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23
LinearQuadratic JumpDiffusion Modeling with Application to Stochastic Volatility
, 2004
"... We aim at accommodating the existing affine jumpdiffusion and quadratic models under the same roof, namely the linearquadratic jumpdiffusion (LQJD) class. We give a complete characterization of the dynamics of this class of models by stating explicitly a list of structural constraints, and comput ..."
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Cited by 26 (1 self)
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We aim at accommodating the existing affine jumpdiffusion and quadratic models under the same roof, namely the linearquadratic jumpdiffusion (LQJD) class. We give a complete characterization of the dynamics of this class of models by stating explicitly a list of structural constraints, and compute standard and extended transforms relevant to asset pricing. We show that the LQJD class can be embedded into the affine class through use of an augmented state vector, and further establish that a onetoone equivalence relationship holds between both classes in terms of transform analysis. An option pricing application to multifactor stochastic volatility models reveals that adding nonlinearity into the model would reduce pricing errors and yield parameter estimates that are more in line with sensible economic interpretation.
Existence, uniqueness and parametrization of Lagrangian invariant subspaces
 SIAM J. Matrix Anal. Appl
"... Abstract. The existence, uniqueness, and parametrization of Lagrangian invariant subspaces for Hamiltonian matrices is studied. Necessary and sufficient conditions and a complete parametrization are given. Some necessary and sufficient conditions for the existence of Hermitian solutions of algebraic ..."
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Cited by 20 (16 self)
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Abstract. The existence, uniqueness, and parametrization of Lagrangian invariant subspaces for Hamiltonian matrices is studied. Necessary and sufficient conditions and a complete parametrization are given. Some necessary and sufficient conditions for the existence of Hermitian solutions of algebraic Riccati equations follow as simple corollaries.
LinearQuadratic Mean Field Games
, 2014
"... As an organic combination of mean field theory in statistical physics and (nonzero sum) stochastic differential games, Mean Field Games (MFGs) has become a very popular research topic in the fields ranging from physical and social sciences to engineering applications, see for example the earlier st ..."
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Cited by 16 (0 self)
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As an organic combination of mean field theory in statistical physics and (nonzero sum) stochastic differential games, Mean Field Games (MFGs) has become a very popular research topic in the fields ranging from physical and social sciences to engineering applications, see for example the earlier studies by Huang, Caines and Malhame ́ (2003), and that by Lasry and Lions (2006a, b and 2007). In this paper, we provide a comprehensive study of a general class of mean field games in the linear quadratic framework. We adopt the adjoint equation approach to investigate the existence and uniqueness of equilibrium strategies of these LinearQuadratic Mean Field Games (LQMFGs). Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forwardbackward ordinary differential equation. For the one dimensional case, we show that the equilibrium strategy always exists uniquely. For dimension greater than one, by choosing a suitable norm and then applying the Banach Fixed Point Theorem, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls and is always satisfied whenever those of the
A.: Efficient strong integrators for linear stochastic systems
 SIAM J. Numer. Anal
"... Abstract. We present numerical schemes for the strong solution of linear stochastic differential equations driven by two Wiener processes and with noncommutative vector fields. These schemes are based on the Neumann and Magnus expansions. We prove that for a sufficiently small stepsize, the half o ..."
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Cited by 13 (6 self)
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Abstract. We present numerical schemes for the strong solution of linear stochastic differential equations driven by two Wiener processes and with noncommutative vector fields. These schemes are based on the Neumann and Magnus expansions. We prove that for a sufficiently small stepsize, the half order Magnus and a new modified order one Magnus integrator are globally more accurate than classical stochastic numerical schemes or Neumann integrators of the corresponding order. These Magnus methods will therefore always be preferable provided the cost of computing the matrix exponential is not significant. Further, for small stepsizes the accurate representation of the Lévy area between the two driving processes dominates the computational cost for all methods of order one and higher. As a consequence, we show that the accuracy of all stochastic integrators asymptotically scales like the squareroot of the computational cost. This has profound implications on the effectiveness of higher order integrators. In particular in terms of efficiency, there are generic scenarios where order one Magnus methods compete with and even outperform higher order methods. We consider the consequences in applications such as linearquadratic optimal control, filtering problems and the pricing of pathdependent financial derivatives.
A differential game approach to formation control
 IEEE Transactions on Control Systems Technology
"... Abstract—This paper presents a differential game approach to formation control of mobile robots. The formation control is formulated as a linearquadratic Nash differential game through the use of graph theory. Finite horizon cost function is discussed under the openloop information structure. An ..."
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Cited by 11 (0 self)
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Abstract—This paper presents a differential game approach to formation control of mobile robots. The formation control is formulated as a linearquadratic Nash differential game through the use of graph theory. Finite horizon cost function is discussed under the openloop information structure. An openloop Nash equilibrium solution is investigated by establishing existence and stability conditions of the solutions of coupled (asymmetrical) Riccati differential equations. Based on the finite horizon openloop Nash equilibrium solution, a receding horizon approach is adopted to synthesize a statefeedback controller for the formation control. Mobile robots with double integrator dynamics are used in the formation control simulation. Simulation results are provided to justify the models and solutions. Index Terms—Formation control, linearquadratic differential game, Nash equilibrium. I.
