I am a Professor in Mathematics at BI Norwegian Business School. I have a Ph.D in Mathematics from the University of Oslo, where I specialized in stochastic analysis and mathematical finance.
My research covers different topics in the field of stochastic analysis: Stochastic optimal control theory, in particular in the non-Markovian case, study of existence and uniqueness of solutions of new kinds of SDEs and BSDEs and modelling based on these new stochastic processes.
I am interested in real-life applications of stochastic models, and how they can improve our understanding of many different fields. I have worked on stochastic models and stochastic control for hydropower management, modelling of infectious diseases, temperature modelling, economic games, financial mathematics, risk assessment at sea, modelling the deterioration of components in a system and connections between stochastic control and reinforcement learning.
Awards:
NRC Young Research Talent project, 2020.
H.R.H. the King of Norway's gold medal for best PhD thesis, 2016.
The Norwegian Computing Center's prize for best master thesis, 2012.
We analyze a stochastic optimal control problem for the PReP vaccine in a model for the spread of HIV. To do so, we use a stochastic model for HIV/AIDS with PReP, where we include jumps in the model. This generalizes previous works in the field. First, we prove that there exists a positive, unique, global solution to the system of stochastic differential equations which makes up the model. Further, we introduce a stochastic control problem for dynamically choosing an optimal percentage of the population to receive PReP. By using the stochastic maximum principle, we derive an explicit expression for the stochastic optimal control. Furthermore, via a generalized Lagrange multiplier method in combination with the stochastic maximum principle, we study two types of budget constraints. We illustrate the results by numerical examples, both in the fixed control case and in the stochastic control case.
Agrell, Christian; Dahl, Kristina Rognlien & Hafver, Andreas (2023)
Optimal sequential decision making with probabilistic digital twins
In this study, we present a formal defnition of the probabilistic digital twin
(PDT). Digital twins are emerging in many industries, typically consisting of
simulation models and data associated with a specifc physical system. In order to
defne probabilistic digital twins, we discuss how epistemic uncertainty can be
treated using measure theory, by modelling epistemic information via sigma-algebras.
A gentle introduction to the necessary mathematical theory is provided
throughout the paper, together with a number of examples to illustrate the core
concepts. We then introduce the problem of optimal sequential decision making.
That is, when the outcome of each decision may inform the next. We discuss how
this problem may be solved theoretically, and the current limitations that prohibit
most practical applications. As a numerically tractable alternative, we propose a
generic approximate solution using deep reinforcement learning together with
neural networks defined on sets. We illustrate the method on a practical problem,
considering optimal information gathering for the estimation of a failure
probability.
Dahl, Kristina Rognlien; Huseby, Arne Bang & Havgar, Marius (2022)
Optimal Reinsurance Contracts under Conditional Value-at-risk
An insurance contract implies that risk is ceded from ordinary policy holders to companies. However, companies do the same thing between themselves, and this is known as reinsurance. The problem of determining reinsurance contracts which are optimal with respect to some reasonable criterion has been studied extensively within actuarial science. Different contract types are considered such as stop-loss contracts where the reinsurance company covers risk above a certain level, and insurance layer contracts where the reinsurance company covers risk within an interval. The contracts are then optimized with respect to some risk measure, such as value-at-risk or conditional value-at- risk. In the present paper we consider the problem of minimizing conditional value-at-risk in the case of multiple stop-loss contracts. Such contracts are known to be optimal in the univariate case, and the optimal contract is easily determined. We show that the same holds in the multivariate case, both with dependent and independent risks. The results are illustrated with some numerical examples.
Eggen, Mari Dahl; Dahl, Kristina Rognlien, Näsholm, Sven Peter & Mæland, Steffen (2022)
This study suggests a stochastic model for time series of daily zonal (circumpolar) mean stratospheric temperature at a given pressure level. It can be seen as an extension of previous studies which have developed stochastic models for surface temperatures. The proposed model is a combination of a deterministic seasonality function and a Lévy-driven multidimensional Ornstein–Uhlenbeck process, which is a mean-reverting stochastic process. More specifically, the deseasonalized temperature model is an order 4 continuous-time autoregressive model, meaning that the stratospheric temperature is modeled to be directly dependent on the temperature over four preceding days, while the model’s longer-range memory stems from its recursive nature. This study is based on temperature data from the European Centre for Medium-Range Weather Forecasts ERA-Interim reanalysis model product. The residuals of the autoregressive model are well represented by normal inverse Gaussian-distributed random variables scaled with a time-dependent volatility function. A monthly variability in speed of mean reversion of stratospheric temperature is found, hence suggesting a generalization of the fourth-order continuous-time autoregressive model. A stochastic stratospheric temperature model, as proposed in this paper, can be used in geophysical analyses to improve the understanding of stratospheric dynamics. In particular, such characterizations of stratospheric temperature may be a step towards greater insight in modeling and prediction of large-scale middle atmospheric events, such as sudden stratospheric warming. Through stratosphere–troposphere coupling, the stratosphere is hence a source of extended tropospheric predictability at weekly to monthly timescales, which is of great importance in several societal and industry sectors.
