TY - JOUR
AU - Kachanovsky, N.A.
PY - 2011/06/30
Y2 - 2024/05/23
TI - Clark-Ocone type formulas in the Meixner white noise analysis
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 3
IS - 1
SE - Scientific articles
DO -
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/3087
SP - 56–72
AB - <p>In the classical Gaussian analysis the Clark-Ocone formula allows to reconstruct an integrand if we know the Itô stochastic integral. This formula can be written in the form $$ F=\mathbf EF+\int\mathbf E\big\{\partial_t F|_{\mathcal F_t}\big\} dW_t, $$ where a function (a random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf E$ $-$ the expectation; $\mathbf E\big\{\circ|_{\mathcal F_t}\big\}$ $-$ the conditional expectation with respect to a full $\sigma$-algebra $\mathcal F_t$ that is generated by the Wiener process $W$ up to the point of time $t$; $\partial_\cdot F$ $-$ the Hida derivative of $F$; $\int\circ (t)dW_t$ $-$ the Itô stochastic integral with respect to the Wiener process.</p><p>In this paper, we explain how to reconstruct an integrand in the case when instead of the Gaussian measure one considers the so-called generalized Meixner measure $\mu$ (depending on parameters, $\mu$ can be the Gaussian, Poissonian, Gamma measure etc.) and obtain corresponding Clark-Ocone type formulas.</p>
ER -