A multiparameter stochastic integral and forward equations
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:\(\psi \to E\int fd\psi (.,w(.))=<h,\psi >\) is a continuous linear functional of \(\psi\). From (1) and the definition of p, the forward equation \(L^*p=h\) is obtained, with \(L^*=L^*_ 1...L^*_ d\), \(L^*_ j=-\partial /\partial x_ j+2^{- 1}\prod_{h\neq j}x_ h\partial^ 2/\partial z^ 2\). For \(d=2\), \(<h,\psi >\) is seen to reduce to the expectation of the sum of \(\psi\) evaluated on an a.s. finite well-defined random set of points, thus proving (2) and giving a probabilistic interpretation to its left-hand member. For \(d\geq 3\), the equation \(L^*p=h\) remains valid, but the paper shows what is the kind of difficulties to extend the conclusion that h is nonnegative.
scientific article; zbMATH DE number 3872412