Trading multiple mean reversion

How should one construct a portfolio from multiple mean-reverting assets? Should one add an asset to portfolio even if the asset has zero mean reversion? We consider a position management problem for an agent trading multiple mean-reverting assets. We solve an optimal control problem for an agent with power utility, and present a semi-explicit solution. The nearly explicit nature of the solution allows us to study the effects of parameter mis-specification, and derive a number of properties of the optimal solution

Revisiting integral functionals of geometric Brownian motion

Research Federation of “Mathématiques des Pays de la Loire

An explicit solution for optimal investment in Heston model

In this paper we consider a variation of the Merton's problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumptions on the underlying process and the utility function. The resulting parabolic PDE is often quite difficult to solve, even when it is linear. The present paper contributes to the pool of explicit solutions for stochastic optimal control problems. Our main result is the exact solution for optimal investment in Heston model

Solving optimal stopping problems for Lévy processes in infinite horizon via AA-transform

We present a method to solve optimal stopping problems in infinite horizon for a L\'evy process when the reward function can be non-monotone. To solve the problem we introduce two new objects. Firstly, we define a random variable $\eta(x)$ which corresponds to the argmax of the reward function. Secondly, we propose a certain integral transform which can be built on any suitable random variable. It turns out that this integral transform constructed from $\eta(x)$ and applied to the reward function produces an easy and straightforward description of the optimal stopping rule. We check the consistency of our method with the existing literature, and further illustrate our results with a new example. The method we propose allows to avoid complicated differential or integro-differential equations which arise if the standard methodology is used

Utility maximization in Wiener-transformable markets

We consider a utility maximization problem in a broad class of markets. Apart from traditional semimartingale markets, our class of markets includes processes with long memory, fractional Brownian motion and related processes, and, in general, Gaussian processes satisfying certain regularity conditions on their covariance functions. Our choice of markets is motivated by the well-known phenomena of the so-called "constant" and "variable depth" memory observed in real world price processes, for which fractional and multifractional models are the most adequate descriptions. We introduce the notion of a Wiener-transformable Gaussian process, and give examples of such processes, and their representations. The representation for the solution of the utility maximization problem in our specific setting is presented for various utility functions