Optional stopping theorem
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Recent developments in prophet inequalities Open
The classic prophet inequality states that, when faced with a finite sequence of non-negative independent random variables, a gambler who knows their distribution and is allowed to stop the sequence at any time, can obtain, in expectation,…
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Integral fluctuation relations for entropy production at stopping times Open
A stopping time T is the first time when a trajectory of a stochastic process satisfies a specific criterion. In this paper, we use martingale theory to derive the integral fluctuation relation for the stochastic entropy production in a st…
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General stopping behaviors of naïve and noncommitted sophisticated agents, with application to probability distortion Open
We consider the problem of stopping a diffusion process with a payoff functional that renders the problem time‐inconsistent. We study stopping decisions of naïve agents who reoptimize continuously in time, as well as equilibrium strategies…
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Equilibrium concepts for time‐inconsistent stopping problems in continuous time Open
A new notion of equilibrium, which we call strong equilibrium , is introduced for time‐inconsistent stopping problems in continuous time. Compared to the existing notions introduced in Huang, Y.‐J., & Nguyen‐Huu, A. (2018, Jan 01). Time‐co…
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Conditional convex orders and measurable martingale couplings Open
Strassen’s classical martingale coupling theorem states that two random vectors are ordered in the convex (resp. increasing convex) stochastic order if and only if they admit a martingale (resp. submartingale) coupling. By analysing topolo…
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An iterated Azéma–Yor type embedding for finitely many marginals Open
We solve the $n$-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures $μ_1,...,μ_n$ which are in convex order and satisfy an additional technical assumption. Our construction is expl…
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On the Optimal Exercise Boundaries of Swing Put Options Open
We use probabilistic methods to characterise time-dependent optimal stopping boundaries in a problem of multiple optimal stopping on a finite time horizon. Motivated by financial applications, we consider a payoff of immediate stopping of …
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Risk-sensitive stopping problems for continuous-time Markov chains Open
In this thesis we consider optimal stopping problems for continuous-time Markov chains, evaluated under a general class of utility functions. Besides the well known dynamic programming approach via HJB equation, these stopping problems are…
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Second order backward SDE with random terminal time Open
Backward stochastic differential equations extend the martingale representation theorem to the nonlinear setting. This can be seen as path-dependent counterpart of the extension from the heat equation to fully nonlinear parabolic equations…
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Global $C^{1}$ regularity of the value function in optimal stopping problems Open
We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopp…
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Discounted optimal stopping problems in continuous hidden Markov models Open
We study a two-dimensional discounted optimal stopping problem related to the pricing of perpetual commodity equities in a model of financial markets in which the behaviour of the underlying asset price follows a generalized geometric Brow…
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Towards a probability-free theory of continuous martingales Open
Without probability theory, we define classes of supermartingales, martingales, and semimartingales in idealized financial markets with continuous price paths. This allows us to establish probability-free versions of a number of standard r…
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An optimal mean-reversion trading rule under a Markov chain model Open
This paper is concerned with a mean-reversion trading rule.In contrast to most market models treated in the literature,the underlying market is solely determined by a two-state Markov chain.The major advantage of such Markov chain modelis …
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Stopping with expectation constraints: 3 points suffice Open
We consider the problem of optimally stopping a one-dimensional regular continuous strong Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the…
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Three-Player Gambler’s Ruin Problem: Some Extensions Open
For calculating the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. However, the martingale approach is not appropriate to determi…
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Optimal Brownian Stopping between radially symmetric marginals in general dimensions Open
Given an initial (resp., terminal) probability measure $μ$ (resp., $ν$) on $\mathbb{R}^d$, we characterize those optimal stopping times $τ$ that maximize or minimize the functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Bro…
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Optimal Stopping under G-expectation Open
We develop a theory of optimal stopping problems under G-expectation framework. We first define a new kind of random times, called G-stopping times, which is suitable for this problem. For the discrete time case with finite horizon, the va…
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Absolute continuity of semimartingales Open
We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weake…
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Optimal stopping under model uncertainty: Randomized stopping times approach Open
In this work we consider optimal stopping problems with conditional convex risk measures called optimised certainty equivalents. Without assuming any kind of time-consistency for the underlying family of risk measures, we derive a novel re…
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Geometry of Distribution-Constrained Optimal Stopping Problems Open
We adapt ideas and concepts developed in optimal transport (and its martingale variant) to give a geometric description of optimal stopping times of Brownian motion subject to the constraint that the distribution of the stopping time is a …
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Stopping spikes, continuation bays and other features of optimal stopping with finite-time horizon Open
We consider optimal stopping problems with finite-time horizon and state-dependent discounting. The underlying process is a one-dimensional linear diffusion and the gain function is time-homogeneous and difference of two convex functions. …
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Local time pushed mixed stopping and smooth fit for time-inconsistent stopping problems Open
We consider the game-theoretic approach to time-inconsistent stopping of a one-dimensional diffusion where the time-inconsistency is due to the presence of a non-exponential (weighted) discount function. In particular, we study (weak) equi…
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Non-zero-sum optimal stopping game with continuous versus periodic exercise opportunities Open
We introduce a new non-zero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modelling the value of an asset, one player observes and can act on the process continuously, while the other playe…
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An algorithm based on an iterative optimal stopping method for Feller processes with applications to impulse control, perturbation, and possibly zero random discount problems Open
In this paper we present an iterative optimal stopping method for general optimal stopping problems for Feller processes. We show using an approximating scheme that the value function of an optimal stopping problem for some general operato…
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Time Consistent Stopping For The Mean-Standard Deviation Problem --- The Discrete Time Case Open
Inspired by Strotz's consistent planning strategy, we formulate the infinite horizon mean-variance stopping problem as a subgame perfect Nash equilibrium in order to determine time consistent strategies with no regret. Equilibria among sto…
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Sample-driven optimal stopping: From the secretary problem to the i.i.d. prophet inequality Open
We take a unifying approach to single selection optimal stopping problems with random arrival order and independent sampling of items. In the problem we consider, a decision maker (DM) initially gets to sample each of $N$ items independent…
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Distribution‐constrained optimal stopping Open
We solve the problem of optimal stopping of a Brownian motion subject to the constraint that the stopping time's distribution is a given measure consisting of finitely many atoms. In particular, we show that this problem can be converted t…
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Dynamic Programming Principles for Optimal Stopping with Expectation Constraint Open
We analyze an optimal stopping problem with a constraint on the expected cost. When the reward function and cost function are Lipschitz continuous in state variable, we show that the value of such an optimal stopping problem is a continuou…
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Optimal stopping and the sufficiency of randomized threshold strategies Open
In a classical optimal stopping problem the aim is to maximize the expected value of a functional of a diffusion evaluated at a stopping time. This note considers optimal stopping problems beyond this paradigm. We study problems in which t…
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A Class of Recursive Optimal Stopping Problems with Applications to Stock Trading Open
In this paper, we introduce and solve a class of optimal stopping problems of recursive type. In particular, the stopping payoff depends directly on the value function of the problem itself. In a multidimensional Markovian setting, we show…