Scaling limits of processes with fast nonlinear mean reversion

Abstract

We derive scaling limits for integral functionals of Itô processes with fast nonlinear meanreversion speeds. In these limits, the fast mean-reverting process is "averaged out" by integrating against its invariant measure. The convergence is uniformly in probability and, under mild integrability conditions, also in S p . These results are a crucial building block for the analysis of portfolio choice models with small superlinear transaction costs, carried out in the companion paper of the present study [11].Mathematics Subject Classification (2010): 60F25, 60H10.Keywords: processes with fast nonlinear mean reversion; scaling limits * We are grateful to two anonymous referees for extremely detailed and helpful comments that have greatly improved the final version of this manuscript.

Trading with small nonlinear price impact

“…For the present study, a number of delicate probabilistic estimates need to be developed from scratch to establish convergence to the limiting problem. This is done in the companion paper of the present study [15].…”

“…We also require that cφ > 0. 9 Under Assumption 2, it follows from [15], Proposition 1.1, that the SDE has a unique strong solution on [0, T ] for all λ > 0, where Wφ ,Q is a Brownian motion under Q. 10 We can now define the trading strategy (…”

“…and define the stopping time (3.10) 10 In the notation of [15], take ε = λ 11 This happens if the trading cost becomes too large, or the target too volatile relative to the risky asset. 12 For proportional or fixed costs, this can be done by a single bulk trade at T , without affecting the asymptotic results at the leading order.…”

“…Their fine structure in turn depends on the degree of activity exhibited byφ. 15 The trading speed (3.6) is determined by the diffusion coefficients c S t and cφ t of the risky asset and the frictionless optimizer, the current trading cost λ t , the risk aversion γ , and the elasticity of the price impact p. In addition, the shape function g p determines how the deviation ϕ λ t =φ t − ϕ λ t of the frictionless optimizer from the actual position is incorporated. For p ↑ 2, the shape function g p converges to g 2 (x) = 2x.…”

“…14 Note that since the stopping time τ ϕ from (3.10) is very close to the terminal time T , ϕ λ can be identified with ϕ λ from (3.5) in the following discussion without loss of generality. 15 For example, if the target strategy is smoother than Brownian motion, then it can be tracked much more closely and with substantially smaller trading costs; compare [56].…”