Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_{\varphi }$ of Wiener measure and for a partition $0=t_0<t_1<\cdots <t_n<t_{n+1}=T$ of $[0,T]$, let $X_{n}C[0,T]\to {ℝ}^{n+1}$ and $X_{n+1}C[0,T]\to {ℝ}^{n+2}$ be given by $X_n\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_n\right))$ and $X_{n+1}\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_{n+1}\right))$, respectively. In this paper, using a simple formula for the conditional $w_{\varphi }$-integral of functions on $C[0,T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_{\varphi }$-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_{\varphi }$-integral...

Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $\parallel Y{\parallel }_W\le \left(2(\alpha +1)²\right)/(2\alpha +1)\parallel X{\parallel }_{L^{\infty }}$. Here W is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin...