StochasticDominance.jl

We present the StochasticDominance.jl package, which provides tools for analyzing higher order stochastic dominance—a method used to establish a partial order between random variables.

Brief overview of stochastic dominance

Stochastic dominance is a concept used to compare decision alternatives based on their cumulative risk profiles, ensuring that one alternative is preferred over another without any trade-offs across values. In the context of portfolio optimization, given a benchmark asset and a portfolio of assets, we seek an optimal allocation that maximizes a chosen objective (e.g., maximizing returns) while satisfying (higher-order) stochastic dominance constraints.

Main features of the package

The StochasticDominance.jl package provides tools to:

Verify higher-order stochastic dominance – Check whether a given portfolio satisfies higher-order dominance criteria relative to a benchmark asset.
Determine the optimal allocation – Find the asset allocation that maximizes a chosen objective (e.g., maximizing returns) while adhering to stochastic dominance constraints. It supports two key objective functions: maximizing expected returns and minimizing higher-order risk measures.

Installation

The package StochasticDominance.jl can be installed in Julia REPL as follows:

using Pkg
Pkg.add("StochasticDominance")
using StochasticDominance

Once you have installed StochasticDominance.jl, we recommend going through the tutorials from beginning to end to understand how to use the package to verify stochastic dominance and determine the optimal allocation between a benchmark asset and a portfolio.

Important functions in the package

The StochasticDominance.jl package provides several important functions, which are explained in detail in the tutorials section. Here, we provide a brief overview of the functions.

verify_dominance: This function checks whether the given benchmark asset, represented as the random variable $X$, and the weighted portfolio asset, represented as the random variable $Y$, exhibit a dominance relationship for the specified stochastic order. This means that $Y$ consistently yields preferable outcomes over $X$ in the specified stochastic order.
optimize_max_return_SD: This function determines the optimal asset allocation that maximizes expected returns for a given stochastic order (SDorder). Additionally, using optimize_max_return_SD(; plots=true), users can generate a pie chart displaying the optimal allocation in percentages, along with the maximized expected returns and benchmark returns. The function also includes the option optimize_max_return_SD(; verbose=true), which allows users to imprint the convergence (or dominance) of the numerical method.
optimize_min_riskreturn_SD: This function determines the optimal asset allocation by minimizing higher-order risk measures for a given stochastic order (SDorder) while also indicating whether dominance is achieved. Additionally, using optimize_min_riskreturn_SD(; plots=true), users can generate a pie chart that visualizes the optimal allocation in percentages, along with the minimizing higher-order risk measure returns. The function also provides the option optimize_min_riskreturn_SD(; verbose=true), allowing users to assess the convergence (or dominance) of the numerical algorithm.

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