93 research outputs found
A Probabilistic Approach to Mean Field Games with Major and Minor Players
We propose a new approach to mean field games with major and minor players.
Our formulation involves a two player game where the optimization of the
representative minor player is standard while the major player faces an
optimization over conditional McKean-Vlasov stochastic differential equations.
The definition of this limiting game is justified by proving that its solution
provides approximate Nash equilibriums for large finite player games. This
proof depends upon the generalization of standard results on the propagation of
chaos to conditional dynamics. Because it is on independent interest, we prove
this generalization in full detail. Using a conditional form of the Pontryagin
stochastic maximum principle (proven in the appendix), we reduce the solution
of the mean field game to a forward-backward system of stochastic differential
equations of the conditional McKean-Vlasov type, which we solve in the Linear
Quadratic setting. We use this class of models to show that Nash equilibriums
in our formulation can be different from those of the formulations contemplated
so far in the literature
The Self-Financing Equation in High Frequency Markets
High Frequency Trading (HFT) represents an ever growing proportion of all
financial transactions as most markets have now switched to electronic order
book systems. The main goal of the paper is to propose continuous time
equations which generalize the self-financing relationships of frictionless
markets to electronic markets with limit order books. We use NASDAQ ITCH data
to identify significant empirical features such as price impact and recovery,
rough paths of inventories and vanishing bid-ask spreads. Starting from these
features, we identify microscopic identities holding on the trade clock, and
through a diffusion limit argument, derive continuous time equations which
provide a macroscopic description of properties of the order book. These
equations naturally differentiate between trading via limit and market orders.
We give several applications (including hedging European options with limit
orders, market maker optimal spread choice, and toxicity indexes) to illustrate
their impact and how they can be used to the benefit of Low Frequency Traders
(LFTs)
A probabilistic weak formulation of mean field games and applications
Mean field games are studied by means of the weak formulation of stochastic
optimal control. This approach allows the mean field interactions to enter
through both state and control processes and take a form which is general
enough to include rank and nearest-neighbor effects. Moreover, the data may
depend discontinuously on the state variable, and more generally its entire
history. Existence and uniqueness results are proven, along with a procedure
for identifying and constructing distributed strategies which provide
approximate Nash equlibria for finite-player games. Our results are applied to
a new class of multi-agent price impact models and a class of flocking models
for which we prove existence of equilibria
Mean field games with common noise
A theory of existence and uniqueness is developed for general stochastic
differential mean field games with common noise. The concepts of strong and
weak solutions are introduced in analogy with the theory of stochastic
differential equations, and existence of weak solutions for mean field games is
shown to hold under very general assumptions. Examples and counter-examples are
provided to enlighten the underpinnings of the existence theory. Finally, an
analog of the famous result of Yamada and Watanabe is derived, and it is used
to prove existence and uniqueness of a strong solution under additional
assumptions
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