SCROLLER - A Stochastic ContROL approach to machine Learning with applications to Environmental Risk models (completed)

• Agrell, Christian & Dahl, Kristina Rognlien (2021). Sequential Bayesian optimal experimental design for structural reliability analysis. Statistics and computing . ISSN 0960-3174.  31  . doi:  10.1007 / s11222-021-10000-2  Full text in knowledge archive .
• Dahl, Kristina Rognlien & Eyjolfsson, Heidar (2021). Self-exciting jump processes as deterioration models, to be published in Proceedings of the 31st European Safety and Reliability Conference. Edited by B. Castanier, M. Cepin, D. Bigaud & C. Berenguer, Research Publishing, Singapore, ISBN: 981-973-0000-00-0. doi: 10.3850 / 981-973-0000-00-0.

• Banos, David Ruiz; Sande, Åsmund Hausken & Sgarra, Carlo (2023). Guaranteed Minimum Maturity Benefits in a Self-Exciting Stochastic Mortality Model: Pricing, Estimation and Calibration. North American Actuarial Journal (NAAJ). ISSN 1092-0277. 28(4), p. 744–771. doi: 10.1080/10920277.2023.2254836. Full text in Research Archive Show summary

  The guaranteed minimum maturity benefit (GMMB) is quite a popular feature embedded in several unit-linked policies offered by insurance companies. The value of this benefit depends on several processes assumed to describe both the mortality and the financial dynamics, typically represented by interest rates and by the fund associated to the unit-linked policy. A large literature is devoted to the valuation of GMMB for different mortality models, in particular when the mortality dynamics is described by affine models of diffusion type. In the present article we assume a self-exciting behavior for the mortality dynamics, described by a Hawkes-type process with exponential kernel. This allows us to keep both the Markov and affine features but introduces jumps with a stochastic intensity. These types of dynamics, exhibiting a jump clustering property, are quite convenient for describing mortality in some critical situations, like epidemics, where contagion phenomena make the probability of jump arrivals higher whenever a jump occurs. By assuming diffusion-type dynamics for both the fund and the interest rate and introducing all of the possible correlations among the diffusion processes necessary to obtain realistic dynamics, we take advantage of the affine features of the model proposed and compute in a semi-explicit form the value of a GMMB. Lastly, we calibrate our model and perform an empirical analysis to determine the effects of excess mortality on GMMB prices as well as a death benefit portfolio.

• Sande, Åsmund Hausken (2023). Convex environmental contours for non-stationary processes. Ocean Engineering. ISSN 0029-8018. 292. doi: 10.1016/j.oceaneng.2023.116615. Full text in Research Archive Show summary

  Environmental contours are tools frequently used in the early design of marine structures. They provide a description of critical design conditions and serve as a means for simplifying expensive long-term response calculations. Here, we consider convex contours based on the assumption of convex failure sets. We provide a rigorous foundation for the existence of such contours when the underlying environmental factors are modelled by a general, possibly non-stationary, process. This constitutes a generalisation of existing theory and is done to properly account for empirically observed increases in extreme sea-states. Two definitions are proposed, based respectively on averages or quantiles of exceedence times, along with minimal conditions on the environmental processes to guarantee existence. In order to illustrate these methods we give two examples, including an empirical study containing a method for constructing contours based on the presented theory.

• Savku, Emel (2023). A Stochastic Control Approach for Constrained Stochastic Differential Games with Jumps and Regimes. Mathematics. 11(14). doi: 10.3390/math11143043. Full text in Research Archive Show summary

  We develop an approach for two-player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different types of constraints, which lead us to find real-valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for a nonzero-sum game problem with the stochastic maximum principle technique. Our application is an example of cooperation between a bank and an insurance company, which is a popular, well-known business agreement type called Bancassurance.

• Eggen, Mari Dahl; Dahl, Kristina Rognlien; Näsholm, Sven Peter & Mæland, Steffen (2022). Stochastic Modeling of Stratospheric Temperature. Mathematical Geosciences. ISSN 1874-8961. 54, p. 651–678. doi: 10.1007/s11004-021-09990-6. Full text in Research Archive Show summary

  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.

