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Anna Aksamit

Publications and source records attributed to Anna Aksamit.

17 recordsLinked to original sources
arXiv2026

Sensitivity analysis of Stochastic Fluid Models: Stationary and transient quantities with applications

We establish results for the first sensitivity analysis of the stochastic fluid models (SFMs). We derive expressions for the sensitivity analysis of the key stationary and transient (time-dependent) quantities of this class of models. We also construct numerical examples to demonstrate the application potential of our methodology in queueing systems, such as deteriorating systems and insurance risk processes. This work forms foundation for the sensitivity analysis of other Markovian modulated models, which are generalisations of the SFMs, and have widespread applications.

math.PR↗
arXiv2026

Entropy and additional utility of a discrete information disclosed progressively in time

The additional information carried by an enlarged filtration and its measurement was studied by several authors. Already Meyer (Sur un theoreme de J. Jacod, 1978) and Yor (Entropie d'une partition, et grossissement initial d'une filtration, 1985), investigated stability of martingale spaces with respect to initial enlargement with atomic sigma-field. We extend these considerations to the case where information is disclosed progressively in time. We define the entropy of such information and we prove that its finiteness is enough for stability of some martingale spaces in progressive setting. Finally we calculate additional logarithmic utility of a discrete information disclosed progressively in time.

math.PR↗
arXiv2025

Switching to a Green and sustainable finance setting: a mean field game approach

We consider a continuum of carbon-emitting firms who seek to maximise their stock price, and a regulator (e.g., Government) who wishes for the economy to flourish, whilst simultaneously punishing firms who behave non-green. Interpreting the regulator as a major player and the firms as the minor players, we model this setting through a mean field game with major and minor players. We extend the stochastic maximum principle derived by Carmona & Zhu [A probabilistic approach to mean field games with major and minor players. Annals of Applied Probability, 2016, 94, 745--788] by relaxing the assumptions on the forms of the minimisers for the Hamiltonians, allowing them to depend on more arguments. This allows the major and representative minor player to interact in a more natural fashion, thereby permitting us to consider more realistic models for our green and sustainable finance problem. Through our stochastic maximum principle, we derive explicit Nash equilibria for a number of examples.

math.PR↗
arXiv2023

Superhedging duality for multi-action options under model uncertainty with information delay

We consider the superhedging price of an exotic option under nondominated model uncertainty in discrete time in which the option buyer chooses some action from an (uncountable) action space at each time step. By introducing an enlarged space we reformulate the superhedging problem for such an exotic option as a problem for a European option, which enables us to prove the pricing-hedging duality. Next, we present a duality result that, when the option buyers action is observed by the seller up to $l$ periods later, the superhedging price equals the model-based price where the option buyer has the power to look into the future for $l$ periods.

q-fin.MF↗
arXiv2023

Random walk on a quadrant: mapping to a one-dimensional level-dependent Quasi-Birth-and-Death process (LD-QBD)

We consider a neighbourhood random walk on a quadrant, $\{(X_1(t),X_2(t),φ(t)):t\geq 0\}$, with state space \begin{eqnarray*} \mathcal{S}&=&\{(n,m,i):n,m=0,1,2,\ldots;i=1,2,\ldots,k(n,m)\}. \end{eqnarray*} Assuming start in state $(n,m,i)$, the process spends exponentially distributed amount of time in $(n,m,i)$ according to some parameter $λ_i^{(n,m)}$. Upon leaving state $(n,m,i)$ the process moves to some state $(n^{'},m^{'},j)$ with $j\in\{1,\ldots,k(n^{'},m^{'})\}$ and $n^{'}\in\{n-1,n,n+1\}$, $m^{'}\in\{m-1,m,m+1\}$, according to some probabilities $(p_{n;a}^{m;b})_{i,j}$ with $a,b\in\{+,-,0\}$. We transform this process into a one-dimensional LD-QBD $\{(Z(t),χ(t)):t\geq 0\}$ with level variable $Z(t)$ and phase variable $χ(t)$. Using this transform we find its transient and stationary analysis using matrix-analytic methods, as well as the distribution at first hitting times.

