# Browsing by Author "Bensoussan, Alain"

Now showing 1 - 17 of 17

- Results Per Page
1 5 10 20 40 60 80 100

- Sort Options
Ascending Descending

Item A Mean-Variance Approach to Capital Investment Optimization(Society for Industrial and Applied Mathematics Publications) Bensoussan, Alain; Hoe, S.; Yan, Z.; 0000-0003-0743-498X (Bensoussan, A); Bensoussan, AlainShow more We develop an improved model of capital investment under uncertainty that incorporates the variance of the capital stock in the payoff functional to manage risk. Our model results in a mean field type control problem that cannot be solved by classical stochastic control methods. We solve our problem using techniques presented in Bensoussan, Frehse, and Yam [Mean Field Games and Mean Field Type Control Theory, Springer, New York, 2013]. The explicit solution is a feedback depending on the initial condition. Moreover, our model can be reduced to Abel's [Amer. Econ. Rev., 73 (1983), pp. 228-233]. Numerical results suggest that the risk reduction optimally exceeds the cost incurred. Following Björk, Khapko, and Murgoci [Finance Stoch., 21 (2017), pp. 331-360], we solve for a time-consistent solution, i.e., the best possible feedback independent of the initial condition. The time-consistent policy discards our risk specification, with the resultant loss of value to the firm. © 2019 Society for Industrial and Applied MathematicsShow more Item A Paradox in Time-Consistency in the Mean-Variance Problem?(Springer Heidelberg, 2018-12-19) Bensoussan, Alain; Wong, Kwok Chuen; Yam, Sheung Chi Phillip; 0000-0003-0743-498X (Bensoussan, A); Bensoussan, AlainShow more We establish new conditions under which a constrained (no short-selling) time-consistent equilibrium strategy, starting at a certain time, will beat the unconstrained counterpart, as measured by the magnitude of their corresponding equilibrium mean-variance value functions. We further show that the pure strategy of solely investing in a risk-free bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. With numerical experiments, we also illustrate that the constrained strategy can dominate the unconstrained one for most of the commencement dates (even more than 90%) of a prescribed planning horizon. Under a precommitment approach, the value function of an investor increases with the size of the admissible sets of strategies. However, this may fail to be true under the game-theoretic paradigm, as the constraint of time-consistency itself affects the value function differently when short-selling is and is not prohibited.Show more Item Analysis of Real Options and Wealth Management Problems Using Non-smooth Variational Inequalities and Asymptotic Methods(2022-05-02) Acharya, Subas; Rachinskiy, Dmitry; Bensoussan, AlainShow more We consider investment problems involving the continuous time stochastic optimization models. The dissertation consists of two parts. In the first part, we consider a real options problem, which is posed as a stochastic optimal control problem. The investment strategy, which plays the role of control, involves a onetime option to expand (invest) and a one-time option to abandon (terminate) the project. The timing and amount of the investment and the termination time are parameters to be optimized in order to maximize the expected value of the profit. This stochastic optimization problem amounts to solving a deterministic variational inequality in dimension one, with the associated obstacle problem. Because we consider both cessation and expansion options and fixed and variable costs of expansion, the obstacle is non-smooth. Due to the lack of smoothness, we use the concept of a weak solution. However, such solutions may not lead to a straightforward investment strategy. Therefore, we further consider strong (C1) solutions based on thresholds. We propose sufficient conditions for the existence of such solutions to the variational inequality with a non-smooth obstacle in dimension one. When applied to the real options problem, these sufficient conditions yield a simple optimal investment strategy with the stopping times defined in terms of the thresholds. In the second part, we develop a dynamic wealth management model for risk-averse investors displaying present bias in the form of hyperbolic discounting. The investor chooses an optimal consumption policy and allocates her funds between a risk-free asset, a traded liquid asset, and a non-traded illiquid asset. We characterize these policies for both sophisticated and naive present-biased investors. There are three results. First, sophisticated investors over-consume more than their naive counterparts if and only if their coefficient of relative risk-aversion is smaller than one. As a result, sophistication is welfare reducing (increasing) when risk-aversion is low (high). Second, increasing asset illiquidity always benefits the sophisticated investor more than the naive investor. Thus, the welfare gap between sophisticated and naive investors is increasing in the proxy for asset illiquidity. Finally, present-biased investors accumulate a larger share of their wealth in the non-traded