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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]

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

[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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Sources

Martin Hutzenthaler, Arnulf Jentzen and Peter E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, §1 (standard reference, not scraped)
Desmond J. Higham, Xuerong Mao and Andrew M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, §3 (standard reference, not scraped)