A note on 𝐿₂-estimates for stable integrals with drift

Kurenok, Vladimir

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

A note on $L_2$-estimates for stable integrals with drift
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by Vladimir Kurenok PDF

Trans. Amer. Math. Soc. 360 (2008), 925-938 Request permission

Abstract:

Let $X$ be of the form $X_t=\int _0^tb_sdZ_s+\int _0^ta_sds, t\ge 0,$ where $Z$ is a symmetric stable process of index $\alpha \in (1,2)$ with $Z_0=0$. We obtain various $L_2$-estimates for the process $X$. In particular, for $m\in \mathbb N, t\ge 0,$ and any measurable, nonnegative function $f$ we derive the inequality \[ {\mathbf E}\int _0^{t\land \tau _m(X)}|b_s|^{\alpha }f(X_s)ds\le N\|f\|_{2,m}.\] 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 $x_0\in \mathbb R$.

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