Introduction to Stochastic Dominance

Stochastic dominance plays a crucial role in decision-making under uncertainty and quantitative finance. It provides a powerful method for comparing random variables using their distribution functions. This concept establishes a structured framework to evaluate the superiority of one investment, policy, or strategy over another in uncertain environments.

By utilizing cumulative distribution functions, stochastic dominance allows decision-makers to assess choices without assuming specific utility functions. This generalization leads to a mathematically rigorous approach to optimization and risk management. The precision and depth of stochastic dominance make it an indispensable tool for analyzing complex probabilistic systems. We refer to stochastic dominance wiki for a quick overview.

Stochastic Dominance of Order $p$

Let $p \in [1,\infty)$ and let $X, Y \in L^p$ be random variables. We say that $X$ is dominated by $Y$ in the stochastic order of $p$, denoted as

\[X \preccurlyeq^{(p)} Y\]

if the condition holds

\[\mathbb{E} (t - X)_+^{p-1} \geq \mathbb{E} (t - Y)_+^{p-1}, \quad \text{for all } t \in \mathbb{R}.\]

This definition provides a precise way to compare random variables under higher order stochastic dominance criteria. From a portfolio optimization perspective, the feasible random variable $Y$ in the above formulation dominates benchmark variables $X$.

For other equivalent formulations and technical details, we refer to Dentcheva and Ruszczyński [1].

Optimal Portfolio Scaling under Stochastic Dominance of Order $p$

Given a fixed benchmark random vector $\xi_0 \in \mathbb{R}^{n}$ and a portfolio return random vector(s) ($d$ vectors) $\xi \in \mathbb{R}^{d \times n}$, we aim to determine an optimal scaling factor $x > 0$ ($d$ dimensional simplex vector) such that the scaled portfolio $x^{\top}\xi$ dominates $\xi_0$ under stochastic dominance of order of $p$. Here, we have $d$ assets and $n$ scenarios.

Goal:

Find $x > 0$ such that:

\[\xi_0 \preccurlyeq^{(p)} x^{\top}\xi \quad \text{or} \quad \mathbb{E}(t - \xi_0)_+^{p-1} \geq \mathbb{E}(t - x^{\top}\xi )_+^{p-1}, \quad \forall t \in \mathbb{R}.\]

Optimization Variants:

1. Maximize Expected Return under Dominance Constraint:

   \[\max_{x > 0} \mathbb{E} x^{\top}\xi \quad \text{subject to} \quad \xi_0 \preccurlyeq^{(p)} x^{\top}\xi.\]

2. Minimize Risk (e.g., Variance) under Dominance Constraint:

   \[\min_{x > 0} \text{Var}(x^{\top}\xi) \quad \text{subject to} \quad \xi_0 \preccurlyeq^{(p)} x^{\top}\xi.\]

Interpretation:

• The condition $\xi_0 \preccurlyeq^{(p)} x^{\top}\xi$ ensures that for all thresholds $t$, the $p^{\text{th}}$ partial moment of $\xi_0$ is at least that of the scaled portfolio $x^{\top}\xi$.
• This allows the investor to scale the portfolio to either increase returns or reduce risk while maintaining or improving the performance relative to the benchmark under higher-order stochastic dominance.

Practical Use:

This approach is especially useful in portfolio optimization where a benchmark (e.g., market index) is fixed, and the investor seeks to optimally scale their portfolio to satisfy risk-return tradeoffs under rigorous dominance conditions.

This package provides tools to analyze stochastic dominance, enabling users to apply these concepts effectively in decision-making and financial optimization. The mathematical background of this package was discussed in Lakshmanan et al. [2].

References