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.
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.
We study a multiplicative transient price impact model for an illiquid
financial market, where trading causes price impact which is multiplicative in
relation to the current price, transient over time with finite rate of
resilience, and non-linear in the order size. We construct explicit solutions
for the optimal control and the value function of singular optimal control
problems to maximize expected discounted proceeds from liquidating a given
asset position. A free boundary problem, describing the optimal control, is
solved for two variants of the problem where admissible controls are monotone
or of bounded variation.Comment: To appear in Applied Mathematics and Optimization. Model assumptions
relaxed; corrections and improvements on referees' suggestion
We investigate Newton's method for complex polynomials of arbitrary degree d, normalized so that all their roots are in the unit disk. For each degree d, we give an explicit set S d of 3.33d log 2 d(1+o(1)) points with the following universal property: for every normalized polynomial of degree d there are d starting points in S d whose Newton iterations find all the roots with a low number of iterations: if the roots are uniformly and independently distributed, we show that with probability at least 1 − 2/d the number of iterations for these d starting points to reach all roots with precision ε is O(d 2 log 4 d + d log | log ε|). This is an improvement of an earlier result in [S2], where the number of iterations is shown to be O(d 4 log 2 d + d 3 log 2 d| log ε|) in the worst case (allowing multiple roots) and O(d 3 log 2 d(log d + log δ) + d log | log ε|) for well-separated (so-called δ-separated) roots.Our result is almost optimal for this kind of starting points in the sense that the number of iterations can never be smaller than O(d 2 ) for fixed ε.
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