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Magnus Perninge

Publications and source records attributed to Magnus Perninge.

18 recordsLinked to original sources
arXiv2026

Feedback stopping rules in path-dependent controller-stopper games

We investigate finite-horizon, zero-sum controller-stopper games in which the stopper observes the state process and implements a feedback stopping rule. Building on a nonlinear Snell envelope representation for a related game established in a companion paper, we prove that our game admits a value by extending the associated first-contact principle. Our approach is purely probabilistic and yields an optimal feedback stopping rule for path-dependent systems while allowing for degeneracy in the underlying stochastic differential equation (SDE). As an application, we consider nonzero-sum controller-stopper games and show that the optimal feedback stopping rule for a zero-sum game appears as a component of a $\varepsilon$-Nash equilibrium for every $\varepsilon>0$.

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arXiv2026

A Nonlinear Snell Envelope Representation for Path-Dependent Controller-Stopper Games

We consider a finite-horizon, zero-sum stochastic differential game in which one player controls a path-dependent stochastic system, while the opponent is given the opportunity to terminate the game prematurely. We introduce a control randomization formulation, which allows us to establish that the upper and lower value functions coincide. Our approach also yields a representation of the common game value in terms of a nonlinear Snell envelope, where the underlying stopped process is given by the unique maximal solution of a backward stochastic differential equation (BSDE) with constrained jumps.

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arXiv2025

Zero-sum Stochastic Differential Games of Impulse Control with Random Intervention Costs

We consider a finite-horizon, zero-sum game in which both players control a stochastic differential equation by invoking impulses. We derive a control randomization formulation of the game and use the existence of a value for the randomized game to show that the upper and lower value functions of the original game coincide. The main contribution of the present work is that we can allow intervention costs that are functions of the state as well as time, and that we do not need to impose any monotonicity assumptions on the involved coefficients.

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arXiv2024

Probabilistic Representation for Viscosity Solutions to Double-Obstacle Quasi-Variational Inequalities

We prove the existence and uniqueness of viscosity solutions to quasi-variational inequalities (QVIs) with both upper and lower obstacles. In contrast to most previous works, we allow all involved coefficients to depend on the state variable and do not assume any type of monotonicity. It is well known that double obstacle QVIs are related to zero-sum games of impulse control, and our existence result is derived by considering a sequence of such games. Full generality is obtained by allowing one player in the game to randomize their control. A by-product of our result is that the corresponding zero-sum game has a value, which is a direct consequence of viscosity comparison. Utilizing recent results for backward stochastic differential equations (BSDEs), we find that the unique viscosity solution to our QVI is related to optimal stopping of BSDEs with constrained jumps and, in particular, to the corresponding non-linear Snell envelope. This gives a new probabilistic representation for double obstacle QVIs. It should be noted that we consider the min-max version (or equivalently the max-min version); however, the conditions under which solutions to the min-max and max-min versions coincide remain unknown and is a topic left for future work.

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arXiv2024

Optimal Stopping of BSDEs with Constrained Jumps and Related Double Obstacle PDEs

We consider partial differential equations (PDEs) characterized by an upper barrier that depends on the solution itself and a fixed lower barrier, while accommodating a non-local driver. First, we show a Feynman-Kac representation for the PDE when the driver is local. Specifically, we relate the non-linear Snell envelope for an optimal stopping problem, where the underlying process is the first component in the solution to a stopped backward stochastic differential equation (BSDE) with jumps and a constraint on the jumps process, to a viscosity solution for the PDE. Leveraging this Feynman-Kac representation, we subsequently prove existence and uniqueness of viscosity solutions in the non-local setting by employing a contraction argument. In addition, the contraction argument yields existence of a new type of non-linear Snell envelope and extends the theory of probabilistic representation for PDEs.

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arXiv2023

Optimal Stopping of BSDEs with Constrained Jumps and Related Zero-Sum Games

In this paper, we introduce a non-linear Snell envelope which at each time represents the maximal value that can be achieved by stopping a BSDE with constrained jumps. We establish the existence of the Snell envelope by employing a penalization technique and the primary challenge we encounter is demonstrating the regularity of the limit for the scheme. Additionally, we relate the Snell envelope to a finite horizon, zero-sum stochastic differential game, where one player controls a path-dependent stochastic system by invoking impulses, while the opponent is given the opportunity to stop the game prematurely. Importantly, by developing new techniques within the realm of control randomization, we demonstrate that the value of the game exists and is precisely characterized by our non-linear Snell envelope.

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arXiv2022

Probabilistic Representation of Viscosity Solutions to Quasi-Variational Inequalities with Non-Local Drivers

We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.

