Model uncertainty stochastic mean-field control

Abstract

We consider the problem of optimal control of a mean-field stochastic differential equation (SDE) under model uncertainty. The model uncertainty is represented by ambiguity about the law L(X(t)) of the state X(t) at time t. For example, it could be the law L P (X(t)) of X(t) with respect to the given, underlying probability measure P. This is the classical case when there is no model uncertainty. But it could also be the law L Q (X(t)) with respect to some other probability measure Q or, more generally, any random measure µ(t) on R with total mass 1. We represent this model uncertainty control problem as a stochastic differential game of a mean-field related type SDE with two players. The control of one of the players, representing the uncertainty of the law of the state, is a measure-valued stochastic process µ(t) and the control of the other player is a classical real-valued stochastic process u(t). This optimal control problem with respect to random probability processes µ(t) in a non-Markovian setting is a new type of stochastic control problems that has not been studied before. By constructing a new Hilbert space M of measures, we obtain a sufficient and a necessary maximum principles for Nash equilibria for such games in the general nonzero-sum case, and for saddle points in zero-sum games.As an application we find an explicit solution of the problem of optimal consumption under model uncertainty of a cash flow described by a mean-field related type SDE. MSC(2010): 60H05, 60H20, 60J75, 93E20, 91G80, 91B70.

Mean-field stochastic control with elephant memory in finite and infinite time horizon

“…Remark 3.5 Using the Itô formula we see that Assumption A3 holds under reasonable smoothness conditions on the coefficients of the equation. A proof for a similar system is given in Lemma 12 in Agram and Øksendal [5]. We omit the details.…”

“…We now define a weighted Sobolev spaces of measures. It is strongly related to the space introduced in Agram and Øksendal [5], [6], but with a different weight, which is more suitable for estimates (see e.g. Lemma 2.4 below):…”

“…We recall the following result, which is proved in Agram and Øksendal [5], Lemma 2.3/Lemma 6: Lemma 2.6 If X(t) is an Itô-Lévy process as in (1.1), then the map t → M(t) : [0, T ] → M 4 0 is absolutely continuous. Hence the derivative…”

Singular Control Optimal Stopping of Memory Mean-Field Processes

“…In this section, we proceed as in Agram and Øksendal [2], [3] and construct a Hilbert space M of random measures on R. It is simpler to work with than the Wasserstein metric space that has been used by many authors previously. See e.g.…”

“…Carmona et al [7], [8], Buckdahn et al [5] and the references therein. Following Agram and Øksendal [2], [3], we now introduce the following Hilbert spaces:…”