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We consider agent trading futures on a market with small transaction costs. Her capital is deposited on a money market account, where compounding is possible. Arithmetic Brownian motion with random coefficients is considered as a model for futures strike price. The coefficients are assumed to be bounded Itô processes with bounded coefficients. Under these assumptions, an almost optimal interval strategy is derived, which almost maximizes expected utility in certain stopping times under hyperbolic absolute risk aversion utility function. Furthermore, under logarithmic utility function the derived strategy almost maximizes expected utility in wide class of (integrable) stopping times.

<p>The thesis consists of four papers on combinatorial auctions and a summary. The first part is more of a practical nature and contains two papers. In the first paper, we study the performance of a caching technique in an optimal algorithm for a multi-unit combinatorial auction.</p><p>In the second paper, we compare the revenues from a second-price combinatorial auction against a second-price one-shot simultaneous auction. In particular, we show that when the synergy parameter is small, the combinatorial auction gives a higher expected revenue than the one-shot. This is in contrast to an earliear result by Krishna and Rosenthal. We also compare the two mechanisms under the assumption that bidders are risk-averse. Such bidders are more sensitive to financial loss (winner's curse) that they tend to bid less aggressively, which leads to lower revenues. Since a direct analytical approach turns out to be difficult, we present numerical results that show which auction mechanism maximizes the seller's revenue depending on the values of synergy and aversion parameter.</p><p>The second part is more theoretical. Here, we analyze the asymptotic performance of a greedy algorithm for a problem inspired by combinatorial auctions. In particular, we consider a special case in which every bid contains exactly 3 items, and use a Poisson process to model an auction with a random (Poisson) No. of bids. For this restricted case, winner determination problem is equivalent to a maximal 3-set packing on a weighted hypergraph, and hence NP-complete. However, the greedy algorithm approximates this special case within a factor of 3.</p><p>In the third paper, we compute the asymptotic expected size of the partial allocation and its corresponding expected total revenue from the greedy algorithm, for some distribution of bid prices.</p><p>In the final paper, we study the case of a deterministic number of bids, which is proportional to the number of distinguishable items in the auction, say M. Then, we prove that the number of bids allocated, suitably normalized, converges to a Normal random variable as M goes to infinity. As a prelude, we also prove that, both the number of bids allocated and those submitted, again suitably normalized, jointly converge in distribution to a continuous 2-dimensional Gaussian process as M goes to infinity.</p>

The thesis consists of four papers on combinatorial auctions and a summary. The first part is more of a practical nature and contains two papers. In the first paper, we study the performance of a caching technique in an optimal algorithm for a multi-unit combinatorial auction. In the second paper, we compare the revenues from a second-price combinatorial auction against a second-price one-shot simultaneous auction. In particular, we show that when the synergy parameter is small, the combinatorial auction gives a higher expected revenue than the one-shot. This is in contrast to an earliear result by Krishna and Rosenthal. We also compare the two mechanisms under the assumption that bidders are risk-averse. Such bidders are more sensitive to financial loss (winner's curse) that they tend to bid less aggressively, which leads to lower revenues. Since a direct analytical approach turns out to be difficult, we present numerical results that show which auction mechanism maximizes the seller's revenue depending on the values of synergy and aversion parameter. The second part is more theoretical. Here, we analyze the asymptotic performance of a greedy algorithm for a problem inspired by combinatorial auctions. In particular, we consider a special case in which every bid contains exactly 3 items, and use a Poisson process to model an auction with a random (Poisson) No. of bids. For this restricted case, winner determination problem is equivalent to a maximal 3-set packing on a weighted hypergraph, and hence NP-complete. However, the greedy algorithm approximates this special case within a factor of 3. In the third paper, we compute the asymptotic expected size of the partial allocation and its corresponding expected total revenue from the greedy algorithm, for some distribution of bid prices. In the final paper, we study the case of a deterministic number of bids, which is proportional to the number of distinguishable items in the auction, say M. Then, we prove that the number of bids allocated, suitably normalized, converges to a Normal random variable as M goes to infinity. As a prelude, we also prove that, both the number of bids allocated and those submitted, again suitably normalized, jointly converge in distribution to a continuous 2-dimensional Gaussian process as M goes to infinity.

