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  1. arXiv:1901.04039 (Published 2019-01-13)

    Change of Variables with Local Time on Surfaces for Jump Processes

    Daniel Wilson
    Comments: 20 pages
    Categories: math.PR

    The `local time on curves' formula of Peskir provides a stochastic change of variables formula for a function whose derivatives may be discontinuous over a time-dependent curve, a setting which occurs often in applications in optimal control and beyond. This formula was further extended to higher dimensions and to include processes with jumps under conditions which may be hard to verify in practice. We build upon the work of Du Toit in weakening the required conditions by allowing semimartingales with jumps. In addition, under vanishing of the sectional first derivative (the so-called `smooth fit' condition), we show that the classical It\^o formula still holds under general conditions.

  2. arXiv:1612.00498 (Published 2016-12-01)

    Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

    Zhe Chen, Lasse Leskelä, Lauri Viitasaari
    Comments: 35 pages
    Categories: math.PR
    Subjects: 60H05, 60G22, 26A33, 60G07, 60G15

    In this article we study the existence of pathwise Stieltjes integrals of the form $\int f(X_t)\, dY_t$ for nonrandom, possibly discontinuous, evaluation functions $f$ and H\"older continuous random processes $X$ and $Y$. We discuss a notion of sufficient variability for the process $X$ which ensures that the paths of the composite process $t \mapsto f(X_t)$ are almost surely regular enough to be integrable. We show that the pathwise integral can be defined as a limit of Riemann-Stieltjes sums for a large class of discontinuous evaluation functions of locally finite variation, and provide new estimates on the accuracy of numerical approximations of such integrals, together with a change of variables formula for integrals of the form $\int f(X_t) \, dX_t$.

  3. arXiv:1303.6452 (Published 2013-03-26)

    Increasing processes and the change of variables formula for non-decreasing functions

    Jean Bertoin, Marc Yor
    Categories: math.PR

    Given an increasing process $(A_t)_{t\geq 0}$, we characterize the right-continuous non-decreasing functions $f: \R_+\to \R_+$ that map $A$ to a pure-jump process. As an example of application, we show for instance that functions with bounded variations belong to the domain of the extended generator of any subordinators with no drift and infinite L\'evy measure.

  4. arXiv:1004.1071 (Published 2010-04-07, updated 2010-05-28)

    When does fractional Brownian motion not behave as a continuous function with bounded variation?

    Ehsan Azmoodeh, Heikki Tikanmäki, Esko Valkeila
    Journal: Statist. Probab. Lett. 80 (2010), no. 19-20, 1543-1550
    Categories: math.PR

    If we compose a smooth function g with fractional Brownian motion B with Hurst index H > 1/2, then the resulting change of variables formula [or It/^o- formula] has the same form as if fractional Brownian motion would be a continuous function with bounded variation. In this note we prove a new integral representation formula for the running maximum of a continuous function with bounded variation. Moreover we show that the analogue to fractional Brownian motion fails.