Stability for gains from large investors’ strategies in $M_{1}$/$J_{1}$ topologies

Abstract

We prove continuity of a controlled SDE solution in Skorokhod's M 1 and J 1 topologies and also uniformly, in probability, as a non-linear functional of the control strategy. The functional comes from a finance problem to model price impact of a large investor in an illiquid market. We show that M 1 -continuity is the key to ensure that proceeds and wealth processes from (self-financing) càdlàg trading strategies are determined as the continuous extensions for those from continuous strategies. We demonstrate by examples how continuity properties are useful to solve different stochastic control problems on optimal liquidation and to identify asymptotically realizable proceeds.

Mean Field Games with Singular Controls

“…Finally, by the linear growth of b (3) , boundedness of l, and compactness of U and A m 0,T , and the uniform bound (4.9), it holds that sup The next result states that the cost functional is continuous on the graph…”

“…Thus, the trader needs to make a decision of the trading rate in order to minimize her trading cost (or maximize her net profit). One can refer to [3,26] among others for liquidation with singular controls. Recently, liquidation models beyond single player especially MFGs of optimal liquidation has drawn a lot of attention; see e.g.…”

“…We assume the price impact to be in a linear form. Hence, the actual market price of the resource follows the dynamic and let (X (1) , X (2) , X (3) , Q, Z (1) , Z (2) ) be the coordinate processes on Ω m , i.e., (2) for each ω = (x (1) , x (2) , x (3) , q, z (1) , z (2) ) ∈ Ω m . Note that by [29,Lemma 3.2] (see also [20, Section 2.1.2]) Q can be identified by an predictable disintegration in the following sense…”

“…Let X o be the coordinate projection onto D 0,T +1 × D 0,T +1 × D 0,T +1 and (F o t ) be the canonical filtration on Ω o . Correspondingly, we extend the time-domain of coefficients from [0, T ] to [0, T + 1], i.e., there exists γ(= b (1) , b (2) , b (3) , σ (1) , σ (2) , λ, l) satisfying the same assumptions as…”

“…In this paper, motivated by optimal portfolio liquidation and optimal exploitation of exhaustible resource (see Section 2) we consider the following MFGs with singular controls                                    1. For fixed probability measures µ := (µ (1) , µ (2) , µ (3) ) in some suitable space solve the optimization probelm : minmize J(u, Z (1) , Z (2) ; µ) such that…”