Sören Christensen

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• Professor
• Professor (Full) at Kiel University

We consider problems of optimal stopping where the driving process is a (one- or multi-dimensional) diffusion. Our approach is motivated by a change of measure techniques and gives a characterization of the optimal stopping set in terms of harmonic functions for one-dimensional diffusions. The generalization to multidimensional diffusions uses the...

Optimal stopping problems form a class of stochastic optimization problems that has a wide range of applications in sequential statistics and mathematical finance. Here we consider a general optimal stopping problem with discounting for autoregressive processes. Our strategy for a solution consists of two steps: First we give elementary conditions...

We characterize the value function and the optimal stopping time for a large class of optimal stopping problems where the underlying process to be stopped is a fairly general Markov process. The main result is inspired by recent findings for L\'evy processes obtained essentially via the Wiener-Hopf factorization. The main ingredient in our approach...

We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these functions. The resulting problem is a linear semi-infinite programming problem, that can be solved using standard...

We introduce a new class of generative diffusion models that, unlike conventional denoising diffusion models, achieve a time-homogeneous structure for both the noising and denoising processes, allowing the number of steps to adaptively adjust based on the noise level. This is accomplished by conditioning the forward process using Doob's $h$-transfo...

We study the (weak) equilibrium problem arising from the problem of optimally stopping a one-dimensional diffusion subject to an expectation constraint on the time until stopping. The weak equilibrium problem is realized with a set of randomized but purely state dependent stopping times as admissible strategies. We derive a verification theorem and...

One of the most classical games for stochastic processes is the zero-sum Dynkin (stopping) game. We present a complete equilibrium solution to a general formulation of this game with an underlying one-dimensional diffusion. A key result is the construction of a characterizable global $\epsilon$-Nash equilibrium in Markovian randomized stopping time...

Over the recent past, data-driven algorithms for solving stochastic optimal control problems in the face of model uncertainty have become an increasingly active area of research. However, for singular controls and underlying diffusion dynamics, the analysis has so far been restricted to the scalar case. In this paper, we fill this gap by studying a...

This paper considers an ergodic version of the bounded velocity follower problem, assuming that the decision maker lacks knowledge of the underlying system parameters and must learn them while simultaneously controlling. We propose algorithms based on moving empirical averages and develop a framework for integrating statistical methods with stochas...

In a probabilistic mean-field game driven by a linear diffusion an individual player aims to minimize an ergodic long-run cost by controlling the diffusion through a pair of –increasing and decreasing– càdlàg processes, while he is interacting with an aggregate of players through the expectation of a similar diffusion controlled by another pair of...

Stochastic optimal control problems have a long tradition in applied probability, with the questions addressed being of high relevance in a multitude of fields. Even though theoretical solutions are well understood in many scenarios, their practicability suffers from the assumption of known dynamics of the underlying stochastic process, raising the...

In optimal stopping problems, a Markov structure guarantees Markovian optimal stopping times (first exit times). Surprisingly, there is no analogous result for Markovian stopping games once randomization is required. This paper addresses this gap by proving the existence of Markov-perfect equilibria in a specific type of stopping game - a general n...

In recent years, there has been an intense debate about how learning in biological neural networks (BNNs) differs from learning in artificial neural networks. It is often argued that the updating of connections in the brain relies only on local information, and therefore a stochastic gradient-descent type optimization method cannot be used. In this...

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 nonexponential (weighted) discount function. In particular, we study (weak) equilibria for this problem in a novel class of mixed (i.e., randomized) stopping times based on a local time constr...

The standard theory of optimal stopping is based on the idealised assumption that the underlying process is essentially known. In this paper, we drop this restriction and study data-driven optimal stopping for a general diffusion process, focusing on investigating the statistical performance of the proposed estimator of the optimal stopping barrier...

Over the recent past data-driven algorithms for solving stochastic optimal control problems in face of model uncertainty have become an increasingly active area of research. However, for singular controls and underlying diffusion dynamics the analysis has so far been restricted to the scalar case. In this paper we fill this gap by studying a multiv...

We study the (weak) equilibrium problem arising from the problem of optimally stopping a one-dimensional diffusion subject to an expectation constraint on the time until stopping. The weak equilibrium problem is realized with a set of randomized but purely state dependent stopping times as admissible strategies. We derive a verification theorem and...

In recent years, there has been an intense debate about how learning in biological neural networks (BNNs) differs from learning in artificial neural networks. It is often argued that the updating of connections in the brain relies only on local information, and therefore a stochastic gradient-descent type optimization method cannot be used. In this...

We study a general formulation of the classical two-player Dynkin game in a Markovian discrete time setting. We show that an appropriate class of mixed, i.e., randomized, strategies in this context are \textit{Markovian randomized stopping times}, which correspond to stopping at any given state with a state-dependent probability. One main result is...

Consider two independent controlled linear diffusions with the same dynamics and the same ergodic controls, the first corresponding to an individual player, the second to the market. Let us also consider a cost function that depends on the first diffusion and the expectation of the second one. In this framework, we study the mean-field game consist...

Let W be a standard Brownian motion with W 0 = 0 and let b : R + → R be a continuous function with b(0) > 0. The first passage time (from below) is then defined as τ := inf{t ≥ 0|W t ≥ b(t)}. It is well-known that the distribution F of τ satisfies a set of Fredholm equations of the first kind. To apply this result, it is fundamental that the distri...

We propose to establish a research direction based on Reinforcement Learning in the scope of Cross Domain Fusion. More precisely, we combine the algorithmic approach of evolutionary rule-based Reinforcement Learning with the efficiency and performance of Deep Reinforcement Learning, while simultaneously developing a sound mathematical foundation. A...

We propose to establish a research direction based on Reinforcement Learning in the scope of Cross Domain Fusion. More precisely, we combine the algorithmic approach of evolutionary rule-based Reinforcement Learning with the efficiency and performance of Deep Reinforcement Learning, while simultaneously developing a sound mathematical foundation. A...

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) equilibria for this problem in a novel class of mixed (i.e., randomized) stopping times based on a local time const...

A moment constraint that limits the number of dividends in an optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz’s...

The Chow–Robbins game is a classical, still partly unsolved, stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide whether you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads $V(n,x) = \sup_{\tau }\mathbb{E} \left...

We call a given American option representable if there exists a European claim which dominates the American payoff at any time and such that the values of the two options coincide in the continuation region of the American option. This concept has interesting implications from a probabilistic, analytic, financial, and numeric point of view. Relying...

This article treats both discrete time and continuous time stopping problems for general Markov processes on the real line with general linear costs as they naturally arise in many problems in sequential decision making. Using an auxiliary function of maximum representation type, conditions are given to guarantee the optimal stopping time to be of...

In this paper, we propose an extension of the forward improvement iteration algorithm, originally introduced in Irle (2006) and recently reconsidered in Miclo and Villeneuve (2021). The main new ingredient is a flexible window parameter describing the look-ahead distance in the improvement step. We consider the framework of a Markovian optimal stop...

For classical finite time horizon stopping problems driven by a Brownian motion \[V(t,x) = \sup_{t\leq\tau\leq0}E_{(t,x)}[g(\tau,W_{\tau})],\] we derive a new class of Fredholm type integral equations for the stopping set. For large problem classes of interest, we show by analytical arguments that the equation uniquely characterizes the stopping bo...

We investigate the impact of Knightian uncertainty on the optimal timing policy of an ambiguity-averse decision-maker in the case where the underlying factor dynamics follow a multidimensional Brownian motion and the exercise payoff depends on either a linear combination of the factors or the radial part of the driving factor dynamics. We present a...

Bruelheide et al. (Diversity and Distributions, 26, 2020, 782) explored repeated habitat mapping data to identify floristic changes over time on the basis of two surveys. Because of the incompleteness of the data, they utilized the Beals' index based on the aggregated data from both surveys as a statistical tool for the analysis. The aim of this no...

Stochastic optimal control problems have a long tradition in applied probability, with the questions addressed being of high relevance in a multitude of fields. Even though theoretical solutions are well understood in many scenarios, their practicability suffers from the assumption of known dynamics of the underlying stochastic process, raising the...

We discuss a class of explicitly solvable mean field control problems/games with a clear economic interpretation. More precisely, we consider long term average impulse control problems with underlying general one-dimensional diffusion processes motivated by optimal harvesting problems in natural resource management. We extend the classical stochast...

This article treats long term average impulse control problems with running costs in the case that the underlying process is a Lévy process. Assuming a maximum representation for the payoff function, we give easy to verify conditions for the control problem to have an s,S strategy as an optimizer. The occurring thresholds are given by the roots of...

A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to choose the intensity function of a Cox process is introduced. A subgame perfect Nash e...

We call a given American option representable if there exists a European claim which dominates the American payoff at any time and such that the values of the two options coincide in the continuation region of the American option. This concept has interesting implications from a probabilistic, analytic, financial, and numeric point of view. Relying...

This article treats both discrete time and continuous time stopping problems for general Markov processes on the real line with general linear costs. Using an auxiliary function of maximum representation type, conditions are given to guarantee the optimal stopping time to be of threshold type. The optimal threshold is then characterized as the root...

For a discrete time Markov chain and in line with Strotz’ consistent planning we develop a framework for problems of optimal stopping that are time-inconsistent due to the consideration of a non-linear function of an expected reward. We consider pure and mixed stopping strategies and a (subgame perfect Nash) equilibrium. We provide different necess...

We consider the discrete time stopping problem \[ V(t,x) = \sup_{\tau}E_{(t,x)}[g(\tau, X_\tau)],\] where $X$ is a random walk. It is well known that the value function $V$ is in general not smooth on the boundary of the continuation set $\partial C$. We show that under some conditions $V$ is not smooth in the interior of $C$ either. More precisely...

For a discrete time Markov chain and in line with Strotz' consistent planning we develop a framework for problems of optimal stopping that are time-inconsistent due to the consideration of a non-linear function of an expected reward. We consider pure and mixed stopping strategies and a (subgame perfect Nash) equilibrium. We provide different necess...

A moment constraint that limits the number of dividends in the optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz'...

This article treats long term average impulse control problems with running costs in the case that the underlying process is a L\'evy process. Under quite general conditions we characterize the value of the control problem as the value of a stopping problem and construct an optimal strategy of the control problem out of an optimizer of the stopping...

One of the fundamental assumptions in stochastic control of continuous time processes is that the dynamics of the underlying (diffusion) process is known. This is, however, usually obviously not fulfilled in practice. On the other hand, over the last decades, a rich theory for nonparametric estimation of the drift (and volatility) for continuous ti...

In den bisherigen Kapiteln haben wir besprochen, wie Daten verdichtet dargestellt werden können und wasWahrscheinlichkeiten sind. In beiden Gebieten kann man auch auf kontrovers diskutierbare Inhalte treffen, aber wir haben bisher ausgeklammert, welche Folgerungen man aus den Ergebnissen zielen sollte. Das ändert sich jetzt und damit geht es ans Ei...

Ein Unternehmen möchte möglichst genau wissen, welcher Mehrabsatz mit einer geplanten Preisrabatt-Aktion verbunden ist. Dies hängt natürlich von verschiedenen Faktoren ab, etwa dem jetzigen Preis und aktuellen Werbeausgaben. Für Fragestellungen dieser Art kann ein Lineares Modell den geeigneten Analyserahmen darstellen. Es handelt sich um einen wah...

warum Statistikkenntnisse in der Informationsgesellschaft eine Schlüsselqualifikation darstellen

Aufgabe der Beschreibenden Statistik ist das Erkennen von Strukturen in einem gegebenen Datensatz. Dazu werden die erhobenen Phänomene bzw. die Eigenschaften des Datensatzes zunächst in Zahlen „übersetzt“.

begründen können, welche Probleme bei einem rein intuitiven Umgang mit Wahrscheinlichkeiten auftreten

The Chow-Robbins game is a classical still partly unsolved stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide if you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads \[V(n,x) = \sup_{\tau }\operatorname{E} \left...

We investigate the impact of Knightian uncertainty on the optimal timing policy of an ambiguity averse decision maker in the case where the underlying factor dynamics follow a multidimensional Brownian motion and the exercise payoff depends on either a linear combination of the factors or the radial part of the driving factor dynamics. We present a...

We consider the impact of ambiguity on the optimal timing of a class of two-dimensional integral option contracts when the exercise payoff is a positively homogeneous measurable function. Hence, the considered class of exercise payoffs includes discontinuous functions as well. We identify a parameterized family of excessive functions generating an...

This paper studies explicitly solvable multidimensional optimal stopping problems of sum- and product-type in discrete and continuous time using the monotone case approach. It gives a review on monotone case stopping using the Doob decomposition, resp. Doob–Meyer decomposition in continuous time, also in its multiplicative versions. The approach vi...

According to conventional wisdom, ambiguity accelerates optimal timing by decreasing the value of waiting in comparison with the unambiguous benchmark case. We study this mechanism in a multidimensional setting and show that in a multifactor model ambiguity does not only influence the rate at which the underlying processes are expected to grow, it...

We study the problem of maximising expected utility of terminal wealth under constant and proportional transaction costs in a multidimensional market with prices driven by a factor process. We show that the value function is the unique viscosity solution of the associated quasi-variational inequalities and construct optimal strategies. While the va...