illiquid asset than in the traded risky stock compared to the neoclassical exponential discounter investor. As a consequence, from the perspective of present-biased investors, the equity premium puzzle and the private equity puzzle are two sides of the same coin.Show more Item Applications of Mean Field Theory in Management Science(2021-07-20) Kim, Joohyun; Bensoussan, AlainShow more The main objective of my PhD study is to understand an aggregate effect arising from a large number of agents who have a similar aspect of decision markings and objectives. The primary idea of mean field approach is that the individual agent makes a decision by considering the distribution of the other agents rather than assuming that all agents’ detailed information on states is collectible. In the first essay of my dissertation, the primary objective is to study the optimal consumption and portfolio selection problem of risk-controlled investors who strive to maximize their utility of both consumption and terminal wealth. Risk is measured by the variance of terminal wealth, which introduces a nonlinear function of the expected value into the control problem, so a standard stochastic control theory is not properly applicable. This control problem is totally open until the discovery of mean field type control. The second essay explores the dynamic competition among a large number of interacting households who own local storage with a self-generated renewable energy system, and each can decide the amount of charging or discharging energy based on the market environment and the level of energy stored. Under the mean field setting, the optimal solution can be interpreted as an optimal policy suggestion by a central planner who is willing to increase the penetration of local storage to enhance the resilience of the grid system. The third essay investigates a new control problem for dealer’s optimal markup and inventory control regarding Over-The-Counter (OTC) trades. The explicit solutions obtained by the mean field approach can contribute to developing a decision support system for dealers willing to coordinate an inter-dealer and investor-dealer market simultaneously. The proposed decision-making rules may facilitate dealers’ responses to imbalances in demand and supply to reduce the possibility of policy intervention about liquidity risk in OTC markets.Show more Item Essays on Corporate Bankruptcy and Debtor-in-possession Financing(2021-12-01T06:00:00.000Z) Fahimi, Mohammad Mahdi; Bensoussan, Alain; Kieschnick, Jr., Robert L.; Saretto, Alessio; Wei, Kelsey D.; Rivera Mesias, AlejandroShow more This dissertation consists of two essays in financial economics. The first essay, included in Chapter 2, concerns the effect of debtor-in-possession (DIP) financing and DIP financing lenders on the outcome of Chapter 11 bankruptcy. When firms file for protection under Chapter 11 bankruptcy, their access to outside financing will be limited. The Bankruptcy Reform Act of 1978 has resolved this issue under section 364 of the US Bankruptcy Code by defining laws for the DIP financing, which is the unique type of financing available to firms filing for Chapter 11 bankruptcy. DIP financing is usually senior to all other securities issued by a firm and violates the absolute priority rule by standing ahead of a company’s existing debts for payment. Among the characteristics of DIP financing, limited attention has been given to the type of lender of the DIP financing. There is not much empirical evidence on whether financing a DIP loan from different types of lenders can lead to different bankruptcy outcomes. In this essay, I investigate the role of DIP financing, especially the DIP lender in the bankruptcy process. I provide evidence for the role of DIP lender, bank versus non-bank, in bankruptcy outcome, while controlling for potential endogeneity of the lender’s type. In order to control for the endogeneity of the DIP lender type, I use an instrumental variable (IV) approach. My results show that even after controlling for the endogeneity of the lender type, the source of the DIP loan still matters for the outcome of the bankruptcy process. More specifically, receiving the DIP loan from banks increases the likelihood of emerging from bankruptcy as a going concern for the bankrupt firm. The second essay, included in Chapter 2, concerns predicting bankruptcy outcome using a machine learning approach and using the bankruptcy outcome predictions to predict firms’ CDS spreads. First, I develop a machine learning model using Extreme Gradient Boosting to predict the outcome of the bankruptcy. I compare the performance of this model with that of a traditional logistics regression model and show that, while both perform well, the machine learning model outperforms the traditional model, mainly because it is able to identify non-linear patterns in the data. I, then, use the predicted probabilities of emerging from bankruptcy, combined with the predicted probabilities of bankruptcy, produced by a second machine learning model, to predict CDS spreads. I show that the predicted probability of bankruptcy and