arXiv subjects

Pietro Siorpaes

Publications and source records attributed to Pietro Siorpaes.

14 recordsLinked to original sources

arXiv2025

An abstract decomposition of measures and its many applications

We consider a little-known abstract decomposition result for positive measures due to Dellacherie, and show that it yields many decompositions of measures, several of which are new. We then extend Dellacherie's result to (controlled) vector measures, and apply it to obtain a decomposition of semimartingales due to Bichteler, on which we improve. Then, we investigate how the outputs of the decomposition depend on its inputs, in particular characterising the two elements of the decomposition as projections in the sense of Riesz spaces and of metric spaces. Finally, we prove a decomposition theorem for strictly positive operators on Riesz spaces which generalises Dellacherie's Theorem.

math.PR↗

arXiv2025

Stretched Brownian Motion: convergence of dual optimising sequences

We consider an irreducible pair $μ\leq_c ν$ of probability measures on $\mathbb{R}^d$ in convex order. In arXiv:2306.11019, Backhoff, Beiglböck, Schachermayer and Tschiderer have shown that the Stretched Brownian Motion from $μ$ to $ν$ is a Bass martingale, that there exists a dual optimiser $ψ_{lim}$, and the following somewhat surprising convergence result: by adding affine functions, one can make any dual optimising sequence $(ψ_n)_n$ (satisfying some minor technical conditions) converge pointwise to $ψ_{lim}$, save possibly on the relative boundary of the convex hull of the support of $ν$. In the present paper we deal with the more delicate issue of convergence on said boundary, showing in particular that $ψ_{lim}$ is $ν$ a.s. finite, and $(ψ_n)_n$ converges to $ψ_{lim}$ in $ν$-measure.

math.PR↗

arXiv2022

How to quantise probabilities while preserving their convex order

We introduce an algorithm which, given probabilities $μ\leq_{\text{cx}} ν$ in convex order and defined on a separable Banach space $B$, constructs finitely-supported approximations $μ_n \to μ, ν_n\to ν$ which are in convex order $μ_n \leq_{\text{cx}} ν_n$. We provide upper-bounds for the speed of convergence, in terms of the Wasserstein distance. We discuss the (dis)advantages of our algorithm and its link with the discretisation of the Martingale Optimal Transport problem, and we illustrate its implementation with numerical examples. We study the operation which, given $μ$/$ν$ and some (finite) partition of $B$, outputs $μ_n$/$ν_n$, showing that applied to a probability $γ$ and to all partitions it outputs the set of all probabilities $ζ\leq_{\text{cx}} γ$.

math.PR↗

arXiv2021

Local times and Tanaka--Meyer formulae for càdlàg paths

Three concepts of local times for deterministic c{à}dl{à}g paths are developed and the corresponding pathwise Tanaka--Meyer formulae are provided. For semimartingales, it is shown that their sample paths a.s. satisfy all three pathwise definitions of local times and that all coincide with the classical semimartingale local time. In particular, this demonstrates that each definition constitutes a legit pathwise counterpart of probabilistic local times. The last pathwise construction presented in the paper expresses local times in terms of normalized numbers of interval crossings and does not depend on the choice of the sequence of grids. This is a new result also for c{à}dl{à}g semimartingales, which may be related to previous results of Nicole El~Karoui and Marc Lemieux.

math.PR↗

arXiv2019

Differentiation of measures on a non-separable space, and the Radon-Nikodym theorem

Given positive measures $ν,μ$ on an arbitrary measurable space $(Ω, \mathcal F)$, we construct a sequence of finite partitions $(π_n)_n$ of $(Ω, \mathcal F)$ s.t. $$ \sum_{A\in π_n: μ(A)>0} 1_{A} \frac{ν(A)}{μ(A)} \longrightarrow \frac{dν^a}{dμ} \quad μ\text{ a.e. as } n\to \infty . $$ As an application, we modify the probabilistic proof of the Radon-Nikodym Theorem so that it uses convergence along a properly chosen sequence (instead of along a net), and so that it does not rely on the martingale convergence theorem (nor any probability theory), obtaining a completely elementary proof.