Quantitative Datenanalyse ist ein unverzichtbares Werkzeug in der digitalen Wissensgesellschaft. Dieses Lehrbuch bietet eine leicht verständliche Einführung in die Thematik für Studium und Berufsalltag. Besondere Aufmerksamkeit wird dabei der Abhängigkeitsmessung gewidmet, da sie typischerweise im Mittelpunkt wirtschaftswissenschaftlicher Anwendung...

Standard Markovian optimal stopping problems are consistent in the sense that the first entrance time into the stopping set is optimal for each initial state of the process. Clearly, the usual concept of optimality cannot in a straightforward way be applied to non-standard stopping problems without this time-consistent structure. This paper is devo...

Nachdem im vorigen Kapitel speziell die Anwendungen der Statistik in Sport und Spiel im Fokus standen, weiten wir jetzt den Blick. Die in diesem Kapitel zusammengetragenen Beispiele sollen aufzeigen, dass Statistik uns im täglichen Leben überall umgibt. Einen großen Teil der modernen Welt kann man nur verstehen, wenn man ein grundsätzliches Verstän...

Wenn Sie dieses Buch von vorn bis hierher durchgelesen haben, haben Sie inzwischen hoffentlich viel Interessantes erfahren. Vielleicht haben Sie auch über einige Beispiele selbst noch etwas nachgedacht. Wurden Sie aber schon richtig selbst aktiv?

Es vergeht wohl kaum eine Nachrichtensendung ohne die Worte „Laut einer aktuellen Studie…“. Studien sind überall. Und in den meisten spielen Zahlen eine entscheidende Rolle.

Verfolgt man die Kommentare von Fußballspielen im Fernsehen, könnte man teilweise auf die Idee kommen, den Zahlenkolonnen eines Buchhalters zu lauschen. Da wird berichtet, dass Spieler X beim letzten Spiel eine Strecke von 13,8 km zurücklegte, Mannschaft Y dabei 28 Torschüsse abgab und Trainer Z in seiner Karriere schon fünfmal auf die Tribüne verb...

Wir hoffen, dass das vorige Kapitel Sie überzeugt hat, dass Mathematik in den unterschiedlichsten Bereichen nützlich ist. Aber mit dieser Kapitelüberschrift lehnen wir uns doch ein wenig zu weit aus dem Fenster, oder? Ästhetische Maßstäbe werden die meisten Leser vermutlich eher selten an Mathematik anlegen.

Satz des Pythagoras, pq-Formel, Binomialkoeffizienten – solche Begriffe lösen bei dem einen oder anderen Leser vielleicht nicht nur positive Assoziationen aus. Vor allem stellt man sich die Frage, wozu man das alles braucht. Und tatsächlich dürften die meisten Erwachsenen höhere Mathematik bewusst eher selten einsetzen und trotzdem gut durchs Leben...

A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. A...

In this paper we demonstrate that the Riesz representation of excessive functions is a useful and enlightening tool to study optimal stopping problems. After a short general discussion of the Riesz representation we concretize to geometric Brownian motions. After this, a classical investment problem, also known as exchange-of-baskets-problem, is st...

Statistik und Mathematik prägen unser Leben so stark wie noch nie. Trotzdem gilt die Welt der Zahlen und Strukturen oft als abstrakt und kompliziert. Dieses Buch tritt den Gegenbeweis an: Die Autoren zeigen auf unterhaltsame Art, wie man – ganz ohne Formeln und besondere Vorbildung – erstaunliche statistische und mathematische Erkenntnisse gewinnen...

In Anscombe’s classical model, the objective is to find the optimal sequential rule for learning about the difference between two alternative treatments and subsequently selecting the superior one. The population for which the procedure is optimized has size N and includes both the patients in the trial and those who are treated with the chosen alt...

In Anscombe's classical model, the objective is to find the optimal sequential rule for learning about the difference between two alternative treatments and subsequently selecting the superior one. The population for which the procedure is optimised has size $N$ and includes both the patients in the trial and those which are treated with the chosen...

We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone str...

We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone str...

Standard Markovian optimal stopping problems are consistent in the sense that the first entrance time into the stopping set is optimal for each initial state of the process. Clearly, the usual concept of optimality cannot in a straightforward way be applied to non-standard stopping problems without this time-consistent structure. This paper is devo...

This paper studies explicitly solvable multidimensional optimal stopping problems of sum- and product-type in discrete and continuous time using the monotone case approach. It gives a review on monotone case stopping using the Doob decomposition, resp. Doob-Meyer decomposition in continuous time, also in its multiplicative versions. The approach vi...

