Optimal Trade Execution in an Order Book Model with Stochastic Liquidity Parameters

Càdlàg semimartingale strategies for optimal trade execution in stochastic order book models

“…1−e −δY t , t ∈ [0, T ], is bounded. Together with (C ≥ε ), (C bdd ) and 0 ≤ L ≤ 1 2 it follows that ψ and η are bounded. One can then show that…”

“…Thus M ⊥,(1) − M ⊥,( 2) is a local martingale starting in 0 with M ⊥,(1) − M ⊥,(2) = 0. It follows from the Burkholder-Davis-Gundy inequality that M ⊥, (1) and M ⊥, (2) are indistinguishable. Then, (76) implies further that…”

“…We define the process Γ = (Γ s ) with f given in (14). Finally, the fact that the discrete-time processes Y h , h ∈ (0, T − t), are (0, 1/2]-valued, which is proved in [1], explains the requirement in (C BSDE ) that Y is [0, 1/2]-valued.…”

“…This and several other interesting qualitative effects are presented in Section 5. We, finally, remark that we study a discrete-time version of the problem in [1], but that study is concentrated on different questions, as the mentioned effects, being purely continuous-time features, cannot be discussed in the framework of [1]. The remainder of this article is organized as follows.…”

“…It then follows from V 0 (x, d) = Y 0 γ 0 (d − γ 0 x) 2 − d 2 2γ 0 and Theorem 2.1 that β (γX * − D * ) + D * = 0 D M -a.e.,(73)where D * = (D * s ) s∈[0,T ] denotes the associated to X * deviation process (see(9)). This yields for the processA * = (A * s ) s∈[0,T ] defined by A * s = X * s − α s D * s , s ∈ [0, T ], that dA * s = dX * s − D * s dα s − α s dD * s − d[α, D * ] s = −D * s dα s + α s ρ s D * s d[M ] s = β s A * s dα s α s − ρ s d[M ] s , s ∈ [0, T ],f denotes driver(14) of BSDE(13):Y t = Y 0 − ⊥,(1) − M ⊥,(2) t = M ⊥,(1) − M ⊥,(2) , s − Z(1) s d M ⊥,(1) − M ⊥,(2) , M s = 0, t ∈ [0, T ].…”