Publications
10 results found
Mostovyi O, Siorpaes P, 2023, Differentiation of measures on an arbitrary measurable space, Journal of Mathematical Analysis and Applications, Vol: 528, ISSN: 0022-247X
Let μ, ν be positive finite measures on an arbitrary measurable space (Ω, F), ν =νa + νs be the Lebesgue decomposition of ν with respect to μ, P be the family ofall finite partitions π ⊆ F of Ω, andfπ(μ) :=A∈π:μ(A)>01Aν(A)μ(A), π ∈ P.We recall that (fπ(μ))π∈P → dνadμ in L1(μ), as is (essentially) known. Here weidentify (πn)n∈N ⊆ P such thatfπn (μ) → dνadμ μ a.s. as n → ∞;in the setting of separable F, a (rather trivial) way to do this was alreadyknown. To do all this, we characterise the case of equality in Jensen’s conditionalinequality (generalising the known case of the standard Jensen inequality), anduse this to determine how, given a p-uniformly integrable martingale (fi)i∈I , onecan identify a sequence (in)n∈N ⊆ I such that (fin )n converges in Lp to somef which closes the whole net (fi)i. We also give a new proof of the (alreadyknown) characterisation of p-uniformly integrable martingales, without relying onthe martingale a.s. convergence theorem.
Łochowski RM, Obłój J, Prömel DJ, et al., 2021, Local times and Tanaka–Meyer formulae for càdlàg paths, Electronic Journal of Probability, Vol: 26, Pages: 1-29, ISSN: 1083-6489
Three concepts of local times for deterministic càdlàg 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 càdlàg semimartingales, which may be related to previous results of Nicole El Karoui [11] and Marc Lemieux [23].
Obłój J, Siorpaes P, 2020, Structure of martingale transports in finite dimensions
We study the structure of martingale transports in finite dimensions. Weconsider the family $\mathcal{M}(\mu,\nu) $ of martingale measures on$\mathbb{R}^N \times \mathbb{R}^N$ with given marginals $\mu,\nu$, andconstruct 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 martingaletransport in $\mathcal{M}(\mu,\nu) $ sends mass from $x$ to within$\overline{C_x}$, $\mu(dx)$--a.e. Our results extend the analogousone-dimensional results of M. Beiglb\"ock and N. Juillet (2016) and M.Beiglb\"ock, M. Nutz, and N. Touzi (2015). We conjecture that the decompositionis canonical and minimal in the sense that it allows to characterise themartingale polar sets, i.e. the sets which have zero mass under all measures in$\mathcal{M}(\mu,\nu)$, and offers the martingale analogue of thecharacterisation of transport polar sets proved in M. Beiglb\"ock, M.Goldstern, G. Maresch, and W. Schachermayer (2009).
Siorpaes P, 2018, Applications of pathwise Burkholder-Davis-Gundy inequalities, Bernoulli, Vol: 24, Pages: 3222-3245, ISSN: 1350-7265
In this paper, after generalizing the pathwise Burkholder–Davis–Gundy (BDG) inequalities from discrete time to cadlag semimartingales, we present several applications of the pathwise inequalities. In particular we show that they allow to extend the classical BDG inequalities1. to the Bessel process of order α ≥ 12. to the case of a random exponent p3. to martingales stopped at a time τ which belongs to a well studied class of random times
Davis M, Obłój J, Siorpaes P, 2018, Pathwise stochastic calculus with local times, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol: 54, Pages: 1-21, ISSN: 0246-0203
We study a notion of local time for a continuous path, defined as a limit ofsuitable discrete quantities along a general sequence of partitions of the timeinterval. Our approach subsumes other existing definitions and agrees with theusual (stochastic) local times a.s. for paths of a continuous semimartingale.We establish pathwise version of the It\^o-Tanaka, change of variables andchange of time formulae. We provide equivalent conditions for existence ofpathwise local time. Finally, we study in detail how the limiting objects, thequadratic variation and the local time, depend on the choice of partitions. Inparticular, we show that an arbitrary given non-decreasing process can beachieved a.s. by the pathwise quadratic variation of a standard Brownian motionfor a suitable sequence of (random) partitions; however, such degeneratebehavior is excluded when the partitions are constructed from stopping times.
Beiglboeck M, Siorpaes P, 2015, Pathwise versions of the Burkholder-Davis-Gundy inequality, Bernoulli, Vol: 21, Pages: 360-373, ISSN: 1350-7265
We present a new proof of the Burkholder–Davis–Gundy inequalities for 1 ≤ p < ∞. 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.
Siorpaes P, 2015, Optimal investment and price dependence in a semi-static market, Finance and Stochastics, Vol: 19, Pages: 161-187, ISSN: 0949-2984
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 modelled as locally bounded semimartingales. Using a general utility function defined on the positive half-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.
Siorpaes P, 2014, Do arbitrage-free prices come from utility maximization?, Mathematical Finance, Vol: 26, Pages: 602-616, ISSN: 0960-1627
In this paper we ask whether, given a stock market and an illiquid derivative, there exists arbitrage-free prices at which a 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.
Siorpaes P, 2014, On a dyadic approximation of predictable processes of finite variation, ELECTRONIC COMMUNICATIONS IN PROBABILITY, Vol: 19, Pages: 1-12, ISSN: 1083-589X
We show that any càdlàg 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.
Beiglboeck M, Siorpaes P, 2013, Riemann-integration and a new proof of the Bichteler-Dellacherie theorem, STOCHASTIC PROCESSES AND THEIR APPLICATIONS, Vol: 124, Pages: 1226-1235, ISSN: 0304-4149
We give a new proof of the celebrated Bichteler–Dellacherie theorem, which states that a process is a good integrator if and only if it is the sum of a local martingale and a finite-variation process. As a corollary, we obtain a characterization of semimartingales along the lines of classical Riemann integrability.
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