Exit Boundaries of Multidimensional SDEs

Russell Lyons

arXiv:1902.06735·math.PR·November 24, 2020

Exit Boundaries of Multidimensional SDEs

Russell Lyons

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TL;DR

This paper proves that solutions to certain multidimensional stochastic differential equations (SDEs) with Lipschitz coefficients do not reach the set where coefficients vanish unless starting there, clarifying boundary behavior.

Contribution

It establishes a boundary non-attainment result for multidimensional SDEs with Lipschitz coefficients, extending understanding of their solution paths.

Findings

01

Solutions never reach the zero-coefficient set unless starting there

02

Provides conditions under which boundary hitting is impossible

03

Enhances theoretical understanding of multidimensional SDE behavior

Abstract

We show that solutions to multidimensional SDEs with Lipschitz coefficients and driven by Brownian motion never reach the set where all coefficients vanish unless the initial position belongs to that set.

Equations25

Λ := {x \in R^{d}; σ (x) = 0, b (x) = 0} .

Λ := {x \in R^{d}; σ (x) = 0, b (x) = 0} .

X_{t} = x + \int_{0}^{t} σ (X_{s}) d B_{s} + \int_{0}^{t} b (X_{s}) d s (t \geq 0) .

X_{t} = x + \int_{0}^{t} σ (X_{s}) d B_{s} + \int_{0}^{t} b (X_{s}) d s (t \geq 0) .

P [\forall t \geq 0 X_{t} \in / Λ] = 1.

P [\forall t \geq 0 X_{t} \in / Λ] = 1.

T_{A}:=\inf\bigl{\{}t\geq 0\,;\;\|\sigma(X_{t})\|^{2}+\|b(X_{t})\|^{2}=A\bigr{\}}.

T_{A}:=\inf\bigl{\{}t\geq 0\,;\;\|\sigma(X_{t})\|^{2}+\|b(X_{t})\|^{2}=A\bigr{\}}.

S_{k} := T_{A / 2^{k + 1}} \land T_{A / 2^{k - 1}} .

S_{k} := T_{A / 2^{k + 1}} \land T_{A / 2^{k - 1}} .

\displaystyle\operatorname{\mathbf{E}\mathopen{}}_{x}\mkern-1.5mu\bigl{[}\|x-X_{t\wedge S_{k}}\|^{2}

\displaystyle\operatorname{\mathbf{E}\mathopen{}}_{x}\mkern-1.5mu\bigl{[}\|x-X_{t\wedge S_{k}}\|^{2}

\displaystyle\leq(m+1)\operatorname{\mathbf{E}\mathopen{}}_{x}\mkern-1.5mu\biggl{[}\int_{0}^{t\wedge S_{k}}\|\sigma(X_{u})\|^{2}\,\mathrm{d}u\biggr{]}+(m+1)\operatorname{\mathbf{E}\mathopen{}}_{x}\mkern-1.5mu\biggl{[}\int_{0}^{t\wedge S_{k}}\|b(X_{u})\|^{2}\,\mathrm{d}u\biggr{]}

\leq (m + 1) \cdot \frac{A}{2 ^{k - 1}} \cdot t

\displaystyle\operatorname{\mathbf{E}\mathopen{}}_{x}\mkern-1.5mu\Bigl{[}

\displaystyle\operatorname{\mathbf{E}\mathopen{}}_{x}\mkern-1.5mu\Bigl{[}

\displaystyle\leq\operatorname{\mathbf{E}\mathopen{}}_{x}\mkern-1.5mu\Bigl{[}\|\sigma(x)+\sigma(X_{t\wedge S_{k}})\|\cdot\|\sigma(x)-\sigma(X_{t\wedge S_{k}})\|

\displaystyle\leq\operatorname{\mathbf{E}\mathopen{}}_{x}\mkern-1.5mu\Bigl{[}\bigl{(}\|\sigma(x)\|+\|\sigma(X_{t\wedge S_{k}})\|+\|b(x)\|+\|b(X_{t\wedge S_{k}})\|\bigr{)}\cdot K\cdot\|x-X_{t\wedge S_{k}}\|\,;\;S_{k}\leq 1\Bigr{]}

\displaystyle\leq 2\cdot\Bigl{(}\frac{A+2A}{2^{k}}\Bigr{)}^{1/2}\cdot K\cdot\operatorname{\mathbf{E}\mathopen{}}_{x}\mkern-1.5mu\bigl{[}\|x-X_{t\wedge S_{k}}\|^{2}\,;\;S_{k}\leq 1\bigr{]}^{1/2},

\bigl{|}\|\sigma(x)\|^{2}+\|b(x)\|^{2}-\|\sigma(X_{t\wedge S_{k}})\|^{2}-\|b(X_{t\wedge S_{k}})\|^{2}\bigr{|}\mathbb{1}_{[S_{k}\leq 1]}\geq\frac{A}{2^{k+1}}.

\bigl{|}\|\sigma(x)\|^{2}+\|b(x)\|^{2}-\|\sigma(X_{t\wedge S_{k}})\|^{2}-\|b(X_{t\wedge S_{k}})\|^{2}\bigr{|}\mathbb{1}_{[S_{k}\leq 1]}\geq\frac{A}{2^{k+1}}.