math.PR↗
arXiv2021

Generalized BSDEs with random time horizon in a progressively enlarged filtration

We study generalized backward stochastic differential equations (BSDEs) up to a random time horizon $\vartheta$, which is not a stopping time, under minimal assumptions regarding the properties of $\vartheta$. In contrast to existing works in this area, we do not impose specific assumptions on the random time $\vartheta$ and we study the existence of solutions to BSDEs and reflected BSDEs with a random time horizon through the method of reduction. In addition, we also examine BSDEs and reflected BSDEs with a làdlàg driver where the driver is allowed to have a finite number of common jumps with the martingale part.

math.PR↗
arXiv2019

Martingale spaces and representations under absolutely continuous changes of probability

In a fully general setting, we study the relation between martingale spaces under two locally absolutely continuous probabilities and prove that the martingale representation property (MRP) is always stable under locally absolutely continuous changes of probability. Our approach relies on minimal requirements, is constructive and, as shown by a simple example, enables us to study situations which cannot be covered by the existing theory.

math.PR↗
arXiv2018

Integral representations of martingales for progressive enlargements of filtrations

We work in the setting of the progressive enlargement $\mathbb G$ of a reference filtration $\mathbb F$ through the observation of a random time $τ$. We study an integral representation property for some classes of $\mathbb G$-martingales stopped at $τ$. In the first part, we focus on the case where $\mathbb F$ is a Poisson filtration and we establish a predictable representation property with respect to three $\mathbb G$-martingales. In the second part, we relax the assumption that $\mathbb F$ is a Poisson filtration and we assume that $τ$ is an $\mathbb F$-pseudo-stopping time. We establish integral representations with respect to some $\mathbb G$-martingales built from $\mathbb F$-martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two $\mathbb G$-martingales.

math.PR↗
arXiv2018

Thin times and random times' decomposition

The paper studies thin times which are random times whose graph is contained in a countable union of the graphs of stopping times with respect to a reference filtration $\mathbb F$. We show that a generic random time can be decomposed into thin and thick parts, where the second is a random time avoiding all $\mathbb F$-stopping times. Then, for a given random time $τ$, we introduce ${\mathbb F}^τ$, the smallest right-continuous filtration containing $\mathbb F$ and making $τ$ a stopping time, and we show that, for a thin time $τ$, each $\mathbb F$-martingale is an ${\mathbb F}^τ$-semimartingale, i.e., the hypothesis $({\mathcal H}^\prime)$ for $(\mathbb F, {\mathbb F}^τ)$ holds. We present applications to honest times, which can be seen as last passage times, showing classes of filtrations which can only support thin honest times, or can accommodate thick honest times as well.

math.PR↗
arXiv2018

Robust framework for quantifying the value of information in pricing and hedging

We investigate asymmetry of information in the context of robust approach to pricing and hedging of financial derivatives. We consider two agents, one who only observes the stock prices and another with some additional information, and investigate when the pricing--hedging duality for the former extends to the latter. We introduce a general framework to express the superhedging and market model prices for an informed agent. Our key insight is that an informed agent can be seen as a regular agent who can restrict her attention to a certain subset of possible paths. We use results of Hou & Obłój on robust approach with beliefs to establish the pricing--hedging duality for an informed agent. Our results cover number of scenarios, including information arriving before trading starts, arriving after static position in European options is formed but before dynamic trading starts or arriving at some point before the maturity. For the latter we show that the superhedging value satisfies a suitable dynamic programming principle, which is of independent interest.

q-fin.MF↗
arXiv2017

Robust pricing--hedging duality for American options in discrete time financial markets

We investigate pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, e.g. a family of European options, only statically. In the first part of the paper we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a non-dominated family of probability measures. Our first insight is that by considering a (universal) enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This may be seen as a weak formulation of the original problem. Our second insight is that lack of duality is caused by the lack of dynamic consistency and hence a different enlargement with dynamic consistency is sufficient to recover duality: it is enough to consider (fictitious) extensions of the market in which all the assets are traded dynamically. In the second part of the paper we study two important examples of robust framework: the setup of Bouchard and Nutz (2015) and the martingale optimal transport setup of Beiglböck et al. (2013), and show that our general results apply in both cases and allow us to obtain pricing-hedging duality for American options.

