p}$ maximal estimate in Lp maximal moment bound for Brownian Itô integrals and the $L^{1}$ maximal estimate $E \sup ∣ \int H d W ∣ \leq 16 \sqrt{m} E (\int ∥ H ∥_{F}^{2})^{1 / 2}$ in An L1 maximal bound for finite-energy Brownian Itô integrals.

[F5]

The polynomial derivative bound implies $∣ b (x) - b (y) ∣ \leq C^{'} (1 + ∣ x ∣^{q} + ∣ y ∣^{q}) ∣ x - y ∣$ by integrating $D b$ along the segment from $x$ to $y$ . This is a consequence of the Statement's derivative hypothesis.

[F6]

Hölder's inequality gives the $L^{p}$ bounds for time integrals and products used in the local-increment estimates. Holder's inequality for integrals, including the endpoint cases

[F7]

If $u (t) \leq A + \int_{0}^{t} B u (s) d s$ with nonnegative continuous $u$ , then $u (t) \leq A e^{B t}$ . Gronwall's integral inequality with variable and constant coefficients

[F8]

[F9]

For each Brownian coordinate $W^{j}$ , a scalar finite-energy Itô integral has a continuous square-integrable martingale version. Each coordinate of a matrix integral is a finite sum of these scalar martingales, hence is a square-integrable martingale by linearity of conditional expectation. Its grid increments therefore have conditional mean zero given the preceding sigma-algebra. The Ito integral process has a continuous martingale version

[F10]

Tonelli's theorem interchanges expectation and time integration for the nonnegative increment and diffusion bounds used below. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[A1]

AC is inherited through the Brownian, resolvent, and Itô interfaces. The Axiom of Choice

Proof

Given: The assumptions and suppliers in the Statement and Facts.

1.1A1F1F2F6givenalgebra

First fix $N$ . Since $R_{h} (- h b (0)) = 0$ , [F1] gives $∣ R_{h} (u) ∣ \leq ∣ u ∣ + h ∣ b (0) ∣$ . Its continuity, linear growth of $σ$ , and the independent Gaussian increments show by finite induction that $Y_{n}$ is $F_{t_{n}}$ -measurable and has every finite moment; the same holds for the comparison sequence. Together with [F2] and Hölder [F6], these moments give finite expected square energy for all stochastic integrands below, including products of a grid error and a diffusion difference. These preliminary bounds may depend on $N$ . For two updates with difference $e_{n} = Y_{n} - {\tilde{Y}}_{n}$ , [F1] and Brownian independence give $\begin{aligned} E (∣ e_{n + 1} ∣^{2} ∣ F_{t_{n}}) & \leq (1 + h μ)^{- 2} E (∣ e_{n} + (σ (Y_{n}) - σ ({\tilde{Y}}_{n})) Δ W_{n} ∣^{2} ∣ F_{t_{n}}) \\ \leq (1 + h L_{σ}^{2}) ∣ e_{n} ∣^{2} . \end{aligned}$ The cross term vanishes because $e_{n}$ and the diffusion difference are $F_{t_{n}}$ -measurable and $Δ W_{n}$ is independent with mean zero; its covariance is $h I_{m}$ . Iterating and using $(1 + h L_{σ}^{2})^{n} \leq e^{L_{σ}^{2} t_{n}}$ proves the second estimate.

1.2F1F3F9givenalgebra

Let $X_{n} = X_{t_{n}}$ and define $d_{n + 1} = \int_{t_{n}}^{t_{n + 1}} (b (X_{s}) - b (X_{n + 1})) d s, m_{n + 1} = \int_{t_{n}}^{t_{n + 1}} (σ (X_{s}) - σ (X_{n})) d W_{s} .$ The SDE integral equation gives $X_{n + 1} - h b (X_{n + 1}) = X_{n} + σ (X_{n}) Δ W_{n} + d_{n + 1} + m_{n + 1}$ . With $R_{h} = (I - h b)^{- 1}$ , the numerical update and [F1] therefore imply, for $e_{n} = Y_{n} - X_{n}$ , $∣ e_{n + 1} ∣ \leq ∣ e_{n} + ξ_{n + 1} - d_{n + 1} ∣, ξ_{n + 1} = \int_{t_{n}}^{t_{n + 1}} (σ (Y_{n}) - σ (X_{s})) d W_{s} .$ The identity for $ξ_{n + 1}$ follows by adding and subtracting $σ (X_{n}) Δ W_{n}$ . Its conditional mean given $F_{t_{n}}$ is zero by [F9].

