Peter Bank

Optimal Control under a Dynamic Fuel Constraint

We present a new approach to solve optimal control problems of the monotone follower type. The key feature of our approach is that it allows us to include an arbitrary dynamic fuel constraint. Instead of dynamic programming, we use the convexity of our cost functional to derive a first order characterization of optimal policies based on the Snell envelope of the objective functionals gradient at the optimum. The optimal control policy is constructed explicitly in terms of the solution to a representation theorem for stochastic processes obtained in Bank and El Karoui (2004), {Ann. Probab.}, 32, pp. 1030--1067. As an illustration, we show how our methodology allows us to extend the scope of the explicit solutions obtained for the classical monotone follower problem and for an irreversible investment problem arising in economics.

American Options, Multi–armed Bandits, and Optimal Consumption Plans: A Unifying View

In this survey, we show that various stochastic optimization problems arising in option theory, in dynamical allocation problems, and in the microeconomic theory of intertemporal consumption choice can all be reduced to the same problem of representing a given stochastic process in terms of running maxima of another process. We describe recent results of Bank and El Karoui (2002) on the general stochastic representation problem, derive results in closed form for Levy processes and diffusions, present an algorithm for explicit computations, and discuss some applications.

A model for a large investor trading at market indifference prices. I: single-period case

We develop a single-period model for a large economic agent who trades with market makers at their utility indifference prices. We compute the sensitivities of these market indifference prices with respect to the size of the investor’s order. It turns out that the price impact of an order is determined both by the market makers’ joint risk tolerance and by the variation of individual risk tolerances. On a technical level, a key role in our analysis is played by a pair of conjugate saddle functions associated with the description of Pareto optimal allocations in terms of the aggregate utility function.

Optimal consumption choice under uncertainty with intertemporal substitution

We extend the analysis of the intertemporal utility maximization problem for Hindy-Huang-Kreps utilities reported in Bank and Riedel (1998) to the stochastic case. Existence and uniqueness of optimal consumption plans are established under arbitrary convex portfolio constraints, including both complete and incomplete markets. For the complete market setting, Kuhn-Tuckerlike necessary and sufficient conditions for optimality are given. Using this characterization, we show that optimal consumption plans are obtained by re- flecting the associated level of satisfaction on a stochastic lower bound. When uncertainty is generated by a Levy process and agents exhibit constant relative risk aversion, closed-form solutions are derived. Depending on the structure of the underlying stochastics, optimal consumption occurs at rates, in gulps, or singular to Lebesgue measure.

Non-time additive utility optimization: The case of certainty

We study the intertemporal utility maximization problem for Hindy-Huang-Kreps utilities. Necessary and sufficient conditions for optimality are given. An explicit solution is provided for a large class of utility functions. In particular, the case of separable power utilities with a finite time horizon is solved explicitly.

Optimal Order Scheduling for Deterministic Liquidity Patterns

We consider a broker who has to place a large order which consumes a sizable part of average daily trading volume. The brokers aim is thus to minimize execution costs he incurs from the adverse impact of his trades on market prices. In contrast to the previous literature (see, e.g., Obizhaeva and Wang [A. Obizhaeva and J. Wang, J. Financial Markets, 16 (2013), pp. 1--32] and Predoiu, Shaikhet, and Shreve [SIAM J. Financial Math., 2 (2011), pp. 183--212]), we allow the liquidity parameters of market depth and resilience to vary deterministically over the course of the trading period. The resulting singular optimal control problem is shown to be tractable by methods from convex analysis, and, under minimal assumptions, we construct an explicit solution to the scheduling problem in terms of some concave envelope of the resilience-adjusted market depth.

Super-replication with nonlinear transaction costs and volatility uncertainty

We study super-replication of contingent claims in an illiquid market with model uncertainty. Illiquidity is captured by nonlinear transaction costs in discrete time and model uncertainty arises as our only assumption on stock price returns is that they are in a range specified by fixed volatility bounds. We provide a dual characterization of super-replication prices as a supremum of penalized expectations for the contingent claims payoff. We also describe the scaling limit of this dual representation when the number of trading periods increases to infinity. Hence, this paper complements the results in [11] and [19] for the case of model uncertainty.

Linear Quadratic Stochastic Control Problems with Stochastic Terminal Constraint

We study a linear quadratic optimal control problem with stochastic coefficients and a terminal state constraint, which may be in force merely on a set with positive, but not necessarily full, probability. Under such a partial terminal constraint, the usual approach via a coupled system of a backward stochastic Riccati equation and a linear backward equation breaks down. As a remedy, we introduce a family of auxiliary problems parametrized by the supersolutions to this Riccati equation alone. The target functional of these problems dominates the original constrained one and allows for an explicit description of both the optimal control policy and the auxiliary problems value in terms of a suitably constructed optimal signal process. This suggests that, for the minimal supersolution of the Riccati equation, the minimizers of the auxiliary problem coincide with those of the original problem, a conjecture that we see confirmed in all examples understood so far.

On a stochastic differential equation arising in a price impact model

We provide sufficient conditions for the existence and uniqueness of solutions to a stochastic differential equation which arises in the price impact model developed by Bank and Kramkov (2011) [1,2]. These conditions are stated as smoothness and boundedness requirements on utility functions or Malliavin differentiability of payoffs and endowments.

Convex duality for stochastic singular control problems.

We develop a general theory of convex duality for certain singular control problems, taking the abstract results by Kramkov and Schachermayer (1999) for optimal expected utility from nonnegative random variables to the level of optimal expected utility from increasing, adapted controls. The main contributions are the formulation of a suitable duality framework, the identification of the problems dual functional as well as the full duality for the primal and dual value functions and their optimizers. The scope of our results is illustrated by an irreversible investment problem and the Hindy-Huang-Kreps utility maximization problem for incomplete financial markets.