• Dordevic, Jasmina & Dahl, Kristina Rognlien (2022). Stochastic optimal control of pre-exposure prophylaxis for HIV infection. Mathematical Medicine and Biology. ISSN 1477-8599. 39(3), p. 197–225. doi: 10.1093/imammb/dqac003. Full text in Research Archive
• Savku, Emel (2022). Fundamentals of Market Making Via Stochastic Optimal Control. In Purutçuoğlu, Vilda; Weber, Gerhard-Wilhelm & Farnoudkia, Hajar (Ed.), Operations Research: New Paradigms and Emerging Applications. CRC Press. ISSN 9781032349268. p. 136–154. Full text in Research Archive
• Dahl, Kristina Rognlien & Eyjolfsson, Heidar (2022). Self-exciting jump processes and their asymptotic behaviour. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2022.2028789. Full text in Research Archive Show summary

  The purpose of this paper is to investigate properties of self-exciting jump processes where the intensity is given by an SDE, which is driven by a finite variation stochastic jump process. The value of the intensity process immediately before a jump may influence the jump size distribution. We focus on properties of this intensity function, and show that for each fixed point in time, t≥0, a scaling limit of the intensity process converges in distribution, and the limit equals the strong solution of the square-root diffusion process (Cox–Ingersoll–Ross process) at t. As a particular example, we study the case of a linear intensity process and derive explicit expressions for the expectation and variance in this case.

• Agrell, Christian & Dahl, Kristina Rognlien (2021). Sequential Bayesian optimal experimental design for structural reliability analysis. Statistics and computing. ISSN 0960-3174. 31. doi: 10.1007/s11222-021-10000-2. Full text in Research Archive Show summary

  Structural reliability analysis is concerned with estimation of the probability of a critical event taking place, described by P(g(X)≤0) for some n-dimensional random variable X and some real-valued function g. In many applications the function g is practically unknown, as function evaluation involves time consuming numerical simulation or some other form of experiment that is expensive to perform. The problem we address in this paper is how to optimally design experiments, in a Bayesian decision theoretic fashion, when the goal is to estimate the probability P(g(X)≤0) using a minimal amount of resources. As opposed to existing methods that have been proposed for this purpose, we consider a general structural reliability model given in hierarchical form. We therefore introduce a general formulation of the experimental design problem, where we distinguish between the uncertainty related to the random variable X and any additional epistemic uncertainty that we want to reduce through experimentation. The effectiveness of a design strategy is evaluated through a measure of residual uncertainty, and efficient approximation of this quantity is crucial if we want to apply algorithms that search for an optimal strategy. The method we propose is based on importance sampling combined with the unscented transform for epistemic uncertainty propagation. We implement this for the myopic (one-step look ahead) alternative, and demonstrate the effectiveness through a series of numerical experiments.

• Dahl, Kristina Rognlien & Eyjolfsson, Heidar (2021). Self-Exciting Jump Processes as Deterioration Models. In Castanier, Bruno; Cepin, Marko; Bigaud, David & Berenguer, Christophe (Ed.), Proceedings of the 31st European Safety and Reliability Conference. Research Publishing Services. ISSN 9789811820168. doi: 10.3850/978-981-18-2016-8_286-cd. Full text in Research Archive Show summary

  Several different approaches to modelling stochastic deterioration for optimising maintenance have been suggested in the reliability literature. These include component lifetime distributions, which have the disadvantage of being binary, in the sense of only telling whether the component has failed or not. Failure rate functions model ageing in a more satisfactory way than lifetime distributions. However, failure rates cannot be observed for a single component, and are therefore not tractable in practical applications. To mitigate this, a theory for modeling deterioration via stochastic processes developed. Various processes have been suggested, such as Brownian motion with drift and compound Poisson processes (CPP) for modeling usage and damage from sporadic shocks and gamma processes to model gradual ageing. However, none of these processes are able to capture jump clustering. To allow for clustering of jumps (failure events), we suggest an alternative approach in this paper: To use self-exciting jump processes to model stochastic deterioration of components in a system where there may be clustering effects in the degradation. Self-exciting processes excite their own intensity, so large shocks are likely to be followed by another shock within a short period of time. Furthermore, self-exciting processes may have both finite and infinite activity. Therefore, we suggest that these processes can be used to model degradation both by sporadic shocks and by gradual wear. We illustrate the use of self-exciting degradation with several numerical examples. In particular, we use Monte Carlo simulation to estimate the expected lifetime of a component with self-exciting degradation. As an illustration, we also estimate the lifetime of a bridge system with independent components with identically distributed self-exciting degradation.