The main idea of this paper is to use the notion of buffered failure probability from probabilistic structural design, first introduced by Rockafellar and Royset (2010), to introduce buffered environmental contours. Classical environmental contours are used in structural design in order to obtain upper bounds on the failure probabilities of a large class of designs. The purpose of buffered failure probabilities is the same. However, in contrast to classical environmental contours, this new concept does not just take into account failure vs. functioning, but also to which extent the system is failing. For example, this is relevant when considering the risk of flooding: We are not just interested in knowing whether a river has flooded. The damages caused by the flooding greatly depends on how much the water has risen above the standard level.
In this article we consider a stochastic optimal control problem where the dynamics of the state process, X(t), is a controlled stochastic differential equation with jumps, delay and noisy memory. The term noisy memory is, to the best of our knowledge, new. By this we mean that the dynamics of X(t) depend on R t t−δ X(s)dB(s) (where B(t) is a Brownian motion). Hence, the dependence is noisy because of the Brownian motion, and it involves memory due to the influence from the previous values of the state process. We derive necessary and sufficient maximum principles for this stochastic control problem in two different ways, resulting in two sets of maximum principles. The first set of maximum principles is derived using Malliavin calculus techniques, while the second set comes from reduction to a discrete delay optimal control problem, and application of previously known results by Øksendal, Sulem and Zhang. The maximum principles also apply to the case where the controller has only partial information, in the sense that the admissible controls are adapted to a sub-σ-algebra of the natural filtration.
Dahl, Kristina Rognlien (2013)
Pricing of Claims in Discrete Time with Partial Information
Kvinner i matematikk: 16 matematikere, 16 portretter. Fotoutstilling med intervju ved Realfagsbiblioteket, UiO.
[Kronikk]
Dahl, Kristina Rognlien (2017)
Intervju med nrk.no i forbindelse med uvanlig lavt antall flyulykker i 2017.
[Kronikk]
Dahl, Kristina Rognlien (2024)
Stochastic optimal control and applications
[Conference Lecture]. Event
Dahl, Kristina Rognlien (2024)
Stochastics, stochastic optimal control and applications
[Conference Lecture]. Event
Eggen, Mari Dahl; Dahl, Kristina Rognlien, Näsholm, Sven Peter & Mæland, Steffen (2022)
Stochastic Modeling of Stratospheric Temperature
[Conference Lecture]. Event
This study suggests a stochastic model for time series of daily zonal (circumpolar) mean stratospheric temperature at a given pressure level. It can be seen as an extension of previous studies which have developed stochastic models for surface temperatures. The proposed model is a combination of a deterministic seasonality function and a Lévy-driven multidimensional Ornstein–Uhlenbeck process, which is a mean-reverting stochastic process. More specifically, the deseasonalized temperature model is an order 4 continuous-time autoregressive model, meaning that the stratospheric temperature is modeled to be directly dependent on the temperature over four preceding days, while the model’s longer-range memory stems from its recursive nature. This study is based on temperature data from the European Centre for Medium-Range Weather Forecasts ERA-Interim reanalysis model product. The residuals of the autoregressive model are well represented by normal inverse Gaussian-distributed random variables scaled with a time-dependent volatility function. A monthly variability in speed of mean reversion of stratospheric temperature is found, hence suggesting a generalization of the fourth-order continuous-time autoregressive model. A stochastic stratospheric temperature model, as proposed in this paper, can be used in geophysical analyses to improve the understanding of stratospheric dynamics. In particular, such characterizations of stratospheric temperature may be a step towards greater insight in modeling and prediction of large-scale middle atmospheric events, such as sudden stratospheric warming. Through stratosphere–troposphere coupling, the stratosphere is hence a source of extended tropospheric predictability at weekly to monthly timescales, which is of great importance in several societal and industry sectors.
