Kristina Rognlien Dahl

Stochastics: An International Journal of Probability and Stochastic Processes Doi: https://doi.org/10.1080/17442508.2022.2028789

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.

Statistics and computing 31 Doi: https://doi.org/10.1007/s11222-021-10000-2

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.

Proceedings of the 31st European Safety and Reliability Conference Doi: https://doi.org/10.3850/978-981-18-2016-8_286-cd

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.

Stochastic Analysis and Applications 38(4) p. 708-729 Doi: https://doi.org/10.1080/07362994.2020.1713810

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.

e-proceedings of the 30th European Safety and Reliability Conference and 15th Probabilistic Safety Assessment and Management Conference (ESREL2020 PSAM15) p. 3233-3240

Mathematical Methods of Operations Research 89 p. 43-71 Doi: https://doi.org/10.1007/s00186-018-00656-4

We consider a stochastic hydroelectric power plant management problem in discrete time with arbitrary scenario space. The inflow to the system is some stochastic process, representing the precipitation to each dam. The manager can control how much water to turbine from each dam at each time. She would like to choose this in a way which maximizes the total profit from the initial time 0 to some terminal time T. The total profit of the hydropower dam system depends on the price of electricity, which is also a stochastic process. The manager must take this price process into account when controlling the draining process. However, we assume that the manager only has partial information of how the price process is formed. She can observe the price, but not the underlying processes determining it. By using the conjugate duality framework, we derive a dual problem to the management problem. This dual problem turns out to be simple to solve in the case where the profit rate process is a martingale or submartingale with respect to the filtration modeling the information of the dam manager. In the case where we only consider a finite number of scenarios, solving the dual problem is computationally more efficient than the primal problem.

Afrika Matematika 30(3-4) p. 663-680 Doi: https://doi.org/10.1007/s13370-019-00674-3

This paper studies a duopoly investment model with uncertainty. There are two alternative irreversible investments. The first firm to invest gets a monopoly benefit for a specified period of time. The second firm to invest gets information based on what happens with the first investor, as well as cost reduction benefits. We describe the payoff functions for both the leader and follower firm. Then, we present a stochastic control game where the firms can choose when to invest, and hence influence whether they become the leader or the follower. In order to solve this problem, we combine techniques from optimal stopping and game theory. For a specific choice of parametres, we show that no pure symmetric subgame perfect Nash equilibrium exists. However, an asymmetric equilibrium is characterized. In this equilibrium, two disjoint intervals of market demand level give rise to preemptive investment behavior of the firms, while the firms otherwise are more reluctant to be the first mover.

Safety and Reliability – Safe Societies in a Changing World. Proceedings of ESREL 2018, June 17-21, 2018, Trondheim, Norway p. 2285-2292

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.

Stochastics: An International Journal of Probability and Stochastic Processes 89(6-7) p. 994-1014 Doi: https://doi.org/10.1080/17442508.2017.1303067

We introduce the concept of singular recursive utility. This leads to a kind of singular backward stochastic differential equation (BSDE) which, to the best of our knowledge, has not been studied before. We show conditions for existence and uniqueness of a solution for this kind of singular BSDE. Furthermore, we analyze the problem of maximizing the singular recursive utility. We derive sufficient and necessary maximum principles for this problem, and connect it to the Skorohod reflection problem. Finally, we apply our results to a specific cash flow. In this case, we find that the optimal consumption rate is given by the solution to the corresponding Skorohod reflection problem. This is an Accepted Manuscript of an article published by Taylor & Francis in Stochastics on 24 Mar 2017, available online: http://www.tandfonline.com/10.1080/17442508.2017.1303067

Stochastic Analysis and Applications 35(2) p. 317-333 Doi: https://doi.org/10.1080/07362994.2016.1255147

We consider the pricing problem facing a seller of a contingent claim. We assume that this seller has some general level of partial information, and that he is not allowed to sell short in certain assets. This pricing problem, which is our primal problem, is a constrained stochastic optimization problem. We derive a dual to this problem by using the conjugate duality theory introduced by Rockafellar. Furthermore, we give conditions for strong duality to hold. This gives a characterization of the price of the claim involving martingale- and super-martingale conditions on the optional projection of the price processes.

Afrika Matematika 27(3-4) p. 555-572 Doi: https://doi.org/10.1007/s13370-015-0360-5

We show how a stochastic version of the Lagrange multiplier method can be combined with the stochastic maximum principle for jump diffusions to solve certain constrained stochastic optimal control problems. Two different terminal constraints are considered; one constraint holds in expectation and the other almost surely. As an application of this method, we study the effects of inflation- and wage risk on optimal consumption. To do this, we consider the optimal consumption problem for a budget constrained agent with a Lévy income process and stochastic inflation. The agent must choose a consumption path such that his wealth process satisfies the terminal constraint. We find expressions for the optimal consumption of the agent in the case of CRRA utility, and give an economic interpretation of the adjoint processes. The final publication is available at Springer via http://dx.doi.org/10.1007/s13370-015-0360-5

Journal of Functional Analysis 271(2) p. 289-329 Doi: https://doi.org/10.1016/j.jfa.2016.04.031

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.

Applied Mathematics and Optimization 68(2) p. 145-155 Doi: https://doi.org/10.1007/s00245-013-9200-x

We consider the pricing problem of a seller with delayed price information. By using Lagrange duality, a dual problem is derived, and it is proved that there is no duality gap. This gives a characterization of the seller’s price of a contingent claim. Finally, we analyze the dual problem, and compare the prices offered by two sellers with delayed and full information respectively. The final publication is available at Springer