2.1F2F3F4F5F6F8F10step 1.1givenalgebra

From [F2], linear growth of $σ$ , and the SDE equation, for $r = 2, 4$ and $s \in [t_{n}, t_{n + 1}]$ , $E ∣ X_{s} - X_{n} ∣^{r} + E ∣ X_{s} - X_{n + 1} ∣^{r} \leq C_{r} h^{r / 2} .$ Indeed the drift increment over either intervening interval has $r$ th moment at most $h^{r - 1} \int_{t_{n}}^{t_{n + 1}} E ∣ b (X_{u}) ∣^{r} d u \leq C h^{r}$ by Hölder and [F10], while [F4] bounds the stochastic increment by $C_{r} E (\int_{t_{n}}^{t_{n + 1}} ∥ σ (X_{u}) ∥_{F}^{2} d u)^{r / 2} \leq C_{r} h^{r / 2}$ . The polynomial local-Lipschitz estimate [F5], all moments in [F2], and Hölder [F6] then give $E ∣ d_{n + 1} ∣^{2} \leq h \int_{t_{n}}^{t_{n + 1}} E ∣ b (X_{s}) - b (X_{n + 1}) ∣^{2} d s \leq C h^{3} .$ For any finite-energy matrix $G$ , writing each coordinate as a sum over the $m$ Brownian coordinates, the scalar isometry [F3] and $∣ \sum_{j} a_{j} ∣^{2} \leq m \sum_{j} ∣ a_{j} ∣^{2}$ give $E {∣ \int_{a}^{b} G_{s} d W_{s} ∣}^{2} \leq m \sum_{i, j} E {∣ \int_{a}^{b} G_{s}^{i j} d W_{s}^{j} ∣}^{2} = m E \int_{a}^{b} ∥ G_{s} ∥_{F}^{2} d s .$ Apply this to $G_{s} = σ (Y_{n}) - σ (X_{s})$ and use global Lipschitzness to obtain $E ∣ ξ_{n + 1} ∣^{2} \leq C h E ∣ e_{n} ∣^{2} + C h^{2} .$ The constants here are independent of $N$ .

3.1F8F9step 1.2step 2.1algebra

Squaring the error inequality in step 1.2 and applying $2 ∣ e_{n} ∣ ∣ d_{n + 1} ∣ \leq h ∣ e_{n} ∣^{2} + h^{- 1} ∣ d_{n + 1} ∣^{2}$ and $2 ∣ ξ_{n + 1} ∣ ∣ d_{n + 1} ∣ \leq ∣ ξ_{n + 1} ∣^{2} + ∣ d_{n + 1} ∣^{2}$ gives $∣ e_{n + 1} ∣^{2} \leq (1 + h) ∣ e_{n} ∣^{2} + 2 ⟨ e_{n}, ξ_{n + 1} ⟩ + 2 ∣ ξ_{n + 1} ∣^{2} + (2 + h^{- 1}) ∣ d_{n + 1} ∣^{2} .$ By [F9], the increments $ξ_{n + 1}$ are martingale differences. Summing their unconditional second-moment bound from step 2.1 gives $\sum_{n < N} E ∣ ξ_{n + 1} ∣^{2} \leq C h \sum_{n < N} E ∣ e_{n} ∣^{2} + C h$ ; step 2.1 also gives $\sum_{n < N} (2 + h^{- 1}) E ∣ d_{n + 1} ∣^{2} \leq C h$ .