math.CA↗

arXiv2017

Structure of martingale transports in finite dimensions

We study the structure of martingale transports in finite dimensions. We consider the family $\mathcal{M}(μ,ν) $ of martingale measures on $\mathbb{R}^N \times \mathbb{R}^N$ with given marginals $μ,ν$, and construct a family of relatively open convex sets $\{C_x:x\in \mathbb{R}^N \}$, which forms a partition of $\mathbb{R}^N$, and such that any martingale transport in $\mathcal{M}(μ,ν) $ sends mass from $x$ to within $\overline{C_x}$, $μ(dx)$--a.e. Our results extend the analogous one-dimensional results of M. Beiglböck and N. Juillet (2016) and M. Beiglböck, M. Nutz, and N. Touzi (2015). We conjecture that the decomposition is canonical and minimal in the sense that it allows to characterise the martingale polar sets, i.e. the sets which have zero mass under all measures in $\mathcal{M}(μ,ν)$, and offers the martingale analogue of the characterisation of transport polar sets proved in M. Beiglböck, M. Goldstern, G. Maresch, and W. Schachermayer (2009).

math.PR↗

arXiv2017

Pathwise Stochastic Calculus with Local Times

We study a notion of local time for a continuous path, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach subsumes other existing definitions and agrees with the usual (stochastic) local times a.s. for paths of a continuous semimartingale. We establish pathwise version of the Itô-Tanaka, change of variables and change of time formulae. We provide equivalent conditions for existence of pathwise local time. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choice of partitions. In particular, we show that an arbitrary given non-decreasing process can be achieved a.s. by the pathwise quadratic variation of a standard Brownian motion for a suitable sequence of (random) partitions; however, such degenerate behavior is excluded when the partitions are constructed from stopping times.

math.PR↗

arXiv2015

Applications of pathwise Burkholder-Davis-Gundy inequalities

We present several applications of the pathwise Burkholder-Davis-Gundy (BDG) inequalities. Most importantly we prove them for cadlag semimartingales and a general function $Φ$, and use this to derive BDG inequalities (non-pathwise ones) for the Bessel process of order $α\geq 1$ and for martingales stopped at $τ$, with $τ$ in a well studied class of random times.

math.PR↗

arXiv2015

Pathwise versions of the Burkholder-Davis-Gundy inequality

We present a new proof of the Burkholder-Davis-Gundy inequalities for $1\leq p<\infty$. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging.

math.PR↗

arXiv2014

A Simple Proof of the Bichteler-Dellacherie Theorem

We give a simple and rather elementary proof of the celebrated Bichteler-Dellacherie-Mokobodzki Theorem, which states that a process S is a good integrator if and only if it is a semimartingale. As a corollary, we obtain a characterization of semimartingales along the lines of classical Riemann integrability.

math.PR↗

arXiv2014

On a dyadic approximation of predictable processes of finite variation

We show that any cadlag predictable process of finite variation is an a.s. limit of elementary predictable processes; it follows that predictable stopping times can be approximated `from below' by predictable stopping times which take finitely many values. We then obtain as corollaries two classical theorems: predictable stopping times are announceable, and an increasing process is predictable iff it is natural.

math.PR↗

arXiv2013

Do arbitrage-free prices come from utility maximization?

In this paper we ask whether, given a stock market and an illiquid derivative, there exists arbitrage-free prices at which an utility-maximizing agent would always want to buy the derivative, irrespectively of his own initial endowment of derivatives and cash. We prove that this is false for any given investor if one considers all initial endowments with finite utility, and that it can instead be true if one restricts to the endowments in the interior. We show however how the endowments on the boundary can give rise to very odd phenomena; for example, an investor with such an endowment would choose not to trade in the derivative even at prices arbitrarily close to some arbitrage price.

q-fin.PM↗

arXiv2013

Optimal Investment with Stocks and Derivatives

This paper studies the problem of maximizing expected utility from terminal wealth combining a static position in derivative securities, which we assume can be traded only at time zero, with a traditional dynamic trading strategy in stocks. We work in the framework of a general semi-martingale model and consider a utility function defined on the positive real line.

q-fin.PM↗

arXiv2013

Optimal investment and price dependence in a semi-static market

This paper studies the problem of maximizing expected utility from terminal wealth in a semi-static market composed of derivative securities, which we assume can be traded only at time zero, and of stocks, which can be traded continuously in time and are modeled as locally-bounded semi-martingales. Using a general utility function defined on the positive real line, we first study existence and uniqueness of the solution, and then we consider the dependence of the outputs of the utility maximization problem on the price of the derivatives, investigating not only stability but also differentiability, monotonicity, convexity and limiting properties.

q-fin.PM↗