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Analysis of a first-order explicit positivity preserving scheme for a class of scalar SDEs
Charles-Edouard Bréhier, David Cohen
TL;DR
The paper addresses how to numerically solve a class of scalar Itô SDEs with nonnegative solutions while preserving positivity at arbitrary step sizes. It constructs an explicit factorization-based scheme and proves that it achieves first-order strong convergence, with pathwise and Stratonovich extensions under stated conditions.
Problem
Standard Euler–Maruyama and Milstein schemes do not preserve positivity for the considered SDEs, motivating a method that preserves nonnegativity while improving on the existing strong order 1/2 scheme.
Method
The paper constructs an explicit scheme from the factorization G(x) = xF(x), g(x) = xf(x), incorporating a correction term and an Itô reformulation for Stratonovich equations.
Results
Strong convergence has order 1, while positivity holds almost surely for every time-step size; pathwise convergence is also obtained at rate 1−ε.
Takeaways & Limitations
The proposed scheme provides a first-order explicit positivity-preserving method for the specified scalar SDE class and can be adapted to Stratonovich equations.
Takeaways & Limitations
The results apply under the paper’s coefficient assumptions, and the Stratonovich extension requires stronger regularity and boundedness conditions on g.
Abstract
from arXiv · showhide
We propose and analyze a first-order numerical scheme for a class of scalar Itô stochastic differential equations with almost surely positive solutions. We construct a new explicit numerical scheme, such that for any choice of the time-step size, the numerical solution is guaranteed to remain almost surely positive. The main result of this article is the first-order strong convergence of the proposed positivity preserving scheme, which is illustrated with numerical experiments and proved rigorously. It is also shown how to adapt the scheme and the results to Stratonovich stochastic differential equations with positive solutions.
1. Introduction
The paper develops an explicit positivity-preserving scheme for scalar Itô SDEs whose solutions are nonnegative, targeting strong convergence of order 1 without restricting the time-step size.
- Motivation: The coefficient condition G(0) = g(0) = 0 ensures nonnegative exact solutions, whereas Euler–Maruyama and Milstein generally do not preserve positivity.Previous positivity-preserving approaches mainly target specific models and use transformations or modified classical schemes.
- Scheme construction: The construction factors the coefficients as G(x) = xF(x) and g(x) = xf(x), then combines this structure with the exact solution of a linear SDE.The resulting interval-wise construction is designed to preserve positivity.
- Scheme construction: The original positivity-preserving scheme remains positive for every time-step size but generally converges with strong order 1/2.The new scheme adds a correction term motivated by achieving higher-order convergence.
- Main results: For any time-step size τ, the new explicit scheme is almost surely positivity preserving and converges strongly with order 1.The result is stated for arbitrary p ∈ [1,∞) and finite T, with the corresponding error bound involving τ.
- Main results: The analysis also establishes pathwise error estimates and almost sure convergence with rate 1−ε, and adapts the scheme to Stratonovich SDEs.Numerical experiments compare the new method with the original positivity-preserving scheme and classical methods that do not preserve positivity.
- Analysis and scope: The proof decomposes the error into auxiliary terms and uses a main cancellation, after first establishing strong error estimates of order 1/2.The article also relates the scalar construction to first-order domain-preserving schemes for higher-dimensional systems.
2. Setting
The setting covers scalar Itô SDEs with nonnegative initial data and coefficients satisfying smoothness, bounded-derivative, and zero-at-the-origin conditions. The formulation factors drift and diffusion by the state variable and explains the additional requirements needed for the Stratonovich extension.
- Itô setting: The initial value is deterministic and nonnegative, while G and g are C2 with bounded first and second derivatives and satisfy G(0) = g(0) = 0.These assumptions imply linear growth bounds for both coefficients.
- Itô setting: Under these assumptions, the Itô SDE has a unique solution that remains nonnegative almost surely for all times and satisfies moment bounds.The proof uses Lipschitz continuity, a comparison principle, BDG estimates, and Grönwall’s lemma.
- Factorized formulation: Because G(0) = g(0) = 0, the coefficients admit the factorization G(x) = xF(x) and g(x) = xf(x).The resulting equivalent SDE writes both drift and diffusion as state-dependent factors multiplied by X(t).
