Growth-optimal portfolios under transaction costs

This paper studies a portfolio optimization problem in a discrete-time Markovian model of a financial market, in which asset price dynamics depend on an external process of economic factors. There are transaction costs with a structure that covers, in particular, the case of fixed plus proportional costs. We prove that there exists a self-financing trading strategy maximizing the average growth rate of the portfolio wealth. We show that this strategy has a Markovian form. Our result is obtained by large deviations estimates on empirical measures of the price process and by a generalization of the vanishing discount method to discontinuous transition operators.Comment: 32 page

Impulse control maximising average cost per unit time: a non-uniformly ergodic case

This paper studies maximisation of an average-cost-per-unit-time ergodic functional over impulse strategies controlling a Feller-Markov process. The uncontrolled process is assumed to be ergodic but, unlike the extant literature, the convergence to invariant measure does not have to be uniformly geometric in total variation norm; in particular, we allow for non-uniform geometric or polynomial convergence. Cost of an impulse may be unbounded, e.g., proportional to the distance the process is shifted. We show that the optimal value does not depend on the initial state and provide optimal or \ve-optimal strategies.Comment: 25 pages; This is an updated version after spinning off two sections of the paper as a basis for arxiv:1607.0601

On utility maximization in discrete-time financial market models

We consider a discrete-time financial market model with finite time horizon and give conditions which guarantee the existence of an optimal strategy for the problem of maximizing expected terminal utility. Equivalent martingale measures are constructed using optimal strategies.Comment: Published at http://dx.doi.org/10.1214/105051605000000089 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cryptographic techniques used to provide integrity of digital content in long-term storage

The main objective of the project was to obtain advanced mathematical methods to guarantee the verification that a required level of data integrity is maintained in long-term storage. The secondary objective was to provide methods for the evaluation of data loss and recovery. Additionally, we have provided the following initial constraints for the problem: a limitation of additional storage space, a minimal threshold for desired level of data integrity and a defined probability of a single-bit corruption. With regard to the main objective, the study group focused on the exploration methods based on hash values. It has been indicated that in the case of tight constraints, suggested by PWPW, it is not possible to provide any method based only on the hash values. This observation stems from the fact that the high probability of bit corruption leads to unacceptably large number of broken hashes, which in turn stands in contradiction with the limitation for additional storage space. However, having loosened the initial constraints to some extent, the study group has proposed two methods that use only the hash values. The first method, based on a simple scheme of data subdivision in disjoint subsets, has been provided as a benchmark for other methods discussed in this report. The second method ("hypercube" method), introduced as a type of the wider class of clever-subdivision methods, is built on the concept of rewriting data-stream into a n-dimensional hypercube and calculating hash values for some particular (overlapping) sections of the cube. We have obtained interesting results by combining hash value methods with error-correction techniques. The proposed framework, based on the BCH codes, appears to have promising properties, hence further research in this field is strongly recommended. As a part of the report we have also presented features of secret sharing methods for the benefit of novel distributed data-storage scenarios. We have provided an overview of some interesting aspects of secret sharing techniques and several examples of possible applications

On Optimal Stopping and Impulse Control with Constraint

The optimal stopping and impulse control problems for a Markov-Feller process are considered when the controls are allowed only when a signal arrives. This is referred to as control problems with constraint. In [28, 29, 30], the HJB equation was solved and an optimal control (for the optimal stopping problem, the discounted impulse control problem and the ergodic impulse control problem, respectively) was obtained, under suitable conditions, including a setting on a compact metric state space. In this work, we extend most of the results to the situation where the state space of the Markov process is locally compact

Discrete time adaptive impulsive control theory

This paper considers both impulsive and adaptive control of discrete time Markov processes. The first part deals with impulsive control with a long run average cost criterion. Under suitable assumptions the optimal impulsive strategies are characterized. In the second part, based on the impulsive control results obtained earlier, the optimal adaptive controller is constructed. With the use of so-called modified maximum likelihood estimation, value consistency is obtained.Feller Markov process * impulsive control * maximum likelihood estimation * martingale stability theorem

On the existence of optimal portfolios for the utility maximization problem in discrete time financial market models

We consider an investor whose preferences are described by a concave nondecreasing function $U:(0,infty)to mathbb{R}$ and prove that in an arbitrage-free discrete-time market model there is a strategy attaining the supremum of expected utility at the terminal date provided that this supremum is finite

Stopping of functionals with discontinuity at the boundary of an open set

We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set . The stopping horizon is either random, equal to the first exit from the set , or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of . Using a generalization of the penalty method, we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or [epsilon]-optimal stopping times.Optimal stopping Feller-Markov process Discontinuous functional Penalty method