arXiv subjects

Daniel Lacker

Publications and source records attributed to Daniel Lacker.

44 records · Page 3Linked to original sources

arXiv2016

Limit theory for controlled McKean-Vlasov dynamics

This paper rigorously connects the problem of optimal control of McKean-Vlasov dynamics with large systems of interacting controlled state processes. Precisely, the empirical distributions of near-optimal control-state pairs for the $n$-state systems, as $n$ tends to infinity, admit limit points in distribution (if the objective functions are suitably coercive), and every such limit is supported on the set of optimal control-state pairs for the McKean-Vlasov problem. Conversely, any distribution on the set of optimal control-state pairs for the McKean-Vlasov problem can be realized as a limit in this manner. Arguments are based on controlled martingale problems, which lend themselves naturally to existence proofs; along the way it is shown that a large class of McKean-Vlasov control problems admit optimal Markovian controls.

math.PR↗

arXiv2016

Law invariant risk measures and information divergences

A one-to-one correspondence is drawn between law invariant risk measures and divergences, which we define as functionals of pairs of probability measures on arbitrary standard Borel spaces satisfying a few natural properties. Divergences include many classical information divergence measures, such as relative entropy and $f$-divergences. Several properties of divergence and their duality with law invariant risk measures are developed, most notably relating their chain rules or additivity properties with certain notions of time consistency for dynamic law invariant risk measures known as acceptance and rejection consistency. These properties are linked also to a peculiar property of the acceptance sets on the level of distributions, analogous to results of Weber on weak acceptance and rejection consistency. Finally, the examples of shortfall risk measures and optimized certainty equivalents are discussed in some detail, and it is shown that the relative entropy is essentially the only divergence satisfying the chain rule.

q-fin.RM↗

arXiv2015

Liquidity, risk measures, and concentration of measure

Expanding on techniques of concentration of measure, we develop a quantitative framework for modeling liquidity risk using convex risk measures. The fundamental objects of study are curves of the form $(ρ(λX))_{λ\ge 0}$, where $ρ$ is a convex risk measure and $X$ a random variable, and we call such a curve a \emph{liquidity risk profile}. The shape of a liquidity risk profile is intimately linked with the tail behavior of the underlying $X$ for some notable classes of risk measures, namely shortfall risk measures. We exploit this link to systematically bound liquidity risk profiles from above by other real functions $γ$, deriving tractable necessary and sufficient conditions for \emph{concentration inequalities} of the form $ρ(λX) \le γ(λ)$, for all $λ\ge 0$. These concentration inequalities admit useful dual representations related to transport inequalities, and this leads to efficient uniform bounds for liquidity risk profiles for large classes of $X$. On the other hand, some modest new mathematical results emerge from this analysis, including a new characterization of some classical transport-entropy inequalities. Lastly, the analysis is deepened by means of a surprising connection between time consistency properties of law invariant risk measures and the tensorization of concentration inequalities.

q-fin.RM↗

arXiv2015

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.

math.PR↗

arXiv2015

Mean field games via controlled martingale problems: Existence of Markovian equilibria

Mean field games are studied in the framework of controlled martingale problems, and general existence theorems are proven in which the equilibrium control is Markovian. The framework is flexible enough to include degenerate volatility, which may depend on both the control and the mean field. The objectives need not be strictly convex, and the mean field interactions considered are nonlocal and Wasserstein-continuous. When the volatility is nondegenerate, continuity assumptions may be weakened considerably. The proofs first use relaxed controls to establish existence. Then, using a convexity assumption and measurable selection arguments, strict (non-relaxed) Markovian equilibria are constructed from relaxed equilibria.

math.PR↗

arXiv2014

Translation invariant mean field games with common noise

This note highlights a special class of mean field games in which the coefficients satisfy a convolution-type structural condition. A mean field game of this type with common noise is related to a certain mean field game without common noise by a simple transformation, which permits a tractable construction of a solution of the problem with common noise from a solution of the problem without.

math.PR↗

arXiv2014

A general characterization of the mean field limit for stochastic differential games

The mean field limit of large-population symmetric stochastic differential games is derived in a general setting, with and without common noise, on a finite time horizon. Minimal assumptions are imposed on equilibrium strategies, which may be asymmetric and based on full information. It is shown that approximate Nash equilibria in the $n$-player games admit certain weak limits as $n$ tends to infinity, and every limit is a weak solution of the mean field game (MFG). Conversely, every weak MFG solution can be obtained as the limit of a sequence of approximate Nash equilibria in the $n$-player games. Thus, the MFG precisely characterizes the possible limiting equilibrium behavior of the $n$-player games. Even in the setting without common noise, the empirical state distributions may admit stochastic limits which cannot be described by the usual notion of MFG solution.

math.PR↗

arXiv2014

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.

math.PR↗