Asymptotics for Small Nonlinear Price Impact: a PDE Approach to the Multidimensional Case

We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small non-linear price impact. In our model cross-impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second order PDE related to an ergodic control problem and a linear parabolic PDE. We illustrate our result on a multivariate geometric Brownian motion price model.Comment: to appear in Mathematical Financ

Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I

The main objective of this paper and the accompanying one \cite{ETZ2} is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work \cite{EKTZ}, focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in \cite{ETZ0}. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the wellposedness results established in \cite{ETZ2}. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path dependent dynamic programming equation.Comment: 42 page

Optimal Stopping under Nonlinear Expectation

Let $X$ be a bounded c\`adl\`ag process with positive jumps defined on the canonical space of continuous paths. We consider the problem of optimal stopping the process $X$ under a nonlinear expectation operator \cE defined as the supremum of expectations over a weakly compact family of nondominated measures. We introduce the corresponding nonlinear Snell envelope. Our main objective is to extend the Snell envelope characterization to the present context. Namely, we prove that the nonlinear Snell envelope is an \cE-supermartingale, and an \cE-martingale up to its first hitting time of the obstacle $X$. This result is obtained under an additional uniform continuity property of $X$. We also extend the result in the context of a random horizon optimal stopping problem. This result is crucial for the newly developed theory of viscosity solutions of path-dependent PDEs as introduced in Ekren et al., in the semilinear case, and extended to the fully nonlinear case in the accompanying papers (Ekren, Touzi, and Zhang, parts I and II).Comment: 36 page

Optimization of a hybrid combination of a photovoltaic syste and a wind energy conversion system: İzmir Institute of Technology Campus area case

Thesis (Master)--Izmir Institute of Technology, Energy Engineering, Izmir, 2003Includes bibliographical references (leaves: 69-75)Text in English; Abstract: Turkish and Englishxii, 90 leavesRenewable energy resources have gained the great importance due to the growing concerns of environmental problems. The integrated utilization of renewable energies such assolar and wind are becoming very attractive, especially, in most of the isolated and remote areas in many parts of the world.In the present study, firstly, wind and solar radiation measurements, made on Izmir Institute of Technology (IZTECH) Campus area, have been analyzed in order to determine the, both, solar and wind energy potentials of the location. The monthly average wind speeds range from 5.7 m/s to 7.7 m/s, the monthly average of daily values of solar radiation range from 2.1 kWh/m2 to 5.7 kWh/m2 at the monitoring station throughout the measurement period of 12 months between 01.01.2002 and 31.12.2002.Secondly, a procedure is described which determines the hybrid system parameters such as photovoltaic (PV) array area and rotor swept area of wind turbine while satisfying a specific load distribution. The employed method is a graphical construction to figure out the optimum configuration of the generators that satisfies the energy demand mentioned above