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Kristoffer Lindensjö

Publications and source records attributed to Kristoffer Lindensjö.

16 recordsLinked to original sources
arXiv2025

Splitting infinity: a de Finetti game with state-dependent profit rates and singular control for diffusions

We study a game of resource extraction of a common good under one-dimensional diffusive dynamics with player actions corresponding to singular stochastic control up to absorption at $0$, implying a trade-off between profitable resource extraction and sustainability. Unsurprisingly, immediate extraction of all available resources is an equilibrium. A main result is that we characterize and prove the existence of non-trivial equilibria that do not result in immediate absorption, but instead are attained with both players extracting resources according to a state-dependent rate of threshold type, corresponding to the presence of control only when the state process is in an interval $(b,\infty)$. The underlying assumption is, roughly, that the drift coefficient of the uncontrolled state process grows sufficiently fast in relation to the discount rate, implying that the value for the corresponding one-player problem is infinite. We also study a generalization of the game that allows a state-dependent profit rate integrated against the control processes. In this game we again characterize and prove the existence of non-trivial equilibria of threshold type. In particular, a main novelty is that we find equilibria where the state process is controlled with its own local time such that we have reflection points with associated initial jumps, as well as other points in the state space where the control processes increase in a singular manner (skew points).

math.PR↗
arXiv2025

Markovian randomized equilibria for general Markovian Dynkin games in discrete time

We study a general formulation of the classical two-player Dynkin game in a discrete time Markovian setting. We identify an appropriate class of mixed strategies -- \textit{Markovian randomized stopping times} -- in which players stop at any given state with a state-dependent probability. One main result is an explicit characterization of Wald-Bellman-type for Nash equilibria based on this notion of randomization. In particular, we derive a novel characterization of randomized equilibria in zero-sum Dynkin games, which we use to (i) establish the existence and explicit construction of Markovian randomized equilibria, (ii) provide necessary and sufficient conditions for the non-existence of pure strategy equilibria, and (iii) construct an example that admits a unique randomized equilibrium but no pure one. We also provide existence and characterization results in the symmetric version of our game. Finally, we establish existence of a characterizable equilibrium in Markovian randomized stopping times for the general game formulation under the assumption that the state space is countable.

math.PR↗
arXiv2025

Time-inconsistent singular control problems: Reflection and Absolutely continuous controls with exploding rates

We study a time-inconsistent singular stochastic control problem for a general one-dimensional diffusion, where time-inconsistency arises from a non-exponential discount function. To address this, we adopt a game-theoretic framework and study the optimality of a novel class of controls that encompasses both traditional singular controls -- responsible for generating multiple jumps and reflective boundaries (strong thresholds) -- and new mild threshold control strategies, which allow for the explosion of the control rate in absolutely continuous controls, thereby creating an inaccessible boundary (mild threshold) for the controlled process. We establish a general verification theorem, formulated in terms of a system of variational inequalities, that provides both necessary and sufficient conditions for equilibrium within the proposed class of control strategies and their combinations. To demonstrate the applicability of our theoretical results, we examine case studies in inventory management. We show that for certain parameter values, the problem admits a strong threshold control equilibrium in the form of Skorokhod reflection. In contrast, for other parameter values, we prove that no such equilibrium exists, necessitating the use of our extended control class. In the latter case, we explicitly construct an equilibrium using a mild threshold control strategy with a discontinuous, increasing, and exploding rate that induces an inaccessible boundary for the optimally controlled process, marking the first example of a singular control problem with such a solution structure.

math.OC↗
arXiv2024

General Markovian randomized equilibrium existence and construction in zero-sum Dynkin games for diffusions

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 $ε$-Nash equilibrium in Markovian randomized stopping times for every $ε> 0$. This is achieved by leveraging the well-known equilibrium structure under a restrictive ordering condition on the payoff functions, leading to a novel approach based on an appropriate notion of randomization that allows for solving the general game without any ordering condition. Additionally, we provide conditions for the existence of pure and randomized Nash equilibria (with $ε=0$). Our results enable explicit identification of equilibrium stopping times and their corresponding values in many cases, illustrated by several examples.

math.PR↗
arXiv2022

Local time pushed mixed stopping and smooth fit for time-inconsistent stopping problems

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 construction of the stopping intensity. For a general formulation of the problem we provide a verification theorem giving sufficient conditions for mixed (and pure) equilibria in terms of a set of variational inequalities, including a smooth fit condition. We apply the theory to prove the existence of (mixed) equilibria in a recently studied real options problem in which no pure equilibria exist.

math.PR↗
arXiv2021

De Finetti's control problem with competition

We investigate the effects of competition in a problem of resource extraction from a common source with diffusive dynamics. In the symmetric version with identical extraction rates we prove the existence of a Nash equilibrium where the strategies are of threshold type, and we characterize the equilibrium threshold. Moreover, we show that increased competition leads to lower extraction thresholds and smaller equilibrium values. For the asymmetric version, where each agent has an individual extraction rate, we provide the existence of an equilibrium in threshold strategies, and we show that the corresponding thresholds are ordered in the same way as the extraction rates.

math.OC↗
arXiv2020

How to detect a salami slicer: a stochastic controller-stopper game with unknown competition

