Kristofer LindensjöAssociate Professor
About me
I am an Associate Professor (Sw: Docent, Universitetslektor) with focus on financial mathematics and mathematical economics. My research is mainly within the theory for stochastic processes, stochastic control, optimal stopping and stochastic games in continuous time, and is typically motivated by problems in mathematical finance, economics and insurance.
PhD student and postdoc positions: probability, stochastic processes, stochastic control, optimal stopping
PhD student positions
From time to time, I have open PhD positions within the general area of probability, stochastic processes, stochastic control, optimal stopping and stochastic dynamic game theory. Positions are usually announced in March/April.
If you are interested in applying in my field then do not forget to specifically mention my name and/or the project title "Stochastic control and optimal stopping for game theory" in your application letter.
PhD student project: Stochastic control and optimal stopping for game theory
General information: PhD studies
Postdoc positions
Are you interested in a postdoc position within stochastic processes, stochastic control, optimal stopping or a related research area? Each year the Lerheden foundation finances several postdoc positions (usually two-tree year durations) at the mathematics department at Stockholm University. If this sounds interesting to you, then you may at any point in time send me an email indicating your interest. No fancy preparation is needed for this, it is enough that you write from where your PhD is, and what your research/PhD thesis is about. When formally applying for the position, you will have to have a research plan (usually circa 5 pages).
Want to write a bachelor or master thesis?
For students at the Stockholm university mathematics department: If you think that you may want to write a thesis (bachelor or master) under my supervision (preferably within stochastic processes, stochastic control, optimal stopping, mathematical finance, or related subjects), then feel free to contact me for an informal discussion. No fancy preparation needed, just send me a short email.
What is stochastic control and optimal stopping theory?
A stochastic process is a collection of random variables indexed by a time variable. The theory of stochastic processes is central to probability theory, as well as several areas of applied mathematics. Optimal stochastic control and stopping theory is a major field within the general research area of stochastic processes.
Suppose we can decide when to stop a stochastic process and that we want to do so according to some optimality criterion, which is usually related to a conditional expected value based on the process. We then face an optimal stopping problem.
Now suppose that we instead can control, to some extent, the evolution of a stochastic process and that we want to do this according to an optimality criterion. We then face a stochastic control problem.
Stochastic control and stopping theory has proven to be extremely useful for modelling dynamic decision making over time in random environments with applications in e.g., biology, engineering, and mathematical economics/finance.
Stochastic dynamic game theory
Stochastic dynamic game theory—i.e., game theory regarding the control of stochastic processes dynamically over time—has recently become an important part of the general field of stochastic control and optimal stopping.
Game theory can be informally motivated as follows. If there are two or more decision-makers, then the notion of an optimal solution is unsuitable in cases where their objectives do not align; they may be able to agree on a solution but it will generally not be optimal from either one of their view-points. In these situations, the game theory approach—dating back to e.g., the Nobel laureate John Nash and the seminal work Nash (1950)—is to instead consider the notion of a (Nash) equilibrium solution.
There are still fundamental open problems remaining within stochastic dynamic game theory. In particular, randomized strategies and fixed point techniques—which are fundamental for Nash equilibrium existence results in classical game theory—are unexplored to a large extent for stochastic dynamic game theory. Moreover, the problem of how to generally define randomized strategies for Markov processes in order to preserve the solution characterization possibilities (which facilitate computation) that these provide for non-randomized strategies is still open.
Research output, background, and teaching
Preprints
[15] Sören Christensen, K. Lindensjö & Berenice Anne Neumann. Markovian randomized equilibria for general Markovian Dynkin games in discrete time (link to arXiv preprint), submitted.
A general formulation of the classical two-player Dynkin (stopping) game is studied in a Markovian discrete time setting. We show that an appropriate class of randomized strategies are Markovian randomized stopping times, which correspond to stopping at any given state with a state-dependent probability. We establish a general equilibrium existence result using a fixed-point technique, as well as equilibrium characterization.
