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Small noise asymptotics for a class of jump-diffusions with heavy tails for large times

Sumith Reddy Anugu, Siva R. Athreya, Vivek S. Borkar

Abstract

In this work, we investigate positive recurrent Lévy diffusions driven by appropriately scaled Brownian motion and $α$-stable process (with $1<α<2$) in the small noise regime. Supposing that in the vanishing noise limit, our Lévy diffusion approaches a deterministic system with a unique asymptotically stable fixed point, we show that the limiting behavior of the one-dimensional marginal distribution at large times is dictated by the optimal value of a deterministic control problem, just as in the classical case of diffusions driven by small variance Brownian motion. In our case, the control is allowed to have two parts: continuous control and impulse control.

Small noise asymptotics for a class of jump-diffusions with heavy tails for large times

Abstract

In this work, we investigate positive recurrent Lévy diffusions driven by appropriately scaled Brownian motion and -stable process (with ) in the small noise regime. Supposing that in the vanishing noise limit, our Lévy diffusion approaches a deterministic system with a unique asymptotically stable fixed point, we show that the limiting behavior of the one-dimensional marginal distribution at large times is dictated by the optimal value of a deterministic control problem, just as in the classical case of diffusions driven by small variance Brownian motion. In our case, the control is allowed to have two parts: continuous control and impulse control.
Paper Structure (9 sections, 13 theorems, 93 equations)

This paper contains 9 sections, 13 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Background and Literature
  3. Main result
  4. Discussion and Outline
  5. Sketch of the proof
  6. Layout of the paper
  7. A "contraction" type principle
  8. Proof of Theorem \ref{['thm-main']}
  9. An optimal control result.

Key Result

Proposition 1.1

Let $(u,v)$ and ${\mathcal{S}}_x(z,T)$ as in Definition def:imp and $I_L$ be as defined in def-IL. Suppose Assumption main-assump holds. Then for every $T>0$, $\{X^{n,\gamma}(T):n\in {\mathbb N}\}$ satisfies WLDP with rate $\log(n)$ and rate function $V_{\gamma,T}:{\mathbb R}^p\rightarrow {\mathbb R

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • ...and 18 more