• Dahl, Kristina Rognlien & Huseby, Arne (2020). Environmental contours and optimal design. In Baraldi, Piero; Maio, Francesco P. Di & Zio, Enrico (Ed.), e-proceedings of the 30th European Safety and Reliability Conference and 15th Probabilistic Safety Assessment and Management Conference (ESREL2020 PSAM15). Research Publishing Services. ISSN 9789811485930. p. 3233–3240. Full text in Research Archive
• Dahl, Kristina Rognlien (2020). Forward-backward stochastic differential equation games with delay and noisy memory. Stochastic Analysis and Applications. ISSN 0736-2994. 38(4), p. 708–729. doi: 10.1080/07362994.2020.1713810. Full text in Research Archive Show summary

  The goal of this paper is to study a stochastic game connected to a system of forward-backward stochastic differential equations (FBSDEs) involving delay and noisy memory. We derive sufficient and necessary maximum principles for a set of controls for the players to be a Nash equilibrium in the game. Furthermore, we study a corresponding FBSDE involving Malliavin derivatives. This kind of equation has not been studied before. The maximum principles give conditions for determining the Nash equilibrium of the game. We use this to derive a closed form Nash equilibrium for an economic model where the players maximize their consumption with respect to recursive utility.

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• Savku, Emel (2023). Stochastic Maximum Principle For A Constraint Nonzero-Sum Game Application:Bancassurance. Full text in Research Archive
• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2023). Heston-Hawkes Stochastic Volatility Model: Change of Measure and Forward Variance. Full text in Research Archive
• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2023). Heston-Hawkes stochastic volatility model: change of measure and forward variance. Full text in Research Archive
• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2023). Heston-Hawkes stochastic volatility model: Change of measure and Thiele's PIDE. Full text in Research Archive
• Savku, Emel (2023). A Nonzero-Sum Regime-Switching Stochastic Differential Game Application with Constraints. Full text in Research Archive
• Savku, Emel (2023). A Stochastic Maximum Principle Approach for a Nash Equilibrium of a Nonzero-Sum Game. Full text in Research Archive
• Savku, Emel (2022). An Application of Nonzero-Sum Stochastic Differential Games in Finance. Full text in Research Archive
• Savku, Emel (2022). An Application of Stochastic Differential Games with Lagrange Multipliers: Bancassurance. Full text in Research Archive
• Savku, Emel (2022). A Constrained Nonzero-Sum Stochastic Differential Game Application. Full text in Research Archive
• Savku, Emel (2022). An Application of Stochastic Differential Games with Lagrange Multipliers:Bancassurance. Full text in Research Archive
• Savku, Emel (2022). A constrained stochastic differential game application: Bancassurance. Full text in Research Archive
• Savku, Emel (2022). A Constrained Nonzero-Sum Game Application: Bancassurance. Full text in Research Archive
• Savku, Emel (2022). An Application of Markov Regime-Switching Models: Bancassurance. Full text in Research Archive
• Dahl, Kristina Rognlien & Eyolfsson, Heidar (2021). Self-exciting jump processes as deterioration models. Full text in Research Archive
• Savku, Emel (2021). An Application of Stochastic Maximum Principle with Regimes and Memory. Full text in Research Archive
• Savku, Emel (2021). A Nonzero-sum Game Formulation for a Markov Regime-Switching Portfolio Strategy. Full text in Research Archive
• Eggen, Mari Dahl; Dahl, Kristina Rognlien; Näsholm, Sven Peter & Mæland, Steffen (2021). Stochastic modelling of stratospheric temperature. Full text in Research Archive
• Savku, Emel (2021). Stochastic Optimal Control Techniques for a Regime-Switching Model with Applications in Finance. Full text in Research Archive
• Savku, Emel (2021). Stochastic Maximum Principle with Regimes and Memory. Full text in Research Archive
• Savku, Emel (2021). Portfolio Strategies via Stochastic Differential Games with Regimes. Full text in Research Archive
• Savku, Emel (2021). Stochastic Differential Games via Dynamic Programming Principle with Regimes. Full text in Research Archive
• Savku, Emel (2021). Stochastic Differential Games within the framework of Regime-Switches. Full text in Research Archive
• Dahl, Kristina Rognlien (2020). The SCROLLER project A Stochastic ContROL approach to machine Learning with applications to Environmental Risk models. Full text in Research Archive
• Dahl, Kristina Rognlien (2020). FBSDE games with delay & noisy memory. Full text in Research Archive
• Dahl, Kristina Rognlien & Huseby, Arne (2020). Environmental contours and optimal design. Full text in Research Archive
• Dahl, Kristina Rognlien (2020). The SCROLLER project and a subproject: Optimal design. Full text in Research Archive

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