Related papers: On Change of Variable Formulas for non-anticipativ…
The constructive martingale representation theorem of functional It\^o calculus is extended, from the space of square integrable martingales, to the space of local martingales. The setting is that of an augmented filtration generated by a…
The main objective consists in generalizing a well-known It{\^o} formula of J. Jacod and A. Shiryaev: given a c{\`a}dl{\`a}g process S, there is an equivalence between the fact that S is a semimartingale with given characteristics (B^k , C,…
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…
We investigate differentiability of functions defined on regions of the real quaternion field and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analog of the Cauchy integral as well as…
A comparison principle for stochastic integro-differential equations driven by Levy processes is proved. This result is obtained via an extension of an Ito formula from [11] for the square of the norm of the positive part of $L_2-$valued,…
We develop the functional It\^o/path-dependent calculus with respect to fractional Brownian motion with Hurst parameter $H> \frac{1}{2}$. Firstly, two types of integrals are studied. The first type is Stratonovich integral, and the second…
We establish noncommutative analogs of some well-known large deviation inequalities for noncommutative random variables. Firstly, for the noncommutative independent case, we characterize the uniformly exponential integrability of random…
This note studies the martingale property of a nonnegative, continuous local martingale Z, given as a nonanticipative functional of a solution to a stochastic differential equation. The condition states that Z is a (uniformly integrable)…
We consider additive functionals as a time and space-dependent function of a diffusion corresponding to nonhomogeneous uniformly elliptic divergence form operator. We show that if the function belongs to natural domain of strong solutions…
Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…
We provide an It\^o's formula for $C^1$-functionals of flows of conditional marginal distributions of continuous semimartingales. This is based on the notion of weak Dirichlet process, and extends the $C^1$-It\^o's formula in Gozzi and…
We consider decompositions of processes of the form $Y=f(t,X_t)$ where $X$ is a semimartingale. The function $f$ is not required to be differentiable, so It\^{o}'s lemma does not apply. In the case where $f(t,x)$ is independent of $t$, it…
Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
We prove new results on the derivative of the Minkowski question mark function. Some of our theorems are non-improvable.
Recently, functional It\=o calculus has been introduced and developed in finite dimension for functionals of continuous semimartingales. With different techniques, we develop a functional It\=o calculus for functionals of Hilbert…
We show that a differential version of the classical Chebyshev-Markov-Stieltjes inequalities holds for a broad family of weight functions. Such a differential version appears to be new. Our results apply to weight functions which are…
Motivated by the theory of large deviations, we introduce a class of non-negative non-linear functionals that have a variational "rate function" representation.
We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians…
We show an It\^ o's formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in It\^o's s formula as…