Naveen Jindal School of Management
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Browsing Naveen Jindal School of Management by Author "0000 0001 1323 1180 (Bensoussan, A)"
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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.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 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 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 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.