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In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model

with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.

This new distance $\widetilde{W}^{2}$ is defined similarly to the classical Wasserstein distance $\widetilde{W}^{2}$ but the set of couplings is restricted to the set of laws of solutions of 2$d$-dimensional stochastic differential equations. We prove that this new distance $\widetilde{W}^{2}$ metrizes the weak topology. Furthermore this distance $\widetilde{W}^{2}$ is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.

In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model

with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional ...

91B70 ; 60H30 ; 60H15 ; 60J60 ; 93E20

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Very large networks linking dynamical agents are now ubiquitous and there is significant interest in their analysis, design and control. The emergence of the graphon theory of large networks and their infinite limits has recently enabled the formulation of a theory of the centralized control of dynamical systems distributed on asymptotically infinite networks [Gao and Caines, IEEE CDC 2017, 2018]. Furthermore, the study of the decentralized control of such systems has been initiated in [Caines and Huang, IEEE CDC 2018] where Graphon Mean Field Games (GMFG) and the GMFG equations are formulated for the analysis of non-cooperative dynamical games on unbounded networks. In this talk the GMFG framework will be first be presented followed by the basic existence and uniqueness results for the GMFG equations, together with an epsilon-Nash theorem relating the infinite population equilibria on infinite networks to that of finite population equilibria on finite networks.

Very large networks linking dynamical agents are now ubiquitous and there is significant interest in their analysis, design and control. The emergence of the graphon theory of large networks and their infinite limits has recently enabled the formulation of a theory of the centralized control of dynamical systems distributed on asymptotically infinite networks [Gao and Caines, IEEE CDC 2017, 2018]. Furthermore, the study of the decentralized ...

49N70 ; 93E20 ; 93E35

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We introduce a new strategy for the solution of Mean Field Games in the presence of major and minor players. This approach is based on a formulation of the fixed point step in spaces of controls. We use it to highlight the differences between open and closed loop problems. We illustrate the implementation of this approach for linear quadratic and finite state space games, and we provide numerical results motivated by applications in biology and cyber-security.

We introduce a new strategy for the solution of Mean Field Games in the presence of major and minor players. This approach is based on a formulation of the fixed point step in spaces of controls. We use it to highlight the differences between open and closed loop problems. We illustrate the implementation of this approach for linear quadratic and finite state space games, and we provide numerical results motivated by applications in biology and ...

93E20 ; 60H10 ; 60K35 ; 49K45

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We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity expansion is irreversible and modeled in terms of timing strategies characterized through threshold rules. Because the impact of changing costs on the producers is asymmetric, we are led to a nonzero-sum timing game describing the transitions among the discrete investment stages. Working in a continuous-time diffusion framework, we characterize and analyze the resulting Nash equilibrium and game values. Our analysis quantifies the dynamic competition effects and yields insight into dynamic preemption and over-investment in a general asymmetric setting. A case-study considering the impact of fluctuating emission costs on power producers investing in nuclear and coal-fired plants is also presented.

We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity expansion is irreversible and modeled in terms of timing strategies characterized through threshold rules. Because the impact of changing costs on the producers is asymmetric, we are led to a nonzero-sum timing game describing the transitions among the ...

93E20 ; 91B38 ; 91A80

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The valuation of American options (a widespread type of financial contract) requires the numerical solution of an optimal stopping problem. Numerical methods for such problems have been widely investigated. Monte-Carlo methods are based on the implementation of dynamic programming principles coupled with regression techniques. In lower dimension, one can choose to tackle the related free boundary PDE with deterministic schemes.

Pricing of American options will therefore be inevitably heavier than the one of European options, which only requires the computation of a (linear) expectation. The calibration (fitting) of a stochastic model to market quotes for American options is therefore an a priori demanding task. Yet, often this cannot be avoided: on exchange markets one is typically provided only with market quotes for American options on single stocks (as opposed to large stock indexes - e.g. S&P500 - for which large amounts of liquid European options are typically available).

In this talk, we show how one can derive (approximate, but accurate enough) explicit formulas - therefore replacing other numerical methods, at least in a low-dimensional case - based on asymptotic calculus for diffusions.

More precisely: based on a suitable representation of the PDE free boundary, we derive an approximation of this boundary close to final time that refines the expansions known so far in the literature. Via the early premium formula, this allows to derive semi-closed expressions for the price of the American put/call. The final product is a calibration recipe of a Dupire's local volatility to American option data.

Based on joint work with Pierre Henry-Labordère.

The valuation of American options (a widespread type of financial contract) requires the numerical solution of an optimal stopping problem. Numerical methods for such problems have been widely investigated. Monte-Carlo methods are based on the implementation of dynamic programming principles coupled with regression techniques. In lower dimension, one can choose to tackle the related free boundary PDE with deterministic schemes.

Pricing of ...

93E20 ; 91G60