3.2F2F4F8F9F10step 1.1step 2.1givenalgebra

Put $v_{j} = E \max_{k \leq j} ∣ e_{k} ∣^{2}$ for $0 \leq j \leq N$ , finite by step 1.1. The partial sums $M_{j} = 2 \sum_{n < j} ⟨ e_{n}, ξ_{n + 1} ⟩$ form a martingale by [F9]. Written as one Brownian Itô integral, its integrand on $(t_{n}, t_{n + 1}]$ is $2 e_{n}^{T} (σ (Y_{n}) - σ (X_{s}))$ , whose expected square energy is finite by the fourth moments in step 1.1 and [F2]. For $1 \leq j \leq N$ , the $L^{1}$ maximal bound [F4] therefore gives $E \max_{l \leq j} ∣ M_{l} ∣ \leq 32 \sqrt{m} E [\max_{n < j} ∣ e_{n} ∣ {(\sum_{n < j} \int_{t_{n}}^{t_{n + 1}} ∥ σ (Y_{n}) - σ (X_{s}) ∥_{F}^{2} d s)}^{1 / 2}] .$ Applying $u v \leq ε u^{2} + (4 ε)^{- 1} v^{2}$ (obtained by expanding $(\sqrt{ε} u - v / (2 \sqrt{ε}))^{2} \geq 0$ ) factors out the error maximum and yields, for any $ε > 0,$ $E \max_{l \leq j} ∣ M_{l} ∣ \leq ε v_{j - 1} + C_{ε} E \sum_{n < j} \int_{t_{n}}^{t_{n + 1}} ∥ σ (Y_{n}) - σ (X_{s}) ∥_{F}^{2} d s .$ By Lipschitzness, $∥ σ (Y_{n}) - σ (X_{s}) ∥_{F}^{2} \leq 2 L_{σ}^{2} (∣ e_{n} ∣^{2} + ∣ X_{s} - X_{n} ∣^{2})$ . Step 2.1 and Tonelli [F10] then give $E \max_{l \leq j} ∣ M_{l} ∣ \leq ε v_{j - 1} + C_{ε} h \sum_{n < j} E ∣ e_{n} ∣^{2} + C_{ε} h,$ since $\sum_{n < j} \int_{t_{n}}^{t_{n + 1}} E ∣ X_{s} - X_{n} ∣^{2} d s \leq C h$ .

4.1A1F7step 1.1step 2.1step 3.1step 3.2algebra∎

Recall $v_{j}$ from step 3.2; $e_{0} = 0$ because both methods start from $ξ$ . Subtract $∣ e_{n} ∣^{2}$ from both sides of step 3.1 and sum from $n = 0$ to $j - 1$ . Taking the maximum over partial endpoints makes the martingale term $\max_{l \leq j} ∣ M_{l} ∣$ ; the remaining nonnegative terms have expectation at most $C h \sum_{k < j} v_{k} + C h$ by the local bounds in steps 2.1--2.2. Use step 3.2 with $ε = 1 / 2$ ; since $v_{j - 1} \leq v_{j}$ , absorb the martingale contribution to obtain $v_{j} \leq C h \sum_{k < j} v_{k} + C h .$ Choose $C_{0} \geq C$ so this implies $v_{j} \leq C_{0} w_{j}$ for $w_{j} = h + h \sum_{k < j} v_{k}$ . Then $w_{j + 1} = w_{j} + h v_{j} \leq (1 + C_{0} h) w_{j}$ and $w_{0} = h$ , so $v_{N} \leq C_{0} h (1 + C_{0} h)^{N} \leq C_{0} e^{C_{0} T} h$ . This proves the first assertion; the stability assertion was proved in step 1.1. AC is exactly [A1].

5 · Examples, counterexamples and false statements

None yet.

Stable sde schemes for superlinear drift

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Locally Lipschitz Itô SDE on an open state domain

Definition

Lp maximal moment bound for Brownian Itô integrals

Statement

Facts & Assumptions

Proof

Positive Lamperti drift-implicit scheme for the CIR process

Definition

Tamed Euler scheme for a superlinear drift

Definition

Local strong existence and uniqueness up to exit for locally Lipschitz SDEs

Statement

Facts & Assumptions

Proof

CIR Lamperti equation and unique positive implicit root

Statement

Facts & Assumptions

Proof

Uniform finite-horizon moments for tamed Euler approximations

Statement

Facts & Assumptions

Proof

Global existence and finite-horizon moments for one-sided Lipschitz SDEs

Statement

Facts & Assumptions

Proof

Drift-implicit Euler–Maruyama with explicit solve-failure status

Definition

CIR paths stay positive and do not explode under the Feller condition

Statement

Facts & Assumptions

Proof

Strong order one-half for tamed Euler under polynomial drift growth

Statement

Facts & Assumptions

Proof

Lamperti drift-implicit CIR scheme converges on finite horizons in probability

Statement

Facts & Assumptions

Proof

Strong order one-half for drift-implicit Euler–Maruyama

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources