Papers of 2022/2023:
Deep-Control of Memory via Stochastic Optimal Control and Deep Learning.
Savku E., Book Chapter, accepted 2023, will appear in Mathematical Methods for Engineering Applications, Springer - Proceedings of ICMASE 2023, Madrid, Spain, July 12–14.
In this survey work, we introduce Stochastic Differential Delay Equations and their impacts on Stochastic Optimal Control problems. We observe time delay in the dynamics of a state process that may correspond to inertia or memory in a financial system. For such systems, we demonstrate two special approaches for handling delayed control problems by applying the Dynamic Programming Principle. Moreover, we clarify the technical challenges rising as a consequence of the conflict between the path-dependent, infinite-dimensional nature of the problem and the necessity of the Markov property. Furthermore, we present two different Deep Learning algorithms to solve targeted delayed control tasks and illustrate the results for a complete memory portfolio optimization problem.
Papers of 2021/2022:
A stochastic control approach for constrained stochastic differential games with jumps and regimes
Savku E, A Stochastic Control Approach for Constrained Stochastic Differential Games with Jumps and Regimes. Mathematics (2023), 11, 3043. DOI:10.3390/math11143043
In this work, we develop techniques to solve stochastic optimal control problems in a Lagrangian game theoretical environment. Both of the zero-sum and nonzero-sum stochastic differential game problems with two specific types of constraints can be approached by dynamic programming principle and stochastic maximum principle within the construction of our theorems. Furthermore, we demonstrate these theorems for a quite extended model of stochastic processes, named Markov regime-switching jump-diffusions. Such models have a wide range of application areas. In our work, we focus on a business agreement, called Bancassurance, between a bank and an insurance company by the methods of stochastic maximum principle for a nonzero-sum stochastic differential game. We investigate the optimal dividend strategy for the company as a best response according to the optimal mean rate of return choice of a bank for its own cash flow and vice versa. We find out a Nash equilibrium for this game and solve the adjoint equations explicitly for each state. It is well known that the timing and the amount of dividend payments are strategic decisions for companies. The announcement of a dividend payment may reduce or increase the stock prices of a company. A high dividend payment may give a message to shareholders and potential investors about the substantial amount of profits achieved by the company. On the other side, it may create an impression that the company does not have a good future project to invest in rather than paying to investors. Furthermore, dividend payments may aim to honor the shareholders' feeling of getting a reward for their trust in the company. From the side of the bank, it is clear that creating a cash flow with high returns would be the main goal. Hence, in our formulation, we provide an insight to both the bank and the insurance company about their best moves in a bancassurance commitment under specified technical conditions.
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 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.
Fundamentals of Market Making via Stochastic Optimal Control
Savku E., Book Chapter, Published in Operations Research - New Paradigms and Emerging Applications, CRC Taylor and Francis (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.