Growth-optimal investments and numeraire portfolios under transaction costs: An analysis based on the von Neumann-Gale model

The numeraire property and long-term growth optimality for drawdown-constrained investments

We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We suggest to use path-wise growth optimality as the decision criterion and encode preferences through restrictions on the class of admissible wealth processes. Specifically, the investor is only interested in strategies which satisfy a given linear drawdown constraint. The paper introduces the numeraire property through the notion of expected relative return and shows that drawdown-constrained strategies with the numeraire property exist and are unique, but may depend on the financia…

计价单位不变的二次对冲与均值-方差投资组合配置

Numeraire-invariant quadratic hedging and mean--variance portfolio allocation

本文在不含无风险资产的半鞅市场中建立计价单位变换下的二次对冲等价性，利用斜投影给出最优策略显式表达式，并简化了纯投资问题有效前沿的计算。

On the existence of shadow prices for optimal investment with random endowment

In this paper, we consider a numéraire-based utility maximization problem under constant proportional transaction costs and random endowment. Assuming that the agent cannot short sell assets and is endowed with a strictly positive contingent claim, a primal optimizer of this utility maximization problem exists. Moreover, we observe that the original market with transaction costs can be replaced by a frictionless shadow market that yields the same optimality. On the other hand, we present an example to show that in some case when these constraints are relaxed, the existence of shadow prices is …

On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets

We study the two-times differentiability of the value functions of the primal and dual optimization problems that appear in the setting of expected utility maximization in incomplete markets. We also study the differentiability of the solutions to these problems with respect to their initial values. We show that the key conditions for the results to hold true are that the relative risk aversion coefficient of the utility function is uniformly bounded away from zero and infinity, and that the prices of traded securities are sigma-bounded under the numéraire given by the optimal wealth process.

A deep learning algorithm for optimal investment strategies

This paper treats the Merton problem how to invest in safe assets and risky assets to maximize an investor's utility, given by investment opportunities modeled by a $d$-dimensional state process. The problem is represented by a partial differential equation with optimizing term: the Hamilton-Jacobi-Bellman equation. The main purpose of this paper is to solve partial differential equations derived from the Hamilton-Jacobi-Bellman equations with a deep learning algorithm: the Deep Galerkin method, first suggested by Sirignano and Spiliopoulos (2018). We then apply the algorithm to get the soluti…