A general method for solving optimal stopping problems for continuous-time Markov processes on the real line is developed. The basic idea is to first study an auxiliary problem for the two-dimensional process consisting of the underlying Markov process and its running maximum. It turns out that this auxiliary problem is much easier to solve using s...

This article contains responses to the four discussion pieces written by experts, commenting on the results presented in Christensen (2017 Christensen, S. (2017). An Effective Method for the Explicit Solution of Sequential Problems on the Real Line, Sequential Analysis 36: 2–18.[Taylor & Francis Online] [Google Scholar]).

This paper establishes existence of optimal controls for a general stochastic impulse control problem. For this, the value function is characterized as the pointwise minimum of a set of superharmonic functions, as the unique continuous viscosity solution of the quasi-variational inequalities (QVIs), and as the limit of a sequence of iterated optima...

In this paper, asymptotic results in a long-term growth rate portfolio optimization model under both fixed and proportional transaction costs are obtained. More precisely, the convergence of the model when the fixed costs tend to zero is investigated. A suitable limit model with purely proportional costs is introduced and an optimal strategy is sho...

In this paper, asymptotic results in a long-term growth rate portfolio optimization model under both fixed and proportional transaction costs are obtained. More precisely, the convergence of the model when the fixed costs tend to zero is investigated. A suitable limit model with purely proportional costs is introduced and an optimal strategy is sho...

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the $d$-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function associated with the problem. Specializing t...

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the $d$-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function associated with the problem. Specializing t...

As a model for an on-line classification setting we consider a stochastic process $(X_{-n},Y_{-n})_{n}$, the present time-point being denoted by 0, with observables $ \ldots,X_{-n},X_{-n+1},\ldots, X_{-1}, X_0$ from which the pattern $Y_0$ is to be inferred. So in this classification setting, in addition to the present observation $X_0$ a number $l...

We study a portfolio optimization problem in a financial market which is under the threat of crashes. At random times, the investor receives warnings that a bubble has formed in the market which may lead to a crash in the risky asset. We propose a regime switching model for the warnings and we make no assumptions about the distribution of the timin...

In their article, Lang, Weiss, Gerstorf, & Wagner, (2013) use the adult life span sample of the national German Socio-Economic Panel (GSOEP) to explore functional outcomes of life satisfaction with regard to hazards of mortality. Their findings suggest that “being overly optimistic [in] predicting a better future than actually observed was associat...

We study the problem of maximizing expected utility of terminal wealth under constant and proportional transactions costs in a multi-asset market in which prices are driven by a multidimensional factor process. We characterize the value function as the pointwise minimum of a set of superharmonic functions and as the unique continuous viscosity solu...

Mit der Durchführung des Zensus 2011 ist die Bundesrepublik Deutschland ihrer Verpflichtung aus der EU-Verordnung 763/2008 vom 9. Juli 2008, eine Volkszählung nach festgelegten Kriterien durchzuführen, nachgekommen. Im Gegensatz zu früheren Volkszählungen (1987 BRD und 1981 DDR) wurde beim Zensus 2011 nicht auf eine Vollerhebung zurückgegriffen, so...

Background In the data report 2014, the German Maternal Convalescence Movement (Müttergenesungswerk) identified a possible discrimination against mothers with a low level of education in health convalescent treatment allocation. Objective The question whether this discrimination is supported by data is investigated. Material and methods Based on a...

Die Anwendung von Statistik ist häufig diffizil und man kann in viele unterschiedliche Fallen tappen. Umso wichtiger ist es, einige Grundphänomene, welche immer wieder auftreten, gut verinnerlicht zu haben. In diesem Kapitel werden, wie immer knapp verpackt, einige der wichtigsten dieser Phänomene dargestellt.

„Brasilien – Kolumbien: 2:1“. So könnte der Inhalt einer sicher tausendfach verschickten SMS aus dem Stadion von Fortaleza nach dem entsprechenden Viertelfinale bei der WM 2014 lauten. Obwohl der Versender der SMS das ganze Spiel mit all seinen Strafraumszenen, Torwartleistungen und Entscheidungen des Schiedsrichters miterlebt hat, verschickt er an...

Einige der zweifelhaften Verwendungen von Statistik, die in diesem Buch bis hierher beschrieben wurden, waren vielleicht eher zum Schmunzeln, andere zum Nachdenken oder auch zum Ärgern. Aber spätestens wenn auf statistischer Grundlage über die Freiheit eines Menschen entschieden wird, ist höchste Sorgfalt geboten. Auch wenn der Gebrauch von Statist...