probability of emerging from bankruptcy can be used to predict firms’ CDS spreads and can improve the prediction power of benchmark models. This study contributes to the bankruptcy and bankruptcy outcome prediction literature by providing empirical evidence of the association between a firm’s characteristics and its bankruptcy outcome. I also show that using machine learning techniques to predict the bankruptcy outcome can help predict CDS spreads more accurately.Show more Item Inventory Control with Fixed Cost and Price Optimization in Continuous Time(Wilmington Scientific Publisher) Bensoussan, Alain; Skaaning, S.; Turi, Jànos; Bensoussan, Alain; Turi, JànosShow more We continue to study the problem of inventory control, with simultaneous pricing optimization in continuous time. In our previous paper [8], we considered the case without set up cost, and established the optimality of the base stock-list price (BSLP) policy. In this paper we consider the situation of fixed price. We prove that the discrete time optimal strategy (see [11]), i.e., the (s; S; p) policy can be extended to the continuous time case using the framework of quasi-variational inequalities (QVIs) involving the value function. In the process we show that an associated second order, nonlinear two-point boundary value problem for the value function has a unique solution yielding the triplet (s; S; p). For application purposes the explicit knowledge of this solution is needed to specify the optimal inventory and pricing strategy. Selecting a particular demand function we are able to formulate and implement a numerical algorithm to obtain good approximations for the optimal strategy.Show more Item Inventory Control with Pricing Optimization in Continuous Time(2016-12) Skaaning, Sonny M; Bensoussan, Alain; Turi, JanosShow more Companies have always sought to find methods to optimize their business platform. An optimal strategy that can achieve these goals is therefore needed. To find the optimal strategy, a constrained optimization problem is introduced. In this dissertation, we study an inventory control problem in continuous time with no fixed cost associated with replenishment. We seek to find the optimal strategy that will maximize the profit functional while having control over the price and the replenishment of the product. We show, through theoretical results, that the policy which optimizes the problem statement is the continuous analog of the discrete time case, known as the Base Stock List Price policy. Firstly, we provide the general setup and theoretical results to the inventory control problem. We set up an analytical problem known as quasi-variational inequalities in a strong format. We show, through transformations, how we arrive at a nonlinear two point boundary value problem, of which the solution satisfies the quasi-variational inequalities. From the solution to the nonlinear two point boundary value problem we build the optimal strategy. Through a verification argument, we establish that the optimal strategy is indeed the value function which constitutes the optimal profit. In order to study the nonlinear two point boundary value problem, we introduce the associated epsilon problem. By use of a limiting process, we show convergence of the epsilon solution to the solution of the original nonlinear two point boundary value problem. The original nonlinear two point boundary value problem will be referenced through the dissertation as the base problem. Secondly, we provide a numerical methodology that solves the nonlinear two point boundary value problem on a semi-infinite domain. We show how theoretical results provide guidance in finding the numerical solution as well as circumvent issues presented. We solve the epsilon problem and show convergence to the base problem when epsilon decreases. The MATLAB solver bvp5c is used to provide the solution which constitutes the optimal strategy. Finally, we study a specific numerical case. We study the case where the average demand function decreases with respect to the price as a power function. We provide the Base Stock List Price policy for our case study as well as numerical results for the epsilon problem which confirms our theoretical findings. Based on our findings, we are able to draw economic conclusions and give guidance as to how a company should maximize their profit margins.Show more Item Joint Inventory-Pricing Optimization with General Demands: An Alternative Approach for Concavity Preservation(Wiley-Blackwell, 2019-05-23) Bensoussan, Alain; Xie, Y.; Yan, H.; 0000-0003-0743-498X (Bensoussan, A); Bensoussan, AlainShow more In this study, we provide an alternative approach for proving the preservation of concavity together with submodularity, and apply it to finite-horizon non-stationary joint inventory-pricing models with general demands. The approach characterizes the optimal price as a function of the inventory level. Further, it employs the Cauchy–Schwarz and arithmetic-geometric mean inequalities to establish a relation between the one-period profit and the profit-to-go function in a dynamic programming setting. With this relation, we