# Summary of publications related to SCROLLER

## Papers of 2020/2021:

##### Sequential Bayesian optimal experimental design for structural reliability analysis

Agrell, C. & Dahl, KR, Statistics and Computing, 2021

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.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. 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.

##### Self-exciting jump processes as deterioration models

Dahl, KR & Eyjolfsson, H., to be published in Proceedings of the 31st European Safety and Reliability Conference, 2021

Several different approaches to modeling stochastic deterioration for optimizing 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 aging 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 aging. 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.

**Fundamentals of Market Making via Stochastic Optimal Control**

Savku E., Book Chapter, Accepted, 2021.

A Market Maker (MM) is an individual or an agent, who actively provides bids

and offers asks in a financial market. Her main goal is to maximize her profit and

loss functional by getting the bid-ask spread. In this work, our purpose is to provide

a literature review to explain dynamics of the market making, impact of the market

orders on the limit order book, adverse selection and inventory risks exposed by

an MM. We present several results of the algorithmic and high frequency trading via stochastic optimal control, especially by Dynamic Programming Principle. We aim to describe the optimal spreads and the corresponding value functions based on the trade of the risky assets and the options.Moreover, market making is a type of High Frequency Trader. Consequently, all technological and regulatory conditions are strongly related to MMs. In this context, the necessity of the new investigations for algorithm developments are shining, such as Reinforcement Learning (RL) techniques. RL is a model-free approach in a close relationship with dynamic programming. It learns from experience without any knowledge of the underlying process and the goal is to maximize the cumulative reward. Hence, we finalize our chapter by giving an insight and outlook for future works from both theoretical and numerical aspects.