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Title: Optimal investment in incomplete markets with multiple Brownian externalities
Authors: Avanesyan, Levon
Advisors: Shkolnikov, Mykhaylo
Sircar, Ronnie
Contributors: Operations Research and Financial Engineering Department
Keywords: eigenvalue equality correlation structure
forward performance process
merton problem
multiple externalities
power mixture
sharpe ratio separable model
Subjects: Applied mathematics
Operations research
Issue Date: 2021
Publisher: Princeton, NJ : Princeton University
Abstract: An investor’s optimal market portfolio is shaped by their investment performance criterion. The latter is largely determined by the investor's idiosyncratic objectives and preferences. This dissertation contributes to the study of optimal investment under the expected utility of terminal wealth (Merton), as well as forward performance criteria. Our set-up is that of a continuous incomplete stock market model, where the incompleteness stems from multiple Brownian externalities. The externalities manifest themselves through the stock return and volatility coefficients, either explicitly by driving observable stochastic factors or implicitly by increasing the market filtration. In a Markovian multifactor market model we introduce the eigenvalue equality (EVE) stock-factor correlation structure, and construct a large class of forward performance processes (FPPs) with power-utility initial data, as well as their corresponding optimal portfolios. This is done by solving the associated non-linear parabolic partial differential equations (PDEs) posed in the ``wrong'' time direction. We establish on domains an explicit form of the generalized Widder's theorem of Nadtochiy and Tehranchi (Math. Financ. 27:438-470, 2015, Theorem 3.12) and rely hereby on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the ``right'' time direction. Next, we consider the Merton problem with power utility in a market model with the stock coefficients adapted to a factor-generated filtration. This setup admits market models with non-semimartingale factors. For models with EVE structure we find the optimal portfolio weights up to the computation of a certain conditional expectation, and explain how to evaluate the latter in affine Volterra factor models. We extend these results to a general stock-factor correlation setting for, what we will call, Sharpe ratio separable (SRS) market models. In our most general market model, we construct a broad class of FPPs with initial conditions of power mixture type, $u(x) = \int_{\mathbb{I}} \frac{x^{1-\gamma}}{1-\gamma }\nu(\dd \gamma)$. We derive the properties of two-power mixture FPPs when the risk aversion coefficients are continuous stochastic processes in (0,1), and provide a full characterization when the coefficients are constants. Finally, we discuss the problem of managing an investment pool of two investors, whose respective preferences evolve as power FPPs.
Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog:
Type of Material: Academic dissertations (Ph.D.)
Language: en
Appears in Collections:Operations Research and Financial Engineering

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