Joint Pricing and Innovation Control in Regulated Recycling-Rate Diffusion

Abstract

We introduce a regulated stochastic diffusion model for the recycling rate and formulate a joint control problem over production and process innovation via the dynamics of recycling investment and product pricing. The resulting stochastic control problem captures the system manager's trade-off between product-price decisions and investment expenditures under an infinite-horizon discounted cost structure. Owing to the recycling-rate specification, we incorporate two regulated state processes, which induce additional policy-driven cost components in the value function consistent with green-economy regulations. We resolve the jointly regulated stochastic production and process-innovation admission control problem by introducing the associated Hamilton-Jacobi-Bellman (HJB) equation and providing rigorous proofs that establish the correspondence between the HJB solution and the value function of the underlying control problem. The HJB equation is analyzed under mild, practically motivated assumptions on the system parameters. We further present numerical experiments and sensitivity analyses to illustrate the tractability of the HJB characterization and to assess the practical relevance of the imposed parameter conditions.

Figures (9)

• Figure 1: Sample path of stochastic differential equation \ref{['eq: sde']}, where the $x$-axis denotes time $t$ and $y$-axis represents the recycling rate $r$.
• Figure 2: Sample path of regulated stochastic differential equation \ref{['eq: sde with reflections']}, where the blue solid line exhibits the recycling rate $r$ with respect to time $t$, the red dashed line $U$ denotes the local time process of $r$ reaching the upper boundary 1, and the green dotted line $L$ represents the local time process of $r$ reaching the lower boundary zero.
• Figure 3: Sample path of optimal state process \ref{['eq: optimal r*']}, where the blue solid line exhibits the optimal state process $r^*$, the red dashed line $U^*$ denotes the optimal local time process of $r^*$ reaching the upper boundary 1, and the green dotted line $L^*$ represents the optimal local time process of $r^*$ reaching the lower boundary zero.
• Figure 4: Sample path of optimal policies \ref{['eq: optimal u^* and p^*']}, where the blue dashed line exhibits the optimal price $p^*$, and the red solid line $u^*$ denotes the optimal recycling investment.
• Figure 5: Sample path of optimal policies \ref{['eq: optimal u^* and p^*']} when $0 < a_1 \leq 1$, where the blue dashed line exhibits the constant optimal price $p^*$, and the red solid line $u^*$ denotes the optimal recycling investment.

Theorems & Definitions (24)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1: Verification Lemma
• proof
• Proposition 2
• Theorem 3
• proof
• Example 1: $a_1 > 1$