demonstrate that the one-dimensional concavity of the price-optimized profit function is preserved as a whole, instead of separately determining the (two-dimensional) joint concavities in price (or mean demand/risk level) and inventory level for the one-period profit and the profit-to-go function in conventional approaches. As a result, we derive the optimality condition for a base-stock, list-price (BSLP) policy for joint inventory-pricing optimization models with general form demand and profit functions. With examples, we extend the optimality of a BSLP policy to cases with non-concave revenue functions in mean demand. We also propose the notion of price elasticity of the slope (PES) and articulate the condition as that in response to a price change of the commodity, the percentage change in the slope of the expected sales is greater than the percentage change in the slope of the expected one-period profit. The concavity preservation conditions for the additive, generalized additive, and location-scale demand models in the literature are unified under this framework. We also obtain the conditions under which a BSLP policy is optimal for the logarithmic and exponential form demand models. © 2019 Production and Operations Management SocietyShow more Item Mathematical Methods for Advanced Problems of Inventory Control(2021-05-01T05:00:00.000Z) Helal, Md Abu; Ramakrishna, Viswanath; Bensoussan, Alain; Izen, Joseph M.; Dragovic, Vladimir; Dabkowski , Mieczyslaw K.; Choudhary, Pankaj K.Show more We study infinite horizon stochastic inventory problems with general demand distributions and piecewise linear concave ordering costs. Such costs arise in the important cases of quantity discounts or multiple suppliers. We consider the case of concave costs involving two linear segments. This corresponds to the case of one supplier with a fixed cost, a variable cost up to a given order quantity, and a quantity discount beyond that, or equivalently, the case of two suppliers, one with a low fixed cost along with a high variable cost and the other with a high fixed cost along with a low variable cost. It is well understood that for a stochastic inventory control problem with a fixed cost and a per-unit variable cost, an (s, S) policy is optimal when there is only one supplier. In this work we address the case of multiple suppliers under several different scenarios. We provide a rigorous mathematical proof of the optimality of several inventory control models, which will help managers make better business decisions regarding procurement policies when facing multiple supply sources and/or quantity discounts for big purchases. Broadly, there are two main areas to explore in the realm of inventory control. The first is lost sales and the second is backlog sales. Our study examines both of these crucial areas. Our analysis is concerned with the generalization of the classical (s, S) policy for general demand distributions under a variety of modifications to the classical work of Scarf [36]. In particular, for the lost sales case, we show that certain three and four parameter generalizations of the classical (s, S) policy are optimal. Our contributions consist of generalizing the demand, solving a functional Bellman equation for the value function that arises in the infinite horizon framework, and providing an explicit solution in the special case of exponential demand density. We also give conditions under which our generalizations of the (s, S) policy reduce to the standard (s, S) policy, even though there are two suppliers involved. Moreover, we provide an explicit solution for the three number policy when the demand distribution is exponential. In the other situation, we are concerned with stochastic inventory control problems with backlog sales during stockout. As was the case for lost sales, we consider both the scenario in which an optimal selection can be made among two suppliers, as well as the scenario in which inventory can be purchased with incremental quantity discounts from a single supplier. We study the problem for arbitrary demand distributions and in infinite horizon. In this case, we first spell out conditions that guarantee the optimization of (s, S) policy for the problem under consideration. If these conditions fail to holds, we also demonstrate that a generalized three parameter policy is optimal in two distinct situations.Show more Item The Maximum Principle for Global Solutions of Stochastic Stackelberg Differential GamesBensoussan, Alain; Chen, Shaokuan; Sethi, Suresh P. (UT Dallas); 0000-0003-0743-498X (Bensoussan, A)Show more For stochastic Stackelberg differential games played by a leader and a follower, there are several solution concepts in terms of the players' information sets. In this paper we derive the maximum principle for the leader's global Stackelberg solution under the adapted closed-loop memoryless information structure, where the term global signifies the leader's domination over the entire game duration. As special cases, we study linear quadratic Stackelberg games under both adapted open-loop and adapted closed-loop memoryless information