Optimal liquidation under stochastic liquidity

Abstract

We solve explicitly a two-dimensional singular control problem of finite fuel type for infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative and transient price impact. Liquidity is stochastic in that the volume effect process, which determines the inter-temporal resilience of the market in spirit of [PSS11], is taken to be stochastic, being driven by own random noise. The optimal control is obtained as the local time of a diffusion process reflected at a non-constant free boundary. To solve the HJB variational inequality and prove optimality, we need a combination of probabilistic arguments and calculus of variations methods, involving Laplace transforms of inverse local times for diffusions reflected at elastic boundaries.

Universal bounds and monotonicity properties of ratios of Hermite and parabolic cylinder functions

“…Since then, inequalities of this form have attracted a lot of attention, and have been proved to be valid for other polynomials such as Hermite (obtained from Hermite functions by taking ν ∈ N), Jacobi, Laguerre or ultraspherical polynomials (see [12,22], among others), and for special functions as (modified) Bessel, Gamma, parabolic cylinder or hypergeometric functions (see [2,4,5,6,7,23], among many others). Applications of Turán type inequalities can be found in many fields, ranging from biophysics (see [3] and the references therein) to information theory (see [15]) and stochastic control (see [8,11]).…”

An Optimal Extraction Problem with Price Impact

“…for the first equality, and Lemma 4.3-(2) with k = 1 for the second equality. Moreover, To prove that the right-derivative U ′ (0+) is negative, we now explain how to employ in our setting the arguments of the proof of Lemma 6.7 in [4]. First of all, we discuss the standing Assumption 2.2 in [4].…”

“…Our mathematical formulation of the previous problem leads to a two-dimensional degenerate finite-fuel singular stochastic control problem (see [8], [20], [21] and [23] as early contributions, and [4] and [15] for recent applications to optimal liquidation problems). The underlying state variable is a two-dimensional process (X, Y ) whose components are the commodity's price and the level of the reservoir (i.e.…”

“…When the price follows an Ornstein-Uhlenbeck dynamics, our proof of the optimality of the constructed candidate value function partly employs arguments developed in the study of an optimal liquidation problem tackled in the recent [4], which shares mathematical similarities with our problem. Indeed, in the case of a "small" marginal cost of extraction, due to the unhandy and implicit form of the increasing eigenfunction of the infinitesimal generator of the Ornstein-Uhlenbeck process, we have not been able to prove via direct means an inequality that the candidate value function needed to satisfy in order to solve the HJB equation.…”

“…Indeed, in the case of a "small" marginal cost of extraction, due to the unhandy and implicit form of the increasing eigenfunction of the infinitesimal generator of the Ornstein-Uhlenbeck process, we have not been able to prove via direct means an inequality that the candidate value function needed to satisfy in order to solve the HJB equation. For this reason, in such a case, we adopted ideas from [4] where an interesting reformulation of the original singular control problem as a calculus of variations approach has been developed. However, it is also worth noticing that when the marginal cost of extraction is "large enough", the approach of [4] is not directly applicable since a fundamental assumption in [4] (cf.…”