math.OC↗
arXiv2016

Projections, Pseudo-Stopping Times and the Immersion Property

Given two filtrations $\mathbb F \subset \mathbb G$, we study under which conditions the $\mathbb F$-optional projection and the $\mathbb F$-dual optional projection coincide for the class of $\mathbb G$-optional processes with integrable variation. It turns out that this property is equivalent to the immersion property for $\mathbb F$ and $\mathbb G$, that is every $\mathbb F$-local martingale is a $\mathbb G$-local martingale, which, equivalently, may be characterised using the class of $\mathbb F$-pseudo-stopping times. We also show that every $\mathbb G$-stopping time can be decomposed into the minimum of two barrier hitting times.

math.PR↗
arXiv2016

Non-Arbitrage under a Class of Honest Times

This paper quantifies the interplay between the non-arbitrage notion of No-Unbounded-Profit-with-Bounded-Risk (NUPBR hereafter) and additional information generated by a random time. This study complements the one of Aksamit/Choulli/Deng/Jeanblanc [1] in which the authors studied similar topics for the case of stopping at the random time instead, while herein we are concerned with the part after the occurrence of the random time. Given that all the literature -up to our knowledge- proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property, while the second main result characterizes the honest times that preserve the NUPBR property for any quasi-left continuous model. Furthermore, we construct explicitly "the-after-tau" local martingale deflators for a large class of initial models (i.e. models in the small filtration) that are already risk-neutralized.

q-fin.PR↗
arXiv2015

Non-Arbitrage Under Additional Information for Thin Semimartingale Models

This paper completes the two studies undertaken in \cite{aksamit/choulli/deng/jeanblanc2} and \cite{aksamit/choulli/deng/jeanblanc3}, where the authors quantify the impact of a random time on the No-Unbounded-Risk-with-Bounded-Profit concept (called NUPBR hereafter) when the stock price processes are quasi-left-continuous (do not jump on predictable stopping times). Herein, we focus on the NUPBR for semimartingales models that live on thin predictable sets only and the progressive enlargement with a random time. For this flow of information, we explain how far the NUPBR property is affected when one stops the model by an arbitrary random time or when one incorporates fully an honest time into the model. This also generalizes \cite{choulli/deng} to the case when the jump times are not ordered in anyway. Furthermore, for the current context, we show how to construct explicitly local martingale deflator under the bigger filtration from those of the smaller filtration.

q-fin.MF↗
arXiv2014

Non-Arbitrage up to Random Horizon for Semimartingale Models

This paper addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon. We focus on No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, which is also known in the literature as the first kind of non-arbitrage. For this non-arbitrage notion, we obtain two principal results. The first result lies in describing the pairs of market model and random time for which the resulting stopped model fulfills NUPBR condition. The second main result characterises the random time models that preserve the NUPBR property after stopping for any market model. These results are elaborated in a very general market model, and we also pay attention to some particular and practical models. The analysis that drives these results is based on new stochastic developments in semimartingale theory with progressive enlargement. Furthermore, we construct explicit martingale densities (deflators) for some classes of local martingales when stopped at random time.

q-fin.PR↗
arXiv2013

Arbitrages in a Progressive Enlargement Setting

This paper completes the analysis of Choulli et al. Non-Arbitrage up to Random Horizons and after Honest Times for Semimartingale Models and contains two principal contributions. The first contribution consists in providing and analysing many practical examples of market models that admit classical arbitrages while they preserve the No Unbounded Profit with Bounded Risk (NUPBR hereafter) under random horizon and when an honest time is incorporated for particular cases of models. For these markets, we calculate explicitly the arbitrage opportunities. The second contribution lies in providing simple proofs for the stability of the No Unbounded Profit with Bounded Risk under random horizon and after honest time satisfying additional important condition for particular cases of models.

math.PR↗