structures, as well as the resulting Riccati equations.Show more Item Mean Field Stackelberg Games: Aggregation of Delayed InstructionsBensoussan, Alain; Chau, M. H. M.; Yam, S. C. P.; 0000000113231180 (Bensoussan, A)Show more In this paper, we consider an N-player interacting strategic game in the presence of a (endogenous) dominating player, who gives direct influence on individual agents, through its impact on their control in the sense of Stackelberg game, and then on the whole community. Each individual agent is subject to a delay effect on collecting information, specifically at a delay time, from the dominating player. The size of his delay is completely known by the agent, while to others, including the dominating player, his delay plays as a hidden random variable coming from a common fixed distribution. By invoking a noncanonical fixed point property, we show that for a general class of finite N-player games, each of them converges to the mean field counterpart which may possess an optimal solution that can serve as an epsilon-Nash equilibrium for the corresponding finite N-player game. Second, we provide, with explicit solutions, a comprehensive study on the corresponding linear quadratic mean field games of small agents with delay from a dominating player. Given the information flow obtained from both the dominating player and the whole community via the mean field term, the filtration to which the control of the representative agent adapted is non-Brownian. Therefore, we propose to utilize backward stochastic dynamics (instead of the common approach through backward stochastic differential equations) for the construction of adjoint process for the resolution of his optimal control. A simple sufficient condition for the unique existence of mean field equilibrium is provided by tackling a class of nonsymmetric Riccati equations. Finally, via a study of a class of forward-backward stochastic functional differential equations, the optimal control of the dominating player is granted given the unique existence of the mentioned mean field equilibrium for small players.Show more Item Parabolic Bellman Equations with Risk Control(Society for Industrial and Applied Mathematics Publications) Bensoussan, Alain; Breit, D.; Frehse, J.; 0000000113231180 (Bensoussan, A); 0000-0003-0743-498X (Bensoussan, A); Bensoussan, AlainShow more We consider stochastic optimal control problems with an additional term representing the variance of the control functions. The latter one may serve as a risk control. We present and treat the problem in a purely analytical way via a Vlasov-McKean functional and Bellman equations with mean field dependence. We obtain global existence and, essentially, optimal global regularity for the solutions of the Bellman equation and the minimizing control. Surprisingly, the risk term simplifies the analysis to a certain extend.Show more Item Sequential Capacity Expansion Options(INFORMS, 2018-10-09) Bensoussan, Alain; Chevalier-Roignant, Benoit; 0000-0003-0743-498X (Bensoussan, A); Bensoussan, AlainShow more This paper considers a firm's capacity expansion decisions under uncertainty. The firm has leeway in timing investments and in choosing how much capacity to install at each investment time. We model this problem as the sequential exercising of compound capacity expansion options with embedded optimal capacity choices. We employ the impulse control methodology and obtain a quasi-variational inequality that involves two state variables: an exogenous, stochastic price process and a controlled capacity process (without a diffusion term). We provide a general verification theorem and identify-and prove the optimality of-a two-dimensional (s, S)-type policy for a specific (admittedly restrictive) choice of the model parameters and of the running profit. The firm delays investment in capacity to ensure that the perpetuity value of newly installed capacity exceeds the total opportunity cost, including the fixed cost component, by a sufficient margin. Our general model for "the option to expand" transcends a single-option exercise and yields predictions of both the optimal investment timing and the optimal scale of production.Show more Item Stochastic Differential Games with a Varying Number of Players(American Institute of Mathematical Sciences) Bensoussan, Alain; Frehse, J.; Grün, C.; 0000 0001 1323 1180 (Bensoussan, A); 2002119562 (Bensoussan, A)Show more We consider a non zero sum stochastic differential game with a maximum n players, where the players control a diffusion in order to minimisena certain cost functional. During the game it is possible that present players may die or new players may appear. The death, respectively the birth time of a player is exponentially distributed with intensities that depend on the diffusion and the controls of the players who are alive. We show how the game is related to a system of partial differential equations with a special coupling in the zero order terms. We provide an existence result for solutions in appropriate spaces that allow to construct Nash optimal feedback controls. The paper is related to a