# Bensoussan, Alain

Permanent URI for this collectionhttps://hdl.handle.net/10735.1/3180

Allain Bensoussan is the Ashbel Smith Professor of Risk and Decision Analysis. He also serves as the Director of the International Center for Decision and Risk Analysis (ICDRiA). In 2017 he was awarded the Lars Magnus Ericsson Chair. His research interests include:

- inventory control with partial information
- Risk management
- Mathematical finance

Learn more about Dr. Bensoussanon his Faculty, Endowed Professorships and Chairs, ICDRiA and Research Explorer pages.

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### Recent Submissions

Item Sequential Capacity Expansion Options(INFORMS, 2018-10-09) Bensoussan, Alain; Chevalier-Roignant, Benoit; 0000-0003-0743-498X (Bensoussan, A); Bensoussan, AlainThis 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.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, AlainWe 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.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, AlainIn 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 SocietyItem 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, AlainWe 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 MathematicsItem 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, AlainThis 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.Item Inventory Control with Fixed Cost and Price Optimization in Continuous Time(Wilmington Scientific Publisher) Bensoussan, Alain; Skaaning, S.; Turi, Jànos; Bensoussan, Alain; Turi, JànosWe 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.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, AlainWe 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.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)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.Item Mean Field Stackelberg Games: Aggregation of Delayed InstructionsBensoussan, Alain; Chau, M. H. M.; Yam, S. C. P.; 0000000113231180 (Bensoussan, A)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.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)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.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)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.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)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.Item Existence and compactness for weak solutions to bellman systems with critical growthBensoussan, Alain; Bulíĉek, M.; Frehse, J.; 0000 0001 1323 1180 (Bensoussan, A); 2002119562 (Bensoussan, A)We deal with nonlinear elliptic and parabolic systems that are the Bellman systems associated to stochastic differential games as a main motivation. We establish the existence of weak solutions in any dimension for an arbitrary number of equations (\players"). The method is based on using a renormalized sub- and super-solution technique. The main novelty consists in the new structure conditions on the critical growth terms with allow us to show weak solvability for Bellman systems to certain classes of stochastic differential games.