1. Optimal stopping: past and present
In this brief introduction to this special issue of Stochastics, it seems appropriate to review briefly the history of Optimal Stopping.
Optimal stopping theory has its roots in classical calculus of variations. The three problem formulations due to Lagrange (18th century), Mayer (19th century) and Bolza (in 1913) were inherited by (stochastic) optimal control theory in the 1940s and 1950s (Bellman). At about the same time, optimal stopping problems emerged from work by Wald in relation to problems of sequential testing. Key principles of optimal stopping were established by Snell in 1952 (Snell's envelope) and Dynkin in 1963 (superharmonic characterization). In the 1950s and 1960s, a fundamental connection between optimal stopping and free boundary problems was discovered by a number of researchers (Mikhalevich, Chernoff, Lindley, McKean, Shiryaev). This equivalence penetrates deeply into the fundamentals of modern probability theory (Einstein, Wiener, Kolmogorov) via exit problems (Kakutani) and their connection with analysis through boundary value problems (Dirichlet, Poisson, Neumann, Cameron, Martin, Girsanov, Feynman, Kac). This connection is one of the most fascinating accomplishments of modern probability theory (even, one might argue, of modern mathematics). Studies of martingales and Markov processes are central to optimal stopping problems and the concepts of filtration (information) and stopping times (non-anticipation) are key to real world applications.
A huge stimulus to the development of optimal stopping theory was provided by option pricing theory, developed in the late 1960s and the 1970s. According to the modern theory of finance, pricing an American option in a complete market is equivalent to solving an optimal stopping problem (with a corresponding generalisation in incomplete markets), the optimal stopping time being the rational time for the option to be exercised. Thanks to the enormous importance of the American price mechanism in finance, this line of research has been intensively pursued in recent times. The most recent breakthrough involves extensions of the classic Itô formula to account for local time on curves and surfaces leading to the so-called “local time-space calculus”. In parallel to the classic problem formulations of Lagrange, Mayer and Bolza mentioned above, in the 1990s several researchers independently started to study problem formulations based on the maximum process (Jacka, Dubins, Shepp, Shiryaev, Graversen, Pedersen, Peskir). Solutions of optimal stopping problems of this type are intimately related to the fundamental inequalities of probability theory (Doob, Hardy–Littlewood, Burkholder–Davis–Gundy) and several deep and intriguing problems are still open. Further studies of optimal stopping presently in train involve optimal prediction, sequential testing, quickest detection, and free-boundary problems for PIDEs (instead of PDEs).
2. Manchester, January 2006
The complexity, breadth and depth of the concepts involved in Optimal Stopping for some time demanded the bringing together of the international community of researchers and aspiring students for dialogue and interaction, as is traditional in mathematics.
This was finally made possible in fairly grand style by the generous financial assistance of the Engineering and Physical Sciences Research Council (EPSRC), the London Mathematical Society (LMS) and the Manchester Institute for Mathematical Sciences (MIMS) from within the UK, and of Advanced Methods for Mathematical Finance (AMAMEF) from The European Science Foundation and École Polytechnique Fédérale de Lausanne (EPFL). Between 17–21 January 2006 a school was hosted by The University of Manchester on the topic of Optimal Stopping with Applications. This was followed on 22–27 January 2006 by an international symposium hosted at the same venue and dedicated to the same theme.
The school was attended by 38 UK based students (from 14 different institutions) and 20 oversees students/participants (from 12 different countries). The school aimed to prepare students for the Symposium and in this vein was highly successful. The symposium was a high-profile meeting on an unprecedented scale in the history of optimal stopping. There were 46 speakers (22 from Denmark, Finland, France, Germany, Italy, the Netherlands, Norway, Poland, Switzerland; 10 from the UK; 8 from the USA; 3 from Russia; and 1 each from Australia, Israel and Uruguay), 24 UK based students (from 10 different institutions), and a total of 95 participants.
3. Special issues of stochastics
To celebrate the overwhelming success of these meetings and the synergy and enthusiasm that was experienced, it was agreed that a special issue of Stochastics would be dedicated to a seminal series of papers on the topic of Optimal Stopping With Applications. Participants at the meeting responded overwhelmingly resulting in articles which will appear in this and the next double issue of Stochastics.
These two double issues have been edited by the Editor-in-chief and the two guest editors, Andreas Kyprianou and Goran Peskir. All papers were refereed and processed in the usual way. As principle organisers of the workshop and symposium in Manchester, we should like to thank all participants at the Manchester events as well as the authors and referees for helping to create something which we believe will be a significant academic point of reference for many years to come in the field of optimal stopping.
