arXiv subjects

Miryana Grigorova

Publications and source records attributed to Miryana Grigorova.

11 recordsLinked to original sources

arXiv2026

Non-linear optimal stopping with Bermudan strategies: the infinite horizon case

In this paper, we consider an optimal stopping problem with infinite horizon, non-negative pay-offs and non-linear evaluations $ρ_{S,τ}$ indexed by two indices: $S$ and $τ$, where $S$ is the time of evaluation and $τ$ is the time when the pay-off is revealed. The agent's stopping strategies are constrained to be in the set of so-called Bermudan stopping times $Θ$. Under suitable assumptions on the non-linear evaluations $ρ$ and on the pay-off, we show that a dynamic programming principle holds in this framework. We investigate the existence of $\varepsilon$-optimal stopping times, as well as the existence of optimal stopping times. We show that an $\varepsilon$-optimal stopping time exists. We also prove that the first time when the value family hits the pay-off is optimal if and only if it is finite. We also provide Doob's type convergence for non-negative \emph{$(Θ, ρ)$}-supermartingales in the case where $ρ_{S,τ}=ρ_S$ depends on the first index only. We provide an example from BSDEs with infinite horizon.

math.PR↗

arXiv2026

European Options in Market Models with Multiple Defaults: the BSDE approach

We study non-linear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and p default martingales. The driver of the BSDE with multiple default jumps can take a generalized form involving an optional finite variation process. We first show existence and uniqueness. We then establish comparison and strict comparison results for these BSDEs, under a suitable assumption on the driver. In the case of a linear driver, we derive an explicit formula for the first component of the BSDE using an adjoint exponential semimartingale. The representation depends on whether the finite variation process is predictable or only optional. We apply our results to the problem of pricing and hedging a European option in a linear complete market with two defaultable assets and in a non-linear complete market with p defaultable assets. Two examples of the latter market model are provided: an example where the seller of the option is a large investor influencing the probability of default of a single asset and an example where the large seller's strategy affects the default probabilities of all p assets.

q-fin.MF↗

arXiv2025

The randomly distorted Choquet integrals with respect to a G-randomly distorted capacity and risk measures

We study randomly distorted Choquet integrals with respect to a capacity c on a measurable space (Ω,F), where the capacity c is distorted by a G-measurable random distortion function (with G a sub-σ-algebra of F). We establish some fundamental properties, including the comonotonic additivity of these integrals under suitable assumptions on the underlying capacity space. We provide a representation result for comonotonic additive conditional risk measures which are monotone with respect to the first-order stochastic dominance relation (with respect to the capacity c) in terms of these randomly distorted Choquet integrals. We also present the case where the random distortion functions are concave. In this case, the G-randomly distorted Choquet integrals are characterised in terms of comonotonic additive conditional risk measures which are monotone with respect to the stop-loss stochastic dominance relation (with respect to the capacity c). We provide examples, extending some well-known risk measures in finance and insurance, such as the Value at Risk and the Average Value at Risk.

math.PR↗

arXiv2025

The non-linear multiple stopping problem: between the discrete and the continuous time

We consider the non-linear optimal multiple stopping problem under general conditions on the non-linear evaluation operators, which might depend on two time indices: the time of evaluation/assessment and the horizon (when the reward or loss is incurred). We do not assume convexity/concavity or cash-invariance. We focus on the case where the agent's stopping strategies are what we call Bermudan stopping strategies, a framework which can be seen as lying between the discrete and the continuous time. We first study the non-linear double optimal stopping problem by using a reduction approach. We provide a necessary and a sufficient condition for optimal pairs, and a result on existence of optimal pairs. We then generalize the results to the non-linear $d$-optimal stopping problem. We treat the symmetric case (of additive and multiplicative reward families) as examples.

math.OC↗

arXiv2023

Non-linear non-zero-sum Dynkin games with Bermudan strategies

In this paper, we study a non-zero-sum game with two players, where each of the players plays what we call Bermudan strategies and optimizes a general non-linear assessment functional of the pay-off. By using a recursive construction, we show that the game has a Nash equilibrium point.

math.OC↗

arXiv2023

Optimal stopping: Bermudan strategies meet non-linear evaluations

We address an optimal stopping problem over the set of Bermudan-type strategies $Θ$ (which we understand in a more general sense than the stopping strategies for Bermudan options in finance) and with non-linear operators (non-linear evaluations) assessing the rewards, under general assumptions on the non-linear operators $ρ$. We provide a characterization of the value family V in terms of what we call the ($ρ$,$Θ$) -Snell envelope of the pay-off family. We establish a Dynamic Programming Principle. We provide an optimality criterion in terms of a ($ρ$,$Θ$) -martingale property of V on a stochastic interval. We investigate the ($ρ$,$Θ$)-martingale structure and we show that the ''first time'' when the value family coincides with the pay-off family is optimal. The reasoning simplifies in the case where there is a finite number n of pre-described stopping times, where n does not depend on the scenario $ω$. We provide examples of non-linear operators entering our framework.

math.OC↗

arXiv2022

Stochastic representation under g-expectation and applications: the discrete time case

In this paper, we address the stochastic representation problem in discrete time under (non-linear) g-expectation. We establish existence and uniqueness of the solution, as well as a characterization of the solution. As an application, we investigate a new approach to the optimal stopping problem under g-expectation and the related pricing of American options under Knightian uncertainty. Our results are also applied to a (non-linear) Skorokhod-type obstacle problem.

math.PR↗

arXiv2018

Optimal stopping with f -expectations: the irregular case

We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $ξ$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $ξ$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $ξ$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American options with irregular pay-off in an imperfect market model.

math.PR↗

arXiv2018

Doubly Reflected BSDEs and ${\cal E}^{f}$-Dynkin games: beyond the right-continuous case

We formulate a notion of doubly reflected BSDE in the case where the barriers $ξ$ and $ζ$ do not satisfy any regularity assumption and with a general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where $ξ$ is right upper-semicontinuous and $ζ$ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding $\mathcal{E}^f$-Dynkin game, i.e. a game problem over stopping times with (non-linear) $f$-expectation, where $f$ is the driver of the doubly reflected BSDE. In the general case where the barriers do not satisfy any regularity assumptions, the solution of the doubly reflected BSDE is related to the value of ''an extension'' of the previous non-linear game problem over a larger set of ''stopping strategies'' than the set of stopping times. This characterization is then used to establish a comparison result and \textit{a priori} estimates with universal constants.

math.PR↗

arXiv2017

Reflected BSDEs when the obstacle is not right-continuous and optimal stopping

In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some tools from the general theory of processes such as Mertens decomposition of optional strong (but not necessarily right-continuous) supermartingales, some tools from optimal stopping theory, as well as an appropriate generalization of It{ô}'s formula due to Gal'chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position $ξ$ is assessed by an $f$-conditional expectation $\mathcal{E}^f(\cdot)$ (where $f$ is a Lipschitz driver). We characterize the "value function" of the problem in terms of the solution to our RBSDE. Under an additional assumption of left upper-semicontinuity on $ξ$, we show the existence of an optimal stopping time. We also provide a generalization of Mertens decomposition to the case of strong $\mathcal{E}^f$-supermartingales.

math.PR↗

arXiv2017

Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs

We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) $g$-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed by g-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamad{è}ne and J. Zhang, we construct a Nash equilibrium point of this game by a recursive procedure. Our results are obtained in the case of a standard Lipschitz driver $g$ without any additional assumption on the driver besides that ensuring the monotonicity of the corresponding $g$-expectation.

math.PR↗