GRASSMANNIAN SPECTRAL SHOOTING
, 2010
"... We present a new numerical method for computing the purepoint spectrum associated with the linear stability of coherent structures. In the context of the Evans function shooting and matching approach, all the relevant information is carried by the flow projected onto the underlying Grassmann manifo ..."
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Cited by 10 (5 self)
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We present a new numerical method for computing the purepoint spectrum associated with the linear stability of coherent structures. In the context of the Evans function shooting and matching approach, all the relevant information is carried by the flow projected onto the underlying Grassmann manifold. We show how to numerically construct this projected flow in a stable and robust manner. In particular, the method avoids representation singularities by, in practice, choosing the best coordinate patch representation for the flow as it evolves. The method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed. For large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and the Ekman boundary layer.
Linearquadratic jumpdiffusion modeling
 Mathematical Finance
, 2007
"... We aim at accommodating the existing affine jumpdiffusion and quadratic models under the same roof, namely the linearquadratic jumpdiffusion (LQJD) class. We give a complete characterization of the dynamics of this class by stating explicitly the structural constraints, as well as the admissibili ..."
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Cited by 9 (1 self)
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We aim at accommodating the existing affine jumpdiffusion and quadratic models under the same roof, namely the linearquadratic jumpdiffusion (LQJD) class. We give a complete characterization of the dynamics of this class by stating explicitly the structural constraints, as well as the admissibility conditions. This allows us to carry out a specification analysis for the 3factor LQJD models. We compute the standard transform of the state vector relevant to asset pricing up to a system of ordinary differential equations. We show that the LQJD class can be embedded into the affine class through use of an augmented state vector. This establishes a onetoone equivalence relationship between both classes in terms of transform analysis. CHENG, P., SCAILLET, Olivier. LinearQuadratic JumpDiffusion Modeling. 2006 Available at:
Economic uncertainty, disagreement, and credit markets, Working paper
, 2010
"... Using an economy populated with agents with heterogeneous beliefs this paper delivers important implications for the role of common and firm specific components of economic uncertainty on credit spreads. We first derive a positive relation between uncertainty and belief heterogeneity in an equilibri ..."
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Cited by 9 (1 self)
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Using an economy populated with agents with heterogeneous beliefs this paper delivers important implications for the role of common and firm specific components of economic uncertainty on credit spreads. We first derive a positive relation between uncertainty and belief heterogeneity in an equilibrium model of credit risk in which agents with different subjective economic uncertainty parameters disagree about the expected growth rate of future cash flows. Then using the model solutions, we obtain testable empirical predictions for the impact of economic uncertainty and differences in beliefs on credit spreads and asset prices. We merge a dataset of firmspecific differences in beliefs and credit spreads and test these predictions empirically. The empirical tests provide 5 new results that are original to the literature: (a) Countercyclical uncertainty is positively related to a common disagreement component about future earning opportunities by financial analysts; (b) The common component of the difference in beliefs in earnings forecasts is the most significant variable in timeseries regressions for credit spreads. At the same time, firmspecific difference in beliefs is the most significant component in the crosssection; (c) Uncertainty induces a significant comovement between credit spreads and stock volatility; (d) During the 2008 Credit Crisis the link between uncertainty and credit spreads was even stronger than in previous crisis periods; (e) Uncertainty and belief heterogeneity have significant explanatory power for noarbitrage violations implied by single factor models in
NonBlowUp Conditions for Riccatitype Matrix Differential and Difference Equations
"... We present several methods to obtain global existence results for solutions of nonsymmetric Riccati matrix differential equations and for generalized or perturbed symmetric Riccati differential equations. One approach is to derive sufficient conditions ensuring that the spectral norm of the solutio ..."
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Cited by 7 (2 self)
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We present several methods to obtain global existence results for solutions of nonsymmetric Riccati matrix differential equations and for generalized or perturbed symmetric Riccati differential equations. One approach is to derive sufficient conditions ensuring that the spectral norm of the solutions remain uniformly bounded in an interval (−∞, t0] or, weaker, that the minimal an the maximal eigenvalue of a hermitian solution remains bounded in (−∞, t0]. If however, there exists a linearizing transformation, as in the case of a nonsymmetric Riccati differential equation, then with the aid of an appropriate Lyapunovtype function we obtain sufficient conditions guaranteeing that no escape finite time can occur. This method also applies to nonsymmetric matrix Riccati difference equations. These results, among others, can then be applied to control problems like H∞control, Markovian Jump Linear Quadratic control, Minimal Cost Variance control and to open loop and memoryless feedback Nash games as well.