previous result in a similar setting for two players leading to a parabolic system of Bellman equations [4]. Here, we study the elliptic case (infinite horizon) and present the generalisation to more than two players.Show more Item The Impact of Competitive Advantage on the Investment Timing in Stackelberg Leader–Follower Game(Taylor and Francis Inc.) Hoe, S.; Yan, Z.; Bensoussan, Alain; 0000 0001 1323 1180 (Bensoussan, A); 0000-0003-0743-498X (Bensoussan, A); Bensoussan, AlainShow more This short note clarifies how the Stackelberg leader’s competitive advantage after the follower’s entry affects the leader’s optimal market entry decision and Stackelberg strategic interactions under uncertainty. Although the Stackelberg leader’s first investment threshold remains constant and coincides with the monopolist’s investment trigger, his second (third) investment threshold, which defines the exit (entry) of the first (second) investment interval, increases with an increased competitive advantage. With an increased competitive advantage, the probability of sequential investment equilibrium (simultaneous investment equilibrium) increases (decreases) irrespective of the level of volatility. Moreover, for a given level of competitive advantage, an increase in the volatility tends to decrease (increase) the probability of simultaneous investment equilibrium (sequential investment equilibrium). For a richer set of results, endogenous firm roles are examined and analyzed as well. The leader’s preemptive threshold is negatively affected by his competitive advantage.Show more Item Threshold-type Policies for Real Options Using Regime-Switching ModelsBensoussan, Alain; Yan, Z.; Yin, G.; 0000 0001 1323 1180 (Bensoussan, A); 2002119562 (Bensoussan, A)Show more To investigate the impact of macroeconomic conditions on irreversible investments under a regime switching model, our main effort in this work is to rigorously justify the existence and uniqueness of optimal threshold-type policies. The underlying cash flow process is modeled as a geometric Brownian motion with return rate and volatility depending on a continuous-time Markov chain. The problem is similar to the American style of call options. When dealing either with American options in a financial market or with real options, a common practice in the literature is to postulate threshold-type strategies and to find the optimal threshold levels as solutions of systems of nonlinear algebraic equations. Although from a computational standpoint, this seems to be a reasonable approach, the issue of existence and uniqueness of solutions has never been addressed to date. Instead of assuming the threshold-type policies, this paper establishes that indeed the threshold-type policies are the right choice. Variational inequalities are used to characterize the optimal strategy by an abstract, nonconstructive reasoning. In addition, numerical simulations are also provided to demonstrate quantitative properties and properties of the systems. Copyright © 2012 by SIAM.Show more Item Time-Consistent Portfolio Selection Under Short-Selling Prohibition: From Discrete to Continuous SettingBensoussan, Alain; Wong, K. C.; Yam, S. C. P.; Yung, S. P.; 0000 0001 1323 1180 (Bensoussan, A); 2002119562 (Bensoussan, A); 0000-0003-0743-498X (Bensoussan, A)Show more In this paper, we study the time consistent strategies in the mean-variance portfolio selection with short-selling prohibition in both discrete and continuous time settings. Recently, [T. Björk, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1-24] considered the problem with state dependent risk aversion in the sense that the risk aversion is inversely proportional to the current wealth, and they showed that the time consistent control is linear in wealth. Considering the counterpart of their continuous time equilibrium control in the discrete time framework, the corresponding "optimal" wealth process can take negative values; and this negativity in wealth will lead the investor to a risk seeker which results in an unbounded value function that is economically unsound; even more, the limiting of the discrete solutions has shown to be their obtained continuous solution in [T. Björk, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1-24]. To deal this limitation, we eliminate the chance of getting nonpositive wealth by prohibiting short-selling. Using backward induction, the equilibrium control in discrete time setting is shown to be linear in wealth. An application of the extended Hamilton-Jacobi-Bellman equation (see [T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, working paper, Stockholm School of Economics, Stockholm, Sweden, 2010]) makes us also conclude that the continuous time equilibrium control is also linear in wealth with investment to wealth ratio satisfying an integral equation uniquely. We also show that the discrete time equilibrium controls converge to that in continuous time setting. Finally, in numerical studies, we illustrate that the constrained strategy in continuous setting can outperform the unconstrained one in some situations as depicted in Figure 8.Show more