A multiparameter stochastic integral and forward equations
type: Item
instance of: scholarly article
title: A multiparameter stochastic integral and forward equations
zbMATH Open document ID: 0547.60057
DOI: 10.1016/0304-4149(84)90162-5
author: Enrique M. Cabaña
published in: Stochastic Processes and their Applications
publication date: 1984-01-01T00:00:00Z
review text: The integral \(\int f(x)d\psi (x,w(x))\) of a nonanticipating process \(f:({\mathbb{R}}^+)^ d\times\Omega \to {\mathbb{R}}\) with respect to the composition of a \(C^{\infty}\) function \(\psi:({\mathbb{R}}^+)^ d\times {\mathbb{R}}\to {\mathbb{R}}\) ((x,z)\(\to\psi (x,z)\), \(x\in {\mathbb{R}}^ d\), \(z\in {\mathbb{R}})\) and a d-parameter Wiener process w, is defined by extending the obvious rule \((\int 1_{(y',y'']}d\psi (.,w(.)))_ x=\square_{(y'\wedge x,y''\wedge x]}\psi (.,w(.))\), where (y',y''] denotes the generalized interval \(\{y=(y_ 1,...,y_ d):\) y'\({}_ 1<y_ 1\leq y''\!_ 1,...,y'\!_ d<y_ d\leq y''\!_ d\}\), \(1_{(y',y'']}\) is its indicator function, \(\square\) denotes the multiple increment, and the infimum (\(\wedge)\) is taken componentwise. The differential operators \(L=L_ 1L_ 2...L_ d\), \(L_ j=\partial /\partial x_ j+2^{-1}\prod_{h\neq j}x_ j\partial^ 2/\partial z^ 2\) are introduced, and it is shown that the integral of nonanticipating f a.s. in \(L^ 2\) satisfies \[ (1)\quad E\int 1_{(0,x]}fd\psi (.,w(.))=E\int 1_{(0,x]}f(y)(L\psi)(y,w(y))dy. \] This particular kind of stochastic integral is used in the paper to obtain a simpler proof of the already known forward inequality \[ (2)\quad (\frac{\partial}{\partial x_ 1}-\frac{1}{2}x_ 2\frac{\partial^ 2}{\partial z^ 2})(\frac{\partial}{\partial x_ 2}- \frac{1}{2}x_ 1\frac{\partial^ 2}{\partial z^ 2})p\geq 0 \] [see the author and \textit{M. Wschebor}, Ann. Probab. 10, 289-302 (1982; Zbl 0532.60072)] for the probability density \(p(x_ 1,x_ 2,z)\) at z of a two-parameter Wiener process \(w(x_ 1,x_ 2)\) killed when it reaches given barriers a,b \((-\infty\leq a<0<b\leq\infty )\). A precise definition of p is as follows: let \(M(x)=\max\{w(y):y\leq x\}\), \(m(x)=\min\{w(y):y\leq x\}\), and \(P(x,z)=P\{w(x)\leq z,M(x)<b\), \(m(x)>a\}\); then \(p(x,z)=(\partial /\partial z)P(x,z).\) It is shown that if f is the indicator function of the random set \(\{M(x)<b,m(x)>a\}\), then h:\(\psi \to E\int fd\psi (.,…
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 3872412
zbMATH Keywords: two-parameter Wiener process
MaRDI profile type: Publication
full work available at URL: 0304-4149(84)90162-5
OpenAlex ID: W2136465158
cites work: The two-parameter Brownian bridge: Kolmogorov inequalities and upper and lower bounds for the distribution of the maximum
cites work: Q5623011
cites work: Martingales and stochastic integrals for processes with a multi-dimensional parameter
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description: scientific article; zbMATH DE number 3872412