Mean-field stochastic differential equations and associated PDEs

By Rainer Buckdahn, Juan Li, Shige Peng, Catherine Rainer. 2014

361 citations recorded by the source corpus.

In this paper we consider a mean-field stochastic differential equation, also called Mc Kean-Vlasov equation, with initial data $(t,x)\in[0,T]\times R^d,$ which coefficients depend on both the solution $X^{t,x}_s$ but also its law. By considering square integrable random variables $\xi$ as initial condition for this equation, we can easily show the flow property of the solution $X^{t,\xi}_s$ of this new equation. Associating it with a process $X^{t,x,P_\xi}_s$ which coincides with $X^{t,\xi}_s$, when one substitutes $\xi$ for $x$, but which has the advantage to depend only on the law $P_\xi$ of $\xi$, we characterise the function $V(t,x,P_\xi)=E[\Phi(X^{t,x,P_\xi}_T,P_{X^{t,\xi}_T})]$ under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a non local PDE of mean-field type, involving the first and second order derivatives of $V$ with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first and second order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding It\^{o} formula. In our approach we use the notion of derivative with respect to a square integrable probability measure introduced in \cite{PL} and we extend it in a direct way to second order derivatives.

Subfields: Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/1141bdb4555ea7bbd99bb0b98f93675173f14107

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Citations in the corpus

Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications (2016, background): “We briefly recall the basic definitions and refer to [14] for the details, see also [12], [21].”
Existence of a solution to an equation arising from the theory of Mean Field Games (2015, background): “Also during the second submission of the paper a referee pointed out preprints [11, 14] which deal with classical solutions of Master Equations for stochastic Mean Field Games.”
Large and Moderate Deviation Principles for McKean-Vlasov SDEs with Jumps (2020, background): “One can refer to [13, 25, 30, 48, 49] and the references therein for the existence and uniqueness of solutions to MVSDEs, [2, 20, 32, 47, 46, 54] for propagation of chaos, [21, 22, 23, 24, 28, 38, 41, 42, 43, 52] for exponential ergodicity and functional inequalities, [1, 19, 27, 40, 53] and the ref”
McKean–Vlasov SDEs under measure dependent Lyapunov conditions (2018, methodology): “Here we follow the argument from [9] explaining how to go from an Itô formula for function of measures only, i.”
Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions (2021, background): “1 in [10], the lifted function of a smooth function on the Wasserstein space may not be smooth on the lifted Hilbert space, and so a viscosity solution in the intrinsic sense may”
Optimal control of path-dependent McKean–Vlasov SDEs in infinite-dimension (2020, background): “without pathdependency of the coefficients, the Wasserstein derivative in the lifted sense of Lions [41], turns out to be a convenient notion of measure derivative when combined with Itô’s formula along the flow of probability measures (see [10]) in order to define the Master equation in mean-field ”
Smoothing properties of McKean–Vlasov SDEs (2017, background): “A result from [8] says that derivatives commute when the mixed derivatives are Lipschitz continuous.”
Freidlin–Wentzell LDP in path space for McKean–Vlasov equations and the functional iterated logarithm law (2017, background): “Mean-field stochastic differential equations and associated PDEs.”
Bismut formula for Lions derivative of distribution dependent SDEs and applications (2018, methodology): “4) but has important applications in solving PDEs with Lions' derivatives, see [5, 17, 20] and references within.”
Strong convergence rates in averaging principle for slow-fast McKean-Vlasov SPDEs (2021, result): “We refer the interested reader to [5, 11, 26, 30, 41] and references therein for more recent results on this topic.”
Bellman equation and viscosity solutions for mean-field stochastic control problem (2015, positive): “…d ), which is then written as: Existence and uniqueness of a solution to (3.7) implies the ﬂow property: Moreover, as pointed out in Remark 3.1 in [10] (see also the remark following (2.3) in [16]), the solution to (3.7) is also unique in law from which it follows that the law of X t,ξ depends on…”
Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics (2016, neutral): “4)) is inspired by [9], where the uncontrolled case is addressed.”
Strong Solutions of Mean-Field Stochastic Differential Equations with irregular drift (2018, background): “For the special case where the mean-field dependence is given via an expectation functional of the form dX t = b(t, X x t ,E[φ(X x t )])dt+ dBt, X x 0 = x ∈ R, t ∈ [0, T ], (9) for some b : [0, T ] × R × R → R, continuous differentiability of X with respect to x can be deduced from [6] under the ass”
Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls (2018, background): “Note that a master equation can rarely have a classical solution, we refer to Buckdahn, Li, Peng, & Rainer [5], Cardaliaguet, Delarue, Lasry, & Lions [8] (defined on torus), Chassagneux, Crisan, & Delarue [12], and Saporito & Zhang [35] for some positive results in this direction.”
Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data (2019, methodology): “We emphasize that all the above global wellposedness results are under certain monotonicity assumption on F and G, with the exception [12] which is linear and thus does not involve controls.”
The Master Equation and the Convergence Problem in Mean Field Games (2015, methodology): “1 and the derivative introduced by Lions in [52] and used (among others) in [16, 22].”
A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria (2014, methodology): “We also observe that, differently from [3], we do not require the coefficients to admit second-order derivatives of the type B(2) μμ.”