\operatorname{\mathbf{P}\mathopen{}}_{\!x}[S_{k}\leq t]\leq\frac{2^{k+1}}{A}\cdot 2\Bigl{(}\frac{3A}{2^{k}}\Bigr{)}^{1/2}\cdot K\cdot\sqrt{(m+1)\cdot\frac{A}{2^{k-1}}\cdot t}=C\sqrt{t}

\operatorname{\mathbf{P}\mathopen{}}_{\!x}[S_{k}\leq t]\leq\frac{2^{k+1}}{A}\cdot 2\Bigl{(}\frac{3A}{2^{k}}\Bigr{)}^{1/2}\cdot K\cdot\sqrt{(m+1)\cdot\frac{A}{2^{k-1}}\cdot t}=C\sqrt{t}

∥ σ (x) ∥^{2} + ∥ b (x) ∥^{2} \geq A / 2^{k} ⟹ P_{x} [T_{A / 2^{k + 1}} \geq t_{0}] \geq 1/2.

∥ σ (x) ∥^{2} + ∥ b (x) ∥^{2} \geq A / 2^{k} ⟹ P_{x} [T_{A / 2^{k + 1}} \geq t_{0}] \geq 1/2.

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\SHORTTITLE

Exit Boundaries of Multidimensional SDEs \TITLEExit Boundaries of Multidimensional SDEs \AUTHORSRussell Lyons111Department of Mathematics, 831 E. 3rd St., Indiana University, Bloomington, IN 47405-7106. \[email protected]. Partially supported by NSF grant DMS-1612363. \KEYWORDSInaccessible; Lipschitz; singular \AMSSUBJ60H10 \SUBMITTEDFeb. 7, 2019 \ACCEPTED \VOLUME0 \YEAR2012 \PAPERNUM0 \DOIvVOL-PID \ABSTRACTWe show that solutions to multidimensional SDEs with Lipschitz coefficients and driven by Brownian motion never reach the set where all coefficients vanish unless the initial position belongs to that set.

The classification of isolated singular points of a 1-dimensional SDE driven by Brownian motion is complete and exhibits several types of behavior: see [1, Fig. 2.2] for a good summary. For example, as has long been known, if $X$ is a (weak) solution to $E_{x}(\sigma,0)$ with $\sigma^{-2}$ being nonzero and locally integrable in some interval $(0,a]$ and $x\in(0,a)$ , then the probability that $X_{t}$ ever reaches 0 is positive (i.e., 0 is accessible) iff $\int_{0}^{a}y\,\sigma(y)^{-2}\,\mathrm{d}y<\infty$ . Much less is known in higher dimensions. In particular, the following theorem that makes the usual assumption of Lipschitz coefficients seems to be new:

Theorem.

Let $d,m\in{\mathbb{N}}^{+}$ . Let $B=(B^{(1)},\dots,B^{(m)})$ be $m$ -dimensional Brownian motion. Let $\sigma\colon{\mathbb{R}}^{d}\to M_{d\times m}({\mathbb{R}})$ and $b\colon{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ be Lipschitz. Write

[TABLE]

Suppose that $X$ solves $E_{x}(\sigma,b)$ , i.e.,

[TABLE]

If $x\notin\Lambda$ , then

[TABLE]

In other words, the set $\Lambda$ is inaccessible.

Proof.

We use the Frobenius norm $\|M\|:=\sqrt{\operatorname{\mathrm{Tr}}(M^{*}M)}$ for a matrix, $M$ . For $A>0$ , define the stopping time

[TABLE]

Fix $A>0$ . For $k\in{\mathbb{N}}^{+}$ , write

[TABLE]

If $x$ is such that $\|\sigma(x)\|^{2}+\|b(x)\|^{2}=A/2^{k}$ , then $\forall t\geq 0$

[TABLE]

and

[TABLE]

where $K$ is a bound for the Lipschitz constants. If, in addition, $t\leq 1$ and $S_{k}\leq t$ , then

[TABLE]

Putting these inequalities together, we obtain $\forall{t\leq 1}$

[TABLE]

for some constant, $C$ , depending only on $m$ and $K$ .

Choose $t_{0}\in(0,1)$ so that $C\sqrt{t_{0}}\leq 1/2$ . Then by the strong Markov property, if $k\in{\mathbb{N}}^{+}$ , $\>A>0$ , and $x\in{\mathbb{R}}^{d}$ ,

[TABLE]

Given $x\notin\Lambda$ , choose $A:=\|\sigma(x)\|^{2}+\|b(x)\|^{2}$ and express the time to reach $\Lambda$ as $\sum_{k\geq 0}\bigl{(}T_{A/2^{k+1}}-T_{A/2^{k}}\bigr{)}$ . By the strong Markov property, infinitely many of these terms are at least $t_{0}$ a.s., whence the total time is infinite a.s. ∎

Acknowledgment. We are grateful to Jean-François Le Gall for showing us a similar idea in a different context.

Bibliography1

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] Alexander S. Cherny and Hans-Jürgen Engelbert, Singular stochastic differential equations , Lecture Notes in Mathematics, 1858 . Springer-Verlag, Berlin, 2005. MR 2112227

TL;DR

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Theorem**.**

Proof.

Theorem.