Related papers: Integration with respect to local time and Ito's f…
We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise…
For $0<\alpha \leq 2$ and $0<H<1$, an $\alpha$-time fractional Brownian motion is an iterated process $Z = \{Z(t)=W(Y(t)), t \ge 0\}$ obtained by taking a fractional Brownian motion $\{W(t), t\in \RR{R} \}$ with Hurst index $0<H<1$ and…
We show that, under certain smoothness conditions, a Brownian martingale at a fixed time can be represented as an exponential of its value at a later time. The time-dependent generator of this exponential operator is equal to one half times…
A generalized It${\hat {\rm o}}$ formula for time dependent functions of two-dimensional continuous semi-martingales is proved. The formula uses the local time of each coordinate process of the semi-martingale, left space and time first…
We give the correct condition for existence of the $k$-th derivative of the intersection local time for fractional Brownian motion, which was originally discussed in [Guo, J., Hu, Y., and Xiao, Y., Higher-order derivative of intersection…
We derive an Ito-formula for the Dawson-Watanabe superprocess, a well-known class of measure-valued processes, extending the classical Ito-formula with respect to two aspects. Firstly, we extend the state-space of the underlying process…
In this paper we give necessary and sufficient conditions for a cylindrical continuous local martingale to be the stochastic integral with respect to a cylindrical Brownian motion. In particular we consider the class of cylindrical…
Let $(W,H,\mu)$ be the classical Wiener space on $\R^d$. Assume that $X=(X_t)$ is a diffusion process satisfying the stochastic differential equation $dX_t=\sigma(t,X)dB_t+b(t,X)dt$, where $\sigma:[0,1]\times C([0,1],\R^n)\to \R^n\otimes…
Let ${\mathscr L}$ be the local time of $G$-Brownian motion $B$. In this paper, we prove the existence of the quadratic covariation $<f(B),B>_{t}$ and the integral $\int_{\mathbb R}f(x){\mathscr L}(dx,t)$. Moreover, a sublinear version of…
We study the local (in time) expansion of a continuous-time process and its conditional moments, including the process' characteristic function. The expansions are conducted by using the properties of the (time-extended) Ito signature, a…
We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic…
In this paper we study the local times of Brownian motion from the point of view of algorithmic randomness. We introduce the notion of effective local time and show that any path which is Martin-L\"of random with respect to the Wiener…
In this paper we generalize a representation formula for the local time of a function of a semimartingale due to Coquet and Ouknine \cite{Ouknine} , our formula being a pointwise equality between two processes we show in addition that the…
The objective of this paper is to study the local time and Tanaka formula of symmetric $G$-martingales. We introduce the local time of $G$-martingales and show that they belong to $G$-expectation space $L_{G}^{2}(\Omega _{T})$. The…
In this paper we compute the $\frac 43$-variation of the derivative of the self-intersection Brownian local time $\gamma_t=\int_0^t \int_0^u \delta '(B_u-B_s)dsdu\,, t\ge 0$, applying techniques from the theory of fractional martingales.
These notes contains an introduction to the theory of Brownian and diffusion local time, as well as its relations to the Tanaka Formula, the extended Ito-Tanaka formula for convex functions, the running maximum process, and the theory of…
We give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. We prove that $ \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon)\wedge t}^+ \indi_{\{X_u \leqslant 0\}} du +…
A peculiar feature of It\^o's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative…
It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct…
For non-anticipative functionals, differentiable in Chitashvili's sense, the It\^o formula for cadlag semimartingales is proved. Relations between different notions of functional derivatives are established.