Publications
[14] Andi Bodnariu & K. Lindensjö. A controller-stopper-game with hidden controller type (link to open access journal paper), Stochastic Processes and their Applications, Volume 173, July 2024, 104361.
A novel controller-stopper game under asymmetric information is studied. A main contribution is that we introduce and study a certain weak formulation for stochastic dynamic games based on diffusions. Another contribution is that we constrain the control process to take values in a finite set, which implies that the problem of the controller can be interpreted as an optimal switching problem, with switching occurring infinitely often.
[13] Andi Bodnariu, Sören Christensen & K. Lindensjö. Local time pushed mixed stopping and smooth fit for time-inconsistent stopping problems (link to arXiv preprint), SIAM Journal on Control and Optimization, 62(2), 2024.
A stopping game corresponding to a time-inconsistent stopping problem is studied. A main contribution is the introduction of a novel class of randomized stopping times based on a local time construction. The developed theory is used to prove the existence of randomized equilibria in a recently studied real options problem in which no non-randomized equilibrium exists. The equilibria are characterized in terms of variational inequalities.
[12] Erik Ekström & K. Lindensjö. De Finetti’s Control Problem with Competition (link to journal, open access), Applied Mathematics and Optimization, 87, 16 (2023).
We study a stochastic dynamic game of resource extraction with diffusive dynamics. In the symmetric case with identical maximal extraction rates we prove the existence of an equilibrium of threshold type which is also characterized. We also provide existence of an equilibrium in threshold strategies using a fixed-point approach for the asymmetric case.
[11] Erik Ekström, K. Lindensjö & Marcus Olofsson. How to detect a salami slicer: a stochastic controller-stopper game with unknown competition (link to arXiv preprint), SIAM Journal on Control and Optimization, 60(1), 2022. https://doi.org/10.1137/21M139044X.
A novel non-zero-sum stochastic dynamic game of controller-stopper type for fraud detection is introduced; it combines stochastic filtering (detection), non-singular control, stopping and asymmetric information. We derive pure as well as randomized Nash equilibria.
[10] Sören Christensen & K. Lindensjö. Moment constrained optimal dividends: precommitment & consistent planning (link to arXiv preprint), Advances in Applied Probability 54(2), 2022. https://doi.org/10.1017/apr.2021.38.
A new type of time-inconsistent stochastic impulse control problem is studied as a stochastic dynamic game. Suitable notions of pure strategies and a (strong) subgame perfect Nash equilibrium are defined. An equilibrium is derived using a new smooth fit condition.
[9] Sören Christensen & K. Lindensjö. On time-inconsistent stopping problems and mixed strategy stopping times (link to journal), Stochastic Processes and their Applications, 130(5), 2886-2917, 2020 (link to arXiv preprint).
The game theory approach to time-inconsistent stopping is studied. A novel class of randomized stopping strategies that allow the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. The equilibrium is characterized using a system of variational (differential) inequalities. Necessary and sufficient equilibrium conditions are established and used in examples.
[8] Hampus Engsner, K. Lindensjö & Filip Lindskog. The value of a liability cash flow in discrete time subject to capital requirements (link to journal, open access), Finance and Stochastics, 24, 125-167, 2020.
We define the market-consistent multi-period value of a liability cash flow in discrete time subject to repeated capital requirements. The value is computed as the solution to a sequence of coupled optimal stopping problems.
[7] Sören Christensen & K. Lindensjö. Time-inconsistent stopping, myopic adjustment & equilibrium stability: with a mean-variance application (link to publication, open access), in Stochastic Modeling and Control, (eds: J. Jakubowski, M. Niewęgłowski, M. Rásonyi and Ł. Stettner), Banach Center Publications (Institute of Mathematics, Polish Academy of Sciences) 122, 53-76, 2020.
[6] K. Lindensjö & Filip Lindskog. Optimal dividends and capital injection under dividend restrictions (link to journal, open access), Mathematical Methods of Operations Research, 92(3):461–487, 2020.
