A note on L2-estimates for stable integrals with drift

Vladimir Kurenok

doi:10.1090/S0002-9947-07-04234-1

A note on L₂-estimates for stable integrals with drift

Vladimir Kurenok

Department of Electrical & Systems Engineering

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

Let X be of the form X_t = ∫^t₀ b _sdZ_s + ∫^t₀ a_sds, t ≥ 0, where Z is a symmetric stable process of index α ε (1, 2) with Z₀ = 0. We obtain various L₂-estimates for the process X. In particular, for m ε ℕ, t ≥ 0, and any measurable, nonnegative function f we derive the inequality E ∫ ^tΛTm(X)₀ |b_s|^αf(X _s)ds ≤ N||f||2,m. As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation dX_t = b(X_t-)dZ_t + a(X_t)dt for any initial value x0 ε ℝ.

Original language	English
Pages (from-to)	925-938
Number of pages	14
Journal	Transactions of the American Mathematical Society
Volume	360
Issue number	2
DOIs	https://doi.org/10.1090/S0002-9947-07-04234-1
State	Published - Feb 2008

Keywords

Bounded drift
Krylov's estimates
One-dimensional stochastic equations
Symmetric stable processes
Weak convergence

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10.1090/S0002-9947-07-04234-1

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title = "A note on L2-estimates for stable integrals with drift",

abstract = "Let X be of the form Xt = ∫t0 b sdZs + ∫t0 asds, t ≥ 0, where Z is a symmetric stable process of index α ε (1, 2) with Z0 = 0. We obtain various L2-estimates for the process X. In particular, for m ε ℕ, t ≥ 0, and any measurable, nonnegative function f we derive the inequality E ∫ tΛTm(X)0 |bs|αf(X s)ds ≤ N||f||2,m. As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation dXt = b(Xt-)dZt + a(Xt)dt for any initial value x0 ε ℝ.",

keywords = "Bounded drift, Krylov's estimates, One-dimensional stochastic equations, Symmetric stable processes, Weak convergence",

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T1 - A note on L2-estimates for stable integrals with drift

AU - Kurenok, Vladimir

PY - 2008/2

Y1 - 2008/2

N2 - Let X be of the form Xt = ∫t0 b sdZs + ∫t0 asds, t ≥ 0, where Z is a symmetric stable process of index α ε (1, 2) with Z0 = 0. We obtain various L2-estimates for the process X. In particular, for m ε ℕ, t ≥ 0, and any measurable, nonnegative function f we derive the inequality E ∫ tΛTm(X)0 |bs|αf(X s)ds ≤ N||f||2,m. As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation dXt = b(Xt-)dZt + a(Xt)dt for any initial value x0 ε ℝ.

AB - Let X be of the form Xt = ∫t0 b sdZs + ∫t0 asds, t ≥ 0, where Z is a symmetric stable process of index α ε (1, 2) with Z0 = 0. We obtain various L2-estimates for the process X. In particular, for m ε ℕ, t ≥ 0, and any measurable, nonnegative function f we derive the inequality E ∫ tΛTm(X)0 |bs|αf(X s)ds ≤ N||f||2,m. As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation dXt = b(Xt-)dZt + a(Xt)dt for any initial value x0 ε ℝ.

KW - Bounded drift

KW - Krylov's estimates

KW - One-dimensional stochastic equations

KW - Symmetric stable processes

KW - Weak convergence

UR - https://www.scopus.com/pages/publications/54049083974

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DO - 10.1090/S0002-9947-07-04234-1

M3 - Article

AN - SCOPUS:54049083974

SN - 0002-9947

VL - 360

SP - 925

EP - 938

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 2

ER -

A note on L₂-estimates for stable integrals with drift

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Keywords

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