We consider a dynamic system whose state is governed by a linear stochastic differential equation with time-dependent coefficients. The control acts additively on the state of the system. Our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process. It is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space-time domain. By using some decomposition arguments, the problems of a two-sided control, i.e. optimal corrections, and the case with constraints on the resources, i.e. finite fuel, can be reduced to a simpler case of only one-sided control, i.e. a monotone follower. These results are applied to solving some examples by the so-called method of similarity solutions.

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Additive control of stochastic linear systems with finite horizon

Relying on the stochastic analysis tools developed in Part I, we will solve the optimal stopping problems for non-linear expectations.

Optimal Stopping for Non-Linear Expectations - Part II

In this chapter, we consider the American call option in a continuous time model of stock prices. The development is similar to that in discrete time and follows our general approach of deriving upper and lower bounds based on the NA principle. We will show that in a complete market, the two bounds coincide.

The American Options

The Skew Brownian motion is of primary importance in modeling diffusion in media with interfaces which arise in many domains ranging from population ecology to geophysics and finance. We show that the maximum likelihood estimator provides a consistent estimator of the parameter of a Skew Brownian motion observed at discrete times. The difficulties are that this process is only null recurrent and has a singular distribution with respect to the one of the Brownian motion. Finally, using the idea of the Expectation-Maximization algorithm, we show that the maximum likelihood estimator can be naturally interpreted as the expected number of positive excursions divided by the expected number of excursions.

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Two consistent estimators for the Skew Brownian motion

On a class of singular stochastic control problems for reflected diffusions

We investigate a class of optimal stopping problems arising in, for example, studies considering the timing of an irreversible investment when the underlying follows a skew Brownian motion. Our results indicate that the local directional predictability modeled by the presence of a skew point for the underlying has a nontrivial and somewhat surprising impact on the timing incentives of the decision maker. We prove that waiting is always optimal at the skew point for a large class of exercise payoffs. An interesting consequence of this finding, which is in sharp contrast with studies relying on ordinary Brownian motion, is that the exercise region for the problem can become unconnected even when the payoff is linear. We also establish that higher skewness increases the incentives to wait and postpones the optimal timing of an investment opportunity. Our general results are explicitly illustrated for a piecewise linear payoff.

Timing in the Presence of Directional Predictability: Optimal Stopping of Skew Brownian Motion

We consider the problem of representing the value of singular stochastic control problems of linear diffusions as expected suprema. Setting the value accrued from following a standard reflection policy equal with the expected value of a unknown function at the running supremum of the underlying is shown to result into a functional equation from which the unknown function can be explicitly derived. We also consider the stopping problem associated with the considered singular stochastic control problem and present a similar representation as an expected supremum in terms of characteristics of the control problem.

Expected Supremum Representation of the Value of a Singular Stochastic Control Problem

We investigate the impact of Knightian uncertainty on the optimal timing policy of an ambiguity averse decision maker in the case where the underlying factor dynamics follow a multidimensional Brownian motion and the exercise payoff depends on either a linear combination of the factors or the radial part of the driving factor dynamics We present a general characterization of the value of the optimal timing policy and the worst case measure in terms of a family of an explicitly identified excessive functions generating an appropriate class of supermartingales In line with previous findings based on linear diffusions, we find that ambiguity accelerates timing in comparison with the unambiguous setting Somewhat surprisingly, we find that ambiguity may result into stationarity in models which typically do not possess stationary behavior In this way, our results indicate that ambiguity may act as a stabilizing mechanism

A Class of Solvable Multidimensional Stopping Problems in the Presence of Knightian Uncertainty

We consider the representation of the value of a class of optimal stopping problems of linear diffusions in a linearized form as an expected supremum of a known function. We establish an explicit integral representation of this representing function by utilizing the explicitly known marginals of the joint probability distribution of the extremal processes. We also delineate circumstances under which the value of a stopping problem induces directly this representation and show how it is connected with the monotonicity of the generator. We compare our findings with existing literature and show, for example, how our representation is linked to the smooth fit principle and how it coincides with the optimal stopping signal representation. The intricacies of the developed integral representation are explicitly illustrated in various examples arising in financial applications of optimal stopping.

Luis H. R. Alvarez E.

Papers

Timing in the Presence of Directional Predictability: Optimal Stopping of Skew Brownian Motion

Expected Supremum Representation of the Value of a Singular Stochastic Control Problem

A Class of Solvable Multidimensional Stopping Problems in the Presence of Knightian Uncertainty

Expected Supremum Representation of a Class of Single Boundary Stopping Problems