We consider a stochastic game of control and stopping specified in terms of a process $X_t=-θΛ_t+W_t$, representing the holdings of Player 1, where $W$ is a Brownian motion, $θ$ is a Bernoulli random variable indicating whether Player 2 is active or not, and $Λ$ is a non-decreasing process representing the accumulated "theft" or "fraud" performed by Player 2 (if active) against Player 1. Player 1 cannot observe $θ$ or $Λ$ directly, but can merely observe the path of the process $X$ and may choose a stopping rule $τ$ to deactivate Player 2 at a cost $M$. Player 1 thus does not know if she is the victim of fraud and operates in this sense under unknown competition. Player 2 can observe both $θ$ and $W$ and seeks to choose the fraud strategy $Λ$ that maximizes the expected discounted amount \[{\mathbb E} \left [θ\int _0^τ e^{-rs} dΛ_s \right ],\] whereas Player 1 seeks to choose the stopping strategy $τ$ so as to minimize the expected discounted cost \[{\mathbb E} \left [θ\int _0^τ e^{-rs} dΛ_s + e^{-rτ}M{\mathbb I}_{\{τ<\infty\}} \right ].\] This non-zero-sum game appears to be novel and is motivated by applications in fraud detection; it combines filtering (detection), non-singular control, stopping, strategic features (games) and asymmetric information. We derive Nash equilibria for this game; for some parameter values we find an equilibrium in pure strategies, and for other parameter values we find an equilibrium by allowing for randomized stopping strategies.

math.OC↗
arXiv2020

Time-inconsistent stopping, myopic adjustment & equilibrium stability: with a mean-variance application

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 necessary and sufficient equilibrium conditions including a verification theorem. Using a fixed point argument we provide equilibrium existence results. We adapt and study the notion of the myopic adjustment process and introduce different kinds of equilibrium stability. We show that neither existence nor uniqueness of equilibria should generally be expected. The developed theory is applied to a mean-variance problem and a variance problem.

math.OC↗
arXiv2020

On time-inconsistent stopping problems and mixed strategy stopping times

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 subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other necessary and sufficient equilibrium conditions including a smooth fit result are proved. Existence and uniqueness are investigated. A mean-variance and a variance problem are studied. The state process is a general one-dimensional Itô diffusion.

math.OC↗
arXiv2019

Moment constrained optimal dividends: precommitment \& consistent planning

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' consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

math.OC↗
arXiv2019

A regular equilibrium solves the extended HJB system

Control problems not admitting the dynamic programming principle are known as time-inconsistent. The game-theoretic approach is to interpret such problems as intrapersonal dynamic games and look for subgame perfect Nash equilibria. A fundamental result of time-inconsistent stochastic control is a verification theorem saying that solving the extended HJB system is a sufficient condition for equilibrium. We show that solving the extended HJB system is a necessary condition for equilibrium, under regularity assumptions. The controlled process is a general Itô diffusion.

math.OC↗
arXiv2019

Optimal dividends and capital injection under dividend restrictions

We study a singular stochastic control problem faced by the owner of an insurance company that dynamically pays dividends and raises capital in the presence of the restriction that the surplus process must be above a given dividend payout barrier in order for dividend payments to be allowed. Bankruptcy occurs if the surplus process becomes negative and there are proportional costs for capital injection. We show that one of the following strategies is optimal: (i) Pay dividends and inject capital in order to reflect the surplus process at an upper barrier and at 0, implying bankruptcy never occurs. (ii) Pay dividends in order to reflect the surplus process at an upper barrier and never inject capital --- corresponding to absorption at 0 --- implying bankruptcy occurs the first time the surplus reaches zero. We show that if the costs of capital injection are low, then a sufficiently high dividend payout barrier will change the optimal strategy from type (i) (without bankruptcy) to type (ii) (with bankruptcy). Moreover, if the costs are high, then the optimal strategy is of type (ii) regardless of the dividend payout barrier. The uncontrolled surplus process is a Wiener process with drift.

math.OC↗
arXiv2018

On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

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 devoted to the solution of time-inconsistent stopping problems with the reward depending on the initial state using an adaptation of Strotz's consistent planning. More precisely, we give a precise equilibrium definition --- of the type subgame perfect Nash equilibrium based on pure Markov strategies. In general, such equilibria do not always exist and if they exist they are in general not unique. We, however, develop an iterative approach to finding equilibrium stopping times for a general class of problems and apply this approach to one-sided stopping problems on the real line. We furthermore prove a verification theorem based on a set of variational inequalities which also allows us to find equilibria. In the case of a standard optimal stopping problem, we investigate the connection between the notion of an optimal and an equilibrium stopping time. As an application of the developed theory we study a selling strategy problem under exponential utility and endogenous habit formation.

math.OC↗
arXiv2018

The value of a liability cash flow in discrete time subject to capital requirements

The aim of this paper is to define the market-consistent multi-period value of an insurance liability cash flow in discrete time subject to repeated capital requirements, and explore its properties. In line with current regulatory frameworks, the approach presented is based on a hypothetical transfer of the original liability and a replicating portfolio to an empty corporate entity whose owner must comply with repeated one-period capital requirements but has the option to terminate the ownership at any time. The value of the liability is defined as the no-arbitrage price of the cash flow to the policyholders, optimally stopped from the owner's perspective, taking capital requirements into account. The value is computed as the solution to a sequence of coupled optimal stopping problems or, equivalently, as the solution to a backward recursion.

q-fin.RM↗
arXiv2017

An explicit formula for optimal portfolios in complete Wiener driven markets: a functional Itô calculus approach

We consider a standard optimal investment problem in a complete financial market driven by a Wiener process and derive an explicit formula for the optimal portfolio process in terms of the vertical derivative from functional It^o calculus. An advantage with this approach compared to the Malliavin calculus approach is that it relies only on an integrability condition.

q-fin.MF↗