4. Appendix
Below is an alphabetical list of talks and poster titles from the Manchester Symposium on Optimal Stopping with Applications.
Alvarez, Luis H.R.: Optimal Harvesting Under Resource Stock and Price Uncertainty (poster).
Attal, Stephane: Stopping Times in Quantum Mechanics.
Boguslavskaya, Elena: On Optimization of Dividends for an Insurance Company in the Presence of Liquidation Value (poster).
Ceci, Claudia: Optimal Stopping Problems with Semicontinuous Reward: Regularity of the Value Function and Viscosity Solutions.
Chen, Xinfu: Free Boundary in First Cross Problem Arising from Risk Management.
Dalang, Robert: A Quickest Detection Problem with an Observation Cost.
Dayanik, Savas: Compound Poisson Disorder Problem.
Davis, Mark H.A.: On the Non-Anticipativity Constraint in Optimal Stopping.
Delbaen, Freddy: Optimising Maximal Monotone Sequences in Poisson Arrivals.
Dupire, Bruno: Free Boundary Solutions of the Skorohod Embedding Problem.
Eberlein, Ernst: Valuation of Floating Range Notes in Lévy Term Structure.
Ekström, Erik: On the Value of Optimal Stopping Games (poster).
El Karoui, Nicole: Supermartingale Decomposition in the Max-Plus Algebra, with Applications to American options, and Constrained Martingale Optimization Problem.
Irle, Albrecht: Solving Problems of Optimal Stopping with Linear Costs.
Kifer, Yuri: Binomial approximations of game options and related probability problems.
Gapeev, Pavel V.: Perpetual Options in Jump-Diffusion Models: Barrier, Credit, Lookback and Switching Options (poster).
Gnedin, Alexander V.: Some Asymptotics in the Problem of Recognising the Last Record.
Hobson, David: The Curious Incident of the Investment in the Market: Real Options, Optimal Stopping and Optimal Control.
Hudson, Robin L.: Stop Times and Some Strong Markov Processes in Quantum Probability.
Jamshidian, Farshid: Numeraire-Invariant American Option Pricing and Minimax Duality of the Snell Envelope.
Kabanov, Yuri: Subtleties in the Theory of Financial Markets with Transaction Costs.
Kühn, Christoph: Callable Puts as Composite Exotic Options (poster).
Kyprianou, Andreas E.: A Conjecture Concerning the Smooth Fit Versus the Continuous Fit Principle.
Lamberton, Damien: Optimal Stopping of a One-Dimensional Diffusion.
Lerche, Hans Rudolf: A Martingale Approach to Optimal Stopping.
Løkka, Arne: Bayesian Detection of a Change Point Before an Observable Event (poster).
Ludkovski, Michael: Optimal Switching for Energy Derivatives (poster).
Mazalov, Vladimir V.: On the Duration Problem on Trajectories.
Mordecki, Ernesto: Ruin Probability and Optimal Stopping for Lévy Processes.
Novikov, Alexander A.: On a Solution of the Optimal Stopping Problem for Random Walks and Lévy Processes.
Obloj, Jan: From an Optimal Stopping Problem via Skorokhod Embeddings to ML-Martingales (poster).
Øksendal, Bernt: Optimal Stopping, Impulse Control and Delayed Reaction.
Peskir, Goran: The Trap of Complacency in Predicting the Maximum.
Pham, Huyen: Explicit Solution to an Optimal Switching Problem in the Two Regime Case.
Pistorius, Martijn: On the Optimal Dividend Problem for a Spectrally Negative Lévy Process (poster).
Pedersen, Jesper Lund: Principle of Smooth Fit at a Single Point (poster).
Rogers, L. Chris G.: Deterministic Stochastic Optimal Control.
Ruschendorf, Ludger: Approximation of Optimal Stopping Problems.
Salminen, Paavo: Solving Optimal Stopping Problems via the Representation Theory of Excessive Functions.
Shepp, Larry: Convexity and Optimal Control Theory.
Shiryaev, Albert N.: Optimal Stopping in the Quickest Detection of the Spontaneously Appearing Effects (Prehistory and the Route of the Solution).
Snell, J. Laurie: The Early Days of Optimal Stopping.
Stockbridge, Richard H.: Optimal Stopping of Singular Stochastic Processes via Linear Programming.
Szajowski, Krzysztof: Correlated Equilibria in Competitive Staff Selection Problem.
Zariphopoulou, Thaleia: Early Exercise, Indifference Valuation and Optimal Investments.
Zervos, Mihail: A Discretionary Stopping Problem with Applications to the Optimal Timing of Investment Decisions.
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