[5] K. Lindensjö. A regular equilibrium solves the extended HJB system (link to journal), Operations Research Letters, 47(5), 427–432, 2019, (link to arXiv preprint).*
An important result of the game theory approach to time-inconsistent stochastic control is that solving a system of PDEs known as the extended HJB system is a sufficient condition for equilibrium. In this paper I show that solving the extended HJB system is also a necessary condition for equilibrium, under regularity assumptions.
[4] Sören Christensen & K. Lindensjö. On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems (link to journal), SIAM Journal on Control and Optimization, 56(6), 4228–4255, 2018, (link to arXiv preprint).
For a general time-inconsistent stopping problem in continuous time we provide mathematical definitions for pure strategies and an equilibrium concept. In the case of a general Markov process, we provide a novel iterative approach to finding equilibrium stopping times. In the case of a general Itô diffusion, we provide a verification theorem based on a set of variational inequalities which also allows us to find equilibria.
[3] K. Lindensjö. Constructive martingale representation in functional Itô calculus: a local martingale extension (link to publication), chapter 9 in S. Silvestrov et al. (eds.), Stochastic Processes and Applications, Springer Proceedings in Mathematics & Statistics 271, 165–172, 2018, (link to arXiv preprint).
[2] K. Lindensjö. Optimal investment and consumption under partial information (link to journal), Mathematical Methods of Operations Research, 83(1):87-107, 2016.
[1] K. Lindensjö. The end of the month option and other embedded options in futures contracts (link to journal), Asia-Pacific Financial Markets, 23(1):69-83, 2016.
Co-authors (alphabetical order): Andi Bodnariu (current PHD student), Berenice Anne Neumann, Sören Christensen, Erik Ekström, Hampus Engsner, Filip Lindskog, Marcus Olofsson, Joanna Tyrcha.
*An early preprint version of this paper had the title Time-inconsistent stochastic control: solving the extended HJB system is a necessary condition for regular equilibria.
Academic positions
Associate Professor at the Department of Mathematics, Stockholm University (Spring semester 2020 – present).
Researcher at the Department of Mathematics, Uppsala University (Fall semester 2019).
Lecturer and postdoc at the Department of Mathematics, Stockholm University (Spring semester 2016 – Spring semester 2019).
Visiting Researcher at the Department of Finance, Copenhagen Business School (Fall semester 2013, during PhD studies).
Visiting Researcher at the Department of Mathematics, ETH Zurich (Fall semester 2010, Spring semester 2012, during PhD studies).
Education
Docent - Department of Mathematics, Stockholm University, Sep. 2020.
PhD - Mathematical Finance, Stockholm School of Economics, Nov. 2013. Supervisor: Tomas Björk, professor of Mathematical Finance. Dissertation title: Essays in Financial Mathematics.
MSc - Mathematics, Stockholm University, June 2006.
MSc - Economics and Business, specialization in Finance, Stockholm School of Economics, June 2006.
Teaching and teaching material (bachelor, master and PhD student level)
I have given courses on bachelor, master and PhD student level in e.g.,: Stochastic processes, Stochastic differential equations, Stochastic analysis, Optimal stopping, Stochastic control, Probability theory, Financial mathematics, Mathematical economics, Economics, Econometrics, and Statistics. I have supervised several Bachelor’s and Master’s Degree projects.
Patrik Andersson, K. Lindensjö & Joanna Tyrcha. Notes in Econometrics, Available at the Department of Mathematics, Stockholm University, 2018.
I teach for example within the following programmes (in Swedish):
Studera kandidatprogrammet i matematisk ekonomi och statistik
Studera masterprogrammet i försäkringsmatematik (aktuarieprogrammet)
Nonacademic positions after PhD studies
Handelsbanken Capital Markets, Stockholm, 2014-2015. I had several positions, the last one as a derivatives trader.
See also: Kristoffer Lindensjö on ResearchGate