Pathwise Uniqueness and Non-explosion Property of Skorohod SDEs with a Class of Non-Lipschitz Coefficients and Non-smoot
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Pathwise Uniqueness and Non-explosion Property of Skorohod SDEs with a Class of Non-Lipschitz Coefficients and Non-smooth Domains Masanori Hino1 · Kouhei Matsuura2 · Misaki Yonezawa3 Received: 2 March 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Here, we study stochastic differential equations with a reflecting boundary condition. We provide sufficient conditions for pathwise uniqueness and non-explosion property of solutions in a framework admitting non-Lipschitz continuous coefficients and nonsmooth domains. Keywords Skorohod SDE · Non-Lipschitz coefficient · Pathwise uniqueness · Non-explosion property Mathematics Subject Classification 60H10
1 Introduction Let w = {w(t)}t≥0 be a one-dimensional Brownian motion on R starting in [0, ∞). A reflecting Brownian motion ξ = {ξ(t)}t≥0 on [0, ∞) is characterized by the solution
This study was supported by JSPS KAKENHI Grant Number JP19H00643.
B
Kouhei Matsuura [email protected] Masanori Hino [email protected] Misaki Yonezawa [email protected]
1
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
2
Institute of Mathematics, University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki 305-8571, Japan
3
Daiwa Securities Co. Ltd., Chiyoda-ku, Tokyo 100-6752, Japan
123
Journal of Theoretical Probability
of the following (pathwise) equation: ⎧ ⎪ ⎨ξ = w + φ, φ is non-decreasing on [0, ∞), φ(0) = 0, and ⎪ t ⎩ φ(t) = 0 1{0} (ξ(s)) dφ(s), t ≥ 0.
(1.1)
Equation (1.1) for a continuous function w = {w(t)}t≥0 with a nonnegative initial value is called the Skorohod problem for ((0, ∞), w). This equation has a unique solution described as w(t), 0 ≤ t ≤ τ, ξ(t) = w(t) − inf{w(s) | τ ≤ s ≤ t}, t > τ, where τ = inf{s > 0 | w(s) < 0}. Given a multidimensional domain D ⊂ Rd and an Rd -valued continuous function w on [0, ∞), the Skorohod problem for (D, w) can be considered similarly to (1.1) (see [10] for a precise formulation). Tanaka [11, Theorem 2.1] showed that the Skorohod problem has a unique solution if D is a convex domain. Saisho [10, Theorem 4.1] extended this result to more general domains satisfying conditions (A) and (B), which are defined in Sect. 2. The class of domains D satisfying these conditions includes all convex domains and domains with a bounded C 2 -boundary and admits some non-smoothness. The Skorohod problem is generalized to a stochastic differential equation (SDE) as follows. Let (Ω, F, P) be a probability space. Let D be a domain of Rd and denote its closure by D. Given an Rd -valued function b and a d × d matrix-valued function σ on [0, ∞) × Ω × D, we are concerned with the following SDE:
dX (t) = σ (t, ·, X (t)) dB(t) + b(t, ·, X (t)) dt + dΦ X (t), t ≥ 0, X (0) ∈ D.
(1.2)
Here, {B(t)}t≥0 denotes a d-dimensional Brownian motion, and Φ X is a reflection term, which is an unknown continuous function of bounded variation with properties (2.2), (2.3), and (2.4), which are presented below. Equation (1.2) is called a Skorohod SDE, which is a natural generalization of
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