Increasing Differences & Supermodularity

• Increasing differences and supermodularity are structural properties that quantify how marginal effects increase with complementary variables, ensuring monotonicity and convexity in diverse models.
• They provide a unified methodological framework for analyzing optimization, stochastic control, and comparative statics through tractable conditions like nonnegative mixed partial derivatives.
• These properties find practical applications in queueing systems, storage models, and economic policy analysis, offering insights into stability, performance bounds, and dynamic decision-making.

Increasing differences and supermodularity constitute central structural properties in mathematical analysis, optimization, probability, economics, and algorithmic design. These properties formalize how the effect of one variable or choice increases with another—typically manifesting as monotonicity, convexity, or complementarity phenomena across continuous, discrete, and stochastic domains.

1. Mathematical Formulation of Increasing Differences and Supermodularity

Supermodularity refers to a function $h : \mathbb{R}^2 \to \mathbb{R}$ satisfying for all $x_1 \leq x_2,\, y_1 \leq y_2$: $$h(x_1, y_1) + h(x_2, y_2) \leq h(x_1, y_2) + h(x_2, y_1).$$ This matrix inequality generalizes to multivariate settings and set functions. When $h$ is twice continuously differentiable, increasing differences are often characterized by nonnegative mixed partial derivatives $\frac{\partial^2 h}{\partial x \partial y} \geq 0$.

For set functions $f:2^V \to \mathbb{R}$, supermodularity aligns with the property that for $A \subseteq B \subseteq V$, $v \notin B$: $$f(A \cup \{v\}) - f(A) \leq f(B \cup \{v\}) - f(B),$$ or, equivalently, that the marginal value of $v$ is greater as the set grows, representing increasing returns.

2. Supermodularity in Stochastically Monotone Markov Processes

The framework presented in (Kella et al., 2020) applies supermodularity to Markov processes beyond classical reflected Lévy models. If $(X_s, X_t)$ are two coordinates of a stochastically monotone process and $h$ is supermodular, then a monotonic ordering of process states ensures: $$\mathbb{E}[h(X_0, X_2)] \leq \mathbb{E}[h(X_1, X_2)].$$ Covariance inequalities follow for nondecreasing functions $f_1, f_2$: $$0 \leq \operatorname{Cov}(f_1(X_0), f_2(X_2)) \leq \operatorname{Cov}(f_1(X_1), f_2(X_2)).$$ This generalizes the well-known properties—nonnegative, nonincreasing, convex autocorrelation—for reflected Lévy processes to broad Markov classes where such structural implications can be deduced directly from the process transition kernel and supermodularity.

Generalization exploits the "generalized inverse" $G(x,u)$ associated to the transition probability $p(x, \cdot)$, with monotonicity of $G(x,u)$ and certain decrease conditions on $G(x,u) - x$ (Condition 1).

3. Extension to Transient Regimes

The results are not confined to stationary Markov processes; transient behavior is rigorously analyzed. When the initial distribution satisfies $X_0 \leq X_t$ almost surely for all $t \geq 0$, monotonicity and regularity are preserved:

• The process is stochastically increasing in time.
• $\mathbb{E}[h(X_s, X_t)]$ is nondecreasing in $s$ if $h$ is supermodular and nondecreasing in the first argument.
• If $h$ is nondecreasing in both coordinates, $\mathbb{E}[h(X_s, X_{s+t})]$ is nondecreasing in $s$.
• Under appropriate conditions and finiteness of means, $\mathbb{E}[X_t]$ becomes nondecreasing and, with Condition 1, concave in $t$.

Concrete examples include Lévy storage models, two-sided reflection, dam processes, and state-dependent random walks, confirming the universality of these transient monotonicity principles.

4. Practical Applications and Interpretive Significance

The utility of the supermodularity property in stochastic process modeling is vast:

• Queueing Systems: Covariance and autocorrelation inherit nonnegativity, nonincreasing, and convexity, aiding in performance analysis and reliability assessments.
• Storage/Dam Processes: Regularity in the expected content or workload underpins stability and optimal resource allocation strategies.
• Comparative Statics in Economics: Increasing differences and supermodularity deliver sharper results for monotonicity and convexity/concavity of value functions and policy mappings.
• Dynamic Programming: The monotonicity imparted by supermodularity informs the structure of optimal policies in processes with monotone transitions.

The framework underscores that many ad-hoc results for specific models arise from fundamental structural properties imposed via supermodularity and increasing differences; thus, these results generalize naturally once such properties are established.

5. Methodological Unification via Generalized Inverses

By representing transition kernels $p(x,\cdot)$ via generalized inverses $G(x,u)$, the analysis distills monotonicity and supermodularity into tractable conditions: $$G(x,u)\ \text{nondecreasing in}\ x;\quad G(x,u) - x\ \text{nonincreasing in}\ x$$ These coordinatewise orderings in generalized inverses operationalize the increasing differences property for transitions, thereby capturing a wide variety of Markov process models—queues, dams, birth–death processes—under a unified mathematical umbrella.

6. Broader Implications and Extensions

• Optimization: Supermodularity shapes solution landscapes in integer and mixed-integer programming. Problems exhibiting increasing differences enable decomposition, convexification, and efficient greedy approximation algorithms.
• Stochastic Control: Monotonicity results for expectations of supermodular functions yield rigorous performance and stability bounds.
• Comparative Analysis: The framework connects the stationarity/transient dichotomy, combinatorial and probabilistic domains, and provides a foundation for further exploration in stochastic dominance theory and the diffusion of monotonicity properties in higher-order models.

Summary Table: Structural Properties Derived from Supermodularity

Property	Stationary Case	Transient Case
Covariance regularity	Nonnegative, convex	Preserved
Monotonicity of means	Nonincreasing (Lévy)	Nondecreasing, concave
Applicability	Reflected Lévy, queues	Dams, birth–death, etc

The methodology and results afford a comprehensive, principled understanding of how supermodularity and increasing differences serve as the backbone of monotonicity, convexity, and comparative statics in modern stochastic process theory, optimization, and beyond. The generalization from reflected Lévy models to broad Markov process classes marks a significant step in revealing the structural underpinnings pervasive in applied probability and mathematical economics (Kella et al., 2020).