Related papers: Pathwise Stieltjes integrals of discontinuously ev…
We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of $p$-th variation along a sequence of time…
We define compositions $\varphi(X)$ of H\"older paths $X$ in $\mathbb{R}^n$ and functions of bounded variation $\varphi$ under a relative condition involving the path and the gradient measure of $\varphi$. We show the existence and…
Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large values of $p$. In this context, rough…
In the existing works, stochastic sets $\mathbb{B}$ of interval type, along with $\mathbb{B}$-stochastic processes, were introduced within the framework of stochastic analysis. In this paper, we undertake the construction of…
We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 + >...))) where the s_n are independent random variables with the same gamma distribution. For every realisation of the sequence, S(t) defines a Stieltjes function.…
We consider the class of stationary-increment harmonizable stable processes with infinite control measure, which most notably includes real harmonizable fractional stable motions. We give conditions for the integrability of the paths of…
Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations -- such as performing a change of the integration path -- one would like to carry out in…
In this work we study Stieltjes differential systems of which the derivators are allowed to change sign. This leads to the definition of the notion of \emph{function of controlled variation}, a characterization of precompact sets of…
Integral equations of the form $$ x(t)=x(t_0)+\int_{t_0}^t d[A]\,x=f(t)-f(t_0)$$ are natural generalizations of systems of linear differential equations. Their main goal is that they admit solutions which need not be absolutely continuous.…
This essay explores the meaning of stochastic differential equations and stochastic integrals. It sets these subjects in a context of Riemann-Stieltjes integration. It is intended as a comment or supplement to \cite{MTRV}.
Using Riemann-Stieltjes methods for integrators of bounded $p$-variation we define a pathwise integral driven by a fractional L\'{e}vy process (FLP). To explicitly solve general fractional stochastic differential equations (SDEs) we…
In this work, we establish pathwise functional It\^o formulas for non-smooth functionals of real-valued continuous semimartingales. Under finite $(p,q)$-variation regularity assumptions in the sense of two-dimensional Young integration…
In this work, we introduce a theory of stochastic integration with respect to symmetric $\alpha$-stable cylindrical L\'evy processes. Since $\alpha$-stable cylindrical L\'evy processes do not enjoy a semi-martingale decomposition, our…
In this paper, motivated by physical considerations, we introduce the notion of modified Riemann sums of Riemann-Stieltjes integrable functions, show that they converge, and compute them explicitely under various assumptions.
We develop a Fourier approach to rough path integration, based on the series decomposition of continuous functions in terms of Schauder functions. Our approach is rather elementary, the main ingredient being a simple commutator estimate,…
We study the question, whether a Riemann-Stieltjes integral of a positive continuous function with respect to a non-negative function of bounded variation is positive.
In this work, we extend the concept of the Stieltjes derivative to encompass left-continuous derivators with bounded variation, thereby relaxing the monotonicity constraint. This generalization necessitates a refined definition of the…
We prove that the stochastic differential equation $$ Y_{s,t}(x) = Y_{s,s}(x) + \int_0^{t-s} f(Y_{s,s+u}(x)) dX_{s+u}, Y_{s,s}(x)=x\in\R^d. $$ driven by a L\'evy process whose paths have finite p-variation almost surely for some $p\in[1,2)$…
We give a new estimate on Stieltjes integrals of H\"older continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with H\"older continuous forcing. We construct stochastic…
Some new integrals involving the Stieltjes constants are developed in this paper.