Risk-sensitive control for a class of diffusions with jumps

Abstract: We consider a class of diffusions controlled through the drift and jump size, and driven by a jump Lévy process and a nondegenerate Wiener process, and we study infinite horizon (ergodic) risk-sensitive control problem for this model. We start with the controlled Dirichlet eigenvalue problem in smooth bounded domains, which also allows us to generalize current results in the literature on exit rate control problems. Then we consider the infinite horizon average risk-sensitive minimization problem and maximization problems on the whole domain. Under suitable hypotheses, we establish existence and uniqueness of a principal eigenfunction for the Hamilton-Jacobi-Bellman (HJB) operator on the whole space, and fully characterize stationary Markov optimal controls as the measurable selectors of this HJB equation.

• The paper introduces a risk-sensitive control framework for diffusions with jumps, establishing existence and uniqueness of the principal eigenvalue and eigenfunction.
• It formulates and analyzes the Hamilton-Jacobi-Bellman equation in both bounded domains and the entire space, leading to measurable selectors for optimal control.
• Numerical results confirm that the optimal risk-sensitive value coincides with the principal eigenvalue, with significant implications for finance and reliability applications.

Risk-Sensitive Control for Diffusions with Jumps

Introduction and Problem Statement

The paper "Risk-sensitive control for a class of diffusions with jumps" (1910.05004) investigates a risk-sensitive control framework for stochastic differential equations (SDEs) characterized by diffusions with jumps. These equations are influenced by a combination of a jump Lévy process and a Wiener process, with control applied through drift and jump size parameters. The key focus of the study is on infinite horizon (ergodic) risk-sensitive control problems which differ from traditional risk-neutral control frameworks by incorporating a nonlinear exponential cost criterion that accounts for the variability of outcomes.

The primary goal is to address the ergodic limit problem of the risk-sensitive criterion. The authors formulate a Hamilton-Jacobi-Bellman (HJB) equation for the controlled system and examine its eigenvalue problem in both bounded domains and the whole space. The study's motivation stems from applications in fields such as finance and reliability theory, where controlling the risk associated with stochastic processes is important.

Methodology

The authors approach the HJB equation's eigenvalue problem by first considering the Dirichlet eigenvalue problem within smooth bounded domains. They then extend these results to infinite horizon risk-sensitive control problems over the entire domain. The fundamental equation governing the dynamic system is formulated using a controlled stochastic differential equation:

$$dX_t = b_0(X_t, Z_t) \, dt + \sigma(X_t) \, dW_t + g(X_t, Z_t, \xi) \, \tilde{N}(dt, d\xi),$$

where $b_0$ represents the drift, $\sigma$ is the diffusion matrix, $g$ is the jump-size, $\tilde{N}$ is a compensated Poisson process, and $Z_t$ is the control vector.

The risk-sensitive control objective is modeled by minimizing a nonlinear functional:

$$\limsup_{T \to \infty} \frac{1}{T} \log \mathbb{E}_x \left[ e^{\int_0^T c(X_s, Z_s) \, ds} \right],$$

where $c(X_s, Z_s)$ is the running cost. The authors explore the structural properties of the principal eigenvalue $λ*$ of the associated HJB operator and provide criteria for existence, uniqueness, and characterization of optimal controls.

Numerical Results and Key Findings

The authors successfully demonstrate the existence and characterization of the principal eigenfunction for the HJB operator. They establish that stationary Markov optimal controls can be represented as measurable selectors of the HJB equation. Numerical tests confirm that the principal eigenvalue coincides with the optimal risk-sensitive value under certain conditions.

Furthermore, the authors provide insight into the uniqueness of the principal eigenvalue and its relation to the monotonicity of eigenfunctions, alongside implications for control problem verification results.

Implications

The findings of this study have important implications for systems governed by stochastic differential equations with jumps, particularly in scenarios where risk sensitivity is critical, such as financial portfolio optimization and queueing networks. The research contributes to the broader understanding of risk-sensitive control in infinite horizon settings, enabling practitioners to design systems that account for risk through structured mathematical frameworks.

Future Directions

The paper opens avenues for exploring risk-sensitive control for jump diffusions where the jump component is significant, yet largely unexplored within this context. Future research could focus on extending these results to more complex systems or alternative stochastic frameworks, considering the continuous evolution of financial and engineering applications. Moreover, investigating the conditions under which the breakdown phenomenon occurs and developing methods for mitigating such occurrences could further enhance robustness in practical implementations.

Conclusion

This paper provides a rigorous examination of risk-sensitive control for a class of diffusions with jumps, establishing theoretical foundations and solving numerous critical questions regarding eigenvalues, eigenfunctions, and control strategy characterization. It sets the stage for further exploration into dynamically optimized risk-sensitive control solutions across various applications.