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arXiv2022

Non-Markovian Impulse Control Under Nonlinear Expectation

We consider a general type of non-Markovian impulse control problems under adverse non-linear expectation or, more specifically, the zero-sum game problem where the adversary player decides the probability measure. We show that the upper and lower value functions satisfy a dynamic programming principle (DPP). We first prove the dynamic programming principle (DPP) for a truncated version of the upper value function in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Following this, we use an approximation based on a combination of truncation and discretization to show that the upper and lower value functions coincide, thus establishing that the game has a value and that the DPP holds for the lower value function as well. Finally, we show that the DPP admits a unique solution and give conditions under which a saddle-point for the game exists. As an example, we consider a stochastic differential game (SDG) of impulse versus classical control of path-dependent stochastic differential equations (SDEs).

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arXiv2021

Zero-sum Stochastic Differential Games of Impulse Versus Continuous Control by FBSDEs

We consider a stochastic differential game in the context of forward-backward stochastic differential equations, where one player implements an impulse control while the opponent controls the system continuously. Utilizing the notion of "backward semigroups" we first prove the dynamic programming principle (DPP) for a truncated version of the problem in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. In particular, this avoids technical constraints imposed in previous works dealing with the same problem. Moreover, our approach allows us to consider impulse costs that depend on the present value of the state process in addition to unbounded coefficients. Using the dynamic programming principle we deduce that the upper and lower value functions are both solutions (in viscosity sense) to the same Hamilton-Jacobi-Bellman-Isaacs obstacle problem. By showing uniqueness of solutions to this partial differential inequality we conclude that the game has a value.

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arXiv2021

Finite Horizon Robust Impulse Control in a Non-Markovian Framework and Related Systems of Reflected BSDEs

We consider a robust impulse control problem in finite horizon where the underlying uncertainty stems from an impulsively and continuously controlled functional stochastic differential equation (FSDE) driven by Brownian motion. We assume that the controller acts upon the system by impulses while the adversary player (nature) acts through continuous controls. We look for a weak solution which leads us to consider a system of sequentially interconnected, obliquely reflected backward stochastic differential equations (RBSDEs) with stochastic Lipschitz coefficients. We show existence of solutions to our system of RBSDEs by applying a Picard iteration approach. Uniqueness then follows by relating the limit to an auxiliary impulse control problem.

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arXiv2021

A Note on Reflected BSDEs in Infinite Horizon with Stochastic Lipschitz Coefficients

We consider an infinite horizon, obliquely reflected backward stochastic differential equation (RBSDE). The main contribution of the present work is that we generalize previous results on infinite horizon reflected BSDEs to the setting where the driver has a stochastic Lipschitz coefficient. As an application we consider robust optimal stopping problems for functional stochastic differential equations (FSDEs) where the driver has linear growth.

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arXiv2020

Infinite Horizon Impulse Control of Stochastic Functional Differential Equations

We consider impulse control of stochastic functional differential equations (SFDEs) driven by L\'evy processes under an additional $L^p$-Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.

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arXiv2019

Discrete-time risk-aware optimal switching with non-adapted costs

We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision maker is risk aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower planning is provided.

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arXiv2019

A Finite Horizon Optimal Switching Problem with Memory and Application to Controlled SDDEs

We consider an optimal switching problem where the terminal reward depends on the entire control trajectory. We show existence of an optimal control by applying a probabilistic technique based on the concept of Snell envelopes. We then apply this result to solve an impulse control problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process. Furthermore, we show that the studied problem arises naturally when maximizing the revenue from operation of a group of hydro-power plants with hydrological coupling.

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arXiv2018

On the Finite Horizon Optimal Switching Problem with Random Lag

We consider an optimal switching problem with random lag and possibility of component failure. The random lag is modeled by letting the operation mode follow a regime switching Markov-model with transition intensities that depend on the switching mode. The possibility of failures is modeled by having absorbing components. We show existence of an optimal control for the problem by applying a probabilistic technique based on the concept of Snell envelopes.

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arXiv2016

A Limited-Feedback Approximation Scheme for Optimal Switching Problems with Execution Delays

We consider a type of optimal switching problems with non-uniform execution delays and ramping. Such problems frequently occur in the operation of economical and engineering systems. We first provide a solution to the problem by applying a probabilistic method. The main contribution is, however, a scheme for approximating the optimal control by limiting the information in the state-feedback. In a numerical example the approximation routine gives a considerable computational performance enhancement, when compared to a conventional algorithm.

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arXiv2015

A Control-variable Regression Monte Carlo Technique for Short-term Electricity Generation Planning

In the day-to-day operation of a power system, the system operator repeatedly solves short-term generation planning problems. When formulating these problems the operators have to weigh the risk of costly failures against increased production costs. The resulting problems are often high-dimensional and various approximations have been suggested in the literature. In this article we formulate the short-term planning problem as an optimal switching problem with delayed reaction. Furthermore, we proposed a control variable technique that can be used in Monte Carlo regression to obtain a computationally efficient numerical algorithm.

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