THE STIGLER-LUCKOCK MODEL FOR A LIMIT ORDER BOOK Abstract One of the types of modern-day markets are so-called order-driven markets whose core component is a database of all incoming buy and sell orders (order book). The main goal of this thesis is to extend the Stigler-Luckock model for order books to give a better insight into the price forming process and behaviour of the market participants themselves. The model introduced in this thesis focuses on a comparison of behaviour and various strategies of market makers who are sophisticated market participants profiting from extensive trading. The market is described using Markov chains, and the strategies are compared using Monte Carlo simulations and game theory. The results showed that market makers' orders should have small spread and large volumes. The final model compares two strategies in which market makers monitor their portfolio. In case of having more cash than asset (or vice versa), they shift prices of their orders to equalise the portfolio. The model recommends checking the market quite often, but acting conservatively, which means not changing prices that frequently and not jumping to conclusions just from a small imbalance in the portfolio.

Hedging has been one of the most important topics in finance. How to effectively hedge the exposed risk draws significant interest from both academicians and practitioners. In a complete financial market, every contingent claim can be hedged perfectly. In an incomplete market, the investor can eliminate his risk exposure by superhedging. However, both perfect hedging and superhedging usually call for a high cost. In some situations, the investor does not have enough capital or is not willing to spend that much to achieve a zero risk position. This brings us to the topic of partial hedging. In this thesis, we establish the risk measure based partial hedging model and study the optimal partial hedging strategies under various criteria. First, we consider two of the most common risk measures known as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We derive the analytical forms of optimal partial hedging strategies under the criterion of minimizing VaR of the investor's total risk exposure. The knock-out call hedging strategy and the bull call spread hedging strategy are shown to be optimal among two admissible sets of hedging strategies. Since VaR risk measure has some undesired properties, we consider the CVaR risk measure and show that bull call spread hedging strategy is optimal under the criterion of minimizing CVaR of the investor's total risk exposure. The comparison between our proposed partial hedging strategies and some other partial hedging strategies, including the well-known quantile hedging strategy, is provided and the advantages of our proposed partial hedging strategies are highlighted. Then we apply the similar approaches in the context of reinsurance. The VaR-based optimal reinsurance strategies are derived under various constraints. Then we study the optimal partial hedging strategies under general risk measures. We provide the necessary and sufficient optimality conditions and use these conditions to study some specific hedging strategies. The robustness of our proposed CVaR-based optimal partial hedging strategy is also discussed in this part. Last but not least, we propose a new method, simulation-based approach, to formulate the optimal partial hedging models. By using the simulation-based approach, we can numerically obtain the optimal partial hedging strategy under various constraints and criteria. The numerical results in the examples in this part coincide with the theoretical results.

The diploma thesis describes portfolio management with proportional transaction costs. The main aim is to describe using of shadow prices to find the optimal Markov policies keeping the proportion of the investor's wealth invested in the risky asset within the corresponding interval in order to maximize the long run geometric growth rate. On the real market, the investor must pay transaction costs when he buys/sells shares. In the diploma thesis we transform these prices into the shadow price; when trading in the shadow price there are no transaction costs. The solution itself is based on Itô formula and the martingal theory. The prices of shares are modeled as geometric Brownian motion. Powered by TCPDF (www.tcpdf.org)

ABSTRACT Usually, real game problems encountered in our daily lives are so complicated that the existing methods are no longer sufficient to deal with them. This motivates us to investigate several kinds of differential game problems, which have not been considered or solved yet, including a pursuit-evasion game with n pursuers and one evader, a problem of guarding a territory with two guarders and two invaders, and a payoff-switching differential game. In this thesis, firstly the geometric method is used to consider the pursuit-evasion game with n pursuers and one evader. Two criteria used to find the solutions of the game in some cases are given. It will be shown that the one-on-one pursuit-evasion game is a special case of this game. Secondly, the problem of guarding a territory with two guarders and two invaders is considered both qualitatively and quantitatively. The investigation of this problem reveals a variety of situations never occurring in the case with one guarder and one invader. An interesting thing found in this investigation is that some invader may play the role as a pursuer for achieving a more favorable payoff in some cases. This will make the problem more complicated and more difficult to be solved. The payoff-switching differential game, first proposed by us, is a kind of differential game with incomplete information. The main difference between this problem and traditional differential games is that in a payoff-switching differential game, any one player at any time may have several choices of payoffs for the future. The optimality in such a problem becomes questionable. Some reasoning mechanisms based on different methods will be provided to determine a reasoning strategy for some player in a payoff-switching differential game. A practical payoff-switching differential game problem, i.e., the guarding three territories with one guarder against one invader, is presented to illustrate the situations of such a game problem. Many computer simulations of this example are given to show the performances of different reasoning strategies. The proposition of the payoff-switching differential game is an important breakthrough in dealing with some kinds of differential games with incomplete information.