- Factorized formulation: The factorized mappings F and f have bounded values and derivatives under the stated assumptions, with a corresponding bound for f′g.These properties support the numerical analysis of the factorized scheme.
- Stratonovich setting: The Stratonovich equation is converted to an equivalent Itô equation with a corrected drift before applying the positivity framework.This conversion preserves consistency because direct interval-wise factorization is not consistent with the Stratonovich interpretation.
- Stratonovich setting: The Stratonovich extension requires stronger regularity and boundedness conditions on g so that the corrected drift is continuous and bounded.For example, g may be assumed C3 with g and g3 bounded.
3. First-order positivity preserving numerical scheme
The paper constructs an explicit positivity-preserving scheme by freezing factored coefficients and solving an auxiliary linear SDE exactly, achieving first-order strong convergence while preserving nonnegativity for any step size.
- Itô scheme: The scheme freezes the auxiliary mappings F and f at Xn and solves the resulting linear SDE on each time interval.The exact auxiliary solution defines the update Xn+1.
- Itô scheme: The basic positivity-preserving scheme (23) preserves nonnegativity for every time-step size but generally has strong order 1/2.This motivates the higher-order construction.
- Main results: The new scheme (26) is positivity preserving almost surely for any time-step size, with Xn ≥ 0 for all grid points.This property follows from the scheme’s construction and is formalized in Proposition 5.
- Main results: Under the stated assumptions and sufficiently small steps, scheme (26) has strong order 1 convergence in mean square.The theorem provides an error bound proportional to τ for the 2p-th moment.
- Stratonovich extension: The method also yields pathwise convergence with rate 1−ε and adapts to Stratonovich equations through the equivalent Itô formulation.The Stratonovich scheme is domain preserving and has first-order strong convergence under appropriate assumptions.
- Numerical experiments: Numerical experiments observe first-order convergence for PP(1.0), while PP(0.5) and Euler–Maruyama converge with strong order 1/2.The first-order behavior matches the Milstein scheme in the reported Itô experiments.
4. Auxiliary results
The analysis introduces an auxiliary continuous-time process matching the numerical solution at grid points, then establishes moment and increment bounds needed for convergence. These bounds hold under a sufficiently small time-step condition and preserve nonnegativity.
- Auxiliary process: The auxiliary process coincides with the numerical solution at grid times and satisfies an SDE with coefficients frozen at the numerical state.On each interval, it solves d rX(t) = rX(t)F(X_n)dt + rX(t)f(X_n)dB(t).
- Regularity and positivity: The auxiliary process has continuous trajectories and remains nonnegative almost surely under the stated assumptions.Its nonnegativity follows from the positivity-preserving property of the scheme.
- Moment bounds: For every finite horizon and moment order, Proposition 9 provides uniform moment bounds for both the numerical solution and the auxiliary process when τ is sufficiently small.The admissible threshold τ_p and bound C_p(T,x_0) depend on the horizon, initial value, and moment order.
- Increment estimates: Proposition 10 bounds exact-solution and auxiliary-process increments by C_p(T,x_0)(t−t_n)^p, and hence by C_p(T,x_0)τ^p.These temporal regularity estimates are established for τ ≤ τ_p.
5. Proof of Theorem 6
The proof decomposes the grid-point error between the numerical and exact solutions, bounds the resulting error terms, and combines these estimates to establish first-order strong convergence. The argument first obtains order 1/2 and then improves it to order 1.
- Error decomposition: The proof represents the grid-point error e_n = X_n−X(t_n) through separate exact-solution and auxiliary-process decompositions.The auxiliary process is used because it solves a frozen-coefficient SDE on each time interval.
- Auxiliary error bounds: Lemmas 11 and 12 provide upper bounds of C_p(T,x_0)τ^2p for the principal groups of decomposed error terms.Lemma 12 additionally requires τ ≤ τ_p.
- Theorem 6: The final theorem yields a strong error estimate with order 1 for all p ∈ [1,∞), under the required time-step restriction.The proof combines the decomposition with the bounds from Lemmas 11, 12, and 13.
- Strong convergence: The convergence proof proceeds in two stages, first establishing strong order 1/2 and then strong order 1.The delicate term involves g′g, which is not assumed globally Lipschitz on R+.