Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization

Abstract

The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. We provide a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter are also used in order to give robust transport-based estimates for the value of having additional information, as well as model sensitivity with respect to the reference measure, for the classical stochastic optimization problems of utility maximization and optimal stopping. Our results have natural extensions to the case of general multidimensional continuous semimartingales.

Extended Mean Field Control Problems: Stochastic Maximum Principle and Transport Perspective

“…π, {∅, C} × F 2 , which we callβ, is well defined. As in [1], one can prove that the process M t := ω t −x 0 − t 0β s (ω)ds is a (µ, F 2 )-martingale. Indeed, taking 0 ≤ s < t ≤ T and h s ∈ L ∞ (F 2 s ), we have where the third equality follows since ω, which is a (γ, F 1 )-martingale, is consequently by causality a (π, F 1 ⊗ F 2 )-martingale.…”

“…To be specific, the optimal transport problem we obtain in the discretization has an additional causality constraint (see e.g. [23,1,4,5]); the numerical analysis of such problems is also having a burst of activity (e.g. [27,28,29]).…”

“…In this section we formulate a variational transport problem on C = C([0, T ]; R d ), the space of R d -valued continuous paths, which is equivalent to finding the weak solutions of the extended Mean Field problem (5.1). This variational formulation is a particular type of transport problem under the so-called causality constraint; see [23,1,4,5]. Here we recall this concept with respect to the filtrations F 1 and F 2 , generated by the first and by the second coordinate process on C × C. Definition 5.3.…”

“…is ≺ cm -monotone. 1 Then the minimization in the extended Mean Field problem (5.1) can be taken over weak closed loop tuples. Moreover, if the infimum is attained, then the optimal control α is of weak closed loop form.…”

“…The proof follows the projection arguments used in [1], which requires the above convexity assumptions. On the other hand, no regularity conditions are required here, unlike in the classical PDE or probabilistic approaches (see Assumptions (I)-(II) in Section 3).…”