Автоматизация складского учета на базе платформы 1С Предприятие является актуальной темой для любого предприятия. Целью данной диссертационной работы является создание программного продукта с использованием новаторских технологий разработки. В ходе работы систематизируются математические результаты и методы вычисления оптимальных стратегий, моделируется процесс принятия оптимальных управленческих решений на основе описанных методов. Объектом данного исследования выступает компания заказчик программного продукта. Предметом исследования является информационно-технологическая система 1С Предприятие. Разработанности темы данной магистерской диссертации характеризуется низким уровнем, поскольку поставленная задача является уникальной для данного вида предприятия. Изученная литература освещает весьма узкий круг вопросов, а также некоторые смежные проблемы, вытекающие или некоторым образом связанные с темой диссертации. / Automation of warehouse account on the basis of "1C Enterprise" platform is a actual topic for any enterprise. Purpose of this dissertation work is creation of a software product with use of innovative technologies of development. During work mathematical results and methods of computation of optimum strategy are systematized, process of acceptance of optimal administrative solutions on the basis of the described methods is modelled. Object of this research is company which is a customer of a software product. An object of research is the information and technological "1C Enterprise" system. Readiness of a subject of this master thesis is characterized by other authors by the low level because objective is unique for this type of enterprise. The studied literature takes up very narrow circle of questions and also some adjacent problems following or to some extent connected to a thesis.

本研究針對退休基金管理的兩項重要議題：提撥政策與資產配置作最適規劃之探討。由於傳統退休基金的評價僅考慮單一期間的離散時間模型，不若多期規劃的效率性，因此，本研究考量連續時間下，利用控制理論觀點，將提撥金額與資產配置視為可調節的因子，以風險最小化為最適定義，提供基金多期管理的有效方法。 首先，為充分反映退休基金管理時所面臨的不確定因素，本研究假設資產價值服從幾何布朗運動，並且經由隨機微分方程式描述退休基金所累積資產與應計負債的動態隨機性質。其次，考量基金管理所面臨的提撥風險與清償風險，給定能夠量化這些風險的評估測度，藉以監督退休基金於管理期間的經營績效，並且利用Bellman方程式求出最適的基金提撥與資產配置策略。 最後以勞動基準法規範下的企業退休金計劃為實證對象，透過動態模擬估計模型中之參數，並且利用數值方法求出所需的函數值，將控制理論與情境模擬連結，藉以檢視現行固定給付退休基金之最適策略。由實證結果可知，透過本研究的方法的確可以有效管理基金同時符合財務清償能力的要求。利用動態規劃所得的最適策略與給定的風險評估函數相關，因此，基金決策者可以依據基金的特性給定適當的風險評估函數，依照不同的投資期限擬定合適的基金策略。 / This study explores two critical issues in pension fund management: funding policy and asset allocation. The traditional valuation of pension fund is restricted in one-period setting under discrete-time framework, and it is not efficient comparing to the continuous-time models. Therefore, in this study, control theory is employed to obtain the optimal strategy based on a specific plan dynamics. Employer's contributions and investment proportions are treated as the controllers in our model. Optimal solutions are obtained by minimizing the given risk performance in monitoring the multi-period fund management. First, the stochastic differential equations are constructed to describe the dynamics of the funding levels and the accrued liabilities. Geometric Brownian motions are used to model the assets held by the fund manager. Secondly, a stochastic control model with given risk measurement is formulated in a continuous-time framework to investigate the optimal decisions. In our approach, the plan's normal costs and accrued liabilities are simulated through plausible scenarios while the optimal contribution and asset allocation are solved through Bellman equation. At last, a specific pension scheme under the regulation of the Taiwan labor standards law is studied for numerical illustrations. A monitoring mechanism linking plausible scenarios and the closed-form solutions are employed to scrutinize the funding policy and asset allocation. The optimal strategies are estimated through dynamic programming under realistic workforce scenario. According to the result, it shows that the methodology in this study can assist the fund manager in obtaining the plan's financial soundness. Meanwhile, the optimal strategy can fully incorporate the given risk measurement. Hence, the policy maker can input certain managerial considerations into the performance measure to investigate the stability and solvency issues.