[Paper Review] Wentzel-Freidlin estimates for jump processes in semi-group theory: upper bound

Rémi Léandre|arXiv (Cornell University)|May 26, 2010

Advanced Operator Algebra Research1 references7 citations

TL;DR

This paper establishes an upper bound for Wentzel-Freidlin estimates in the context of jump processes using semi-group theory, translating large deviation principles from stochastic analysis into the framework of Feller generators and Hamilton-Jacobi-Bellman equations. The key result shows that the logarithmic decay rate of transition probabilities for a jump process is bounded above by the infimum of a time-integrated action functional, defined via the Legendre transform of the generator's symbol.

ABSTRACT

We give a translation in semi-group theory of Wentzel-Freidlin estimates for Poisson process. We consider the case of the upper bound.

Motivation & Objective

• To extend Wentzel-Freidlin large deviation theory to jump processes using semi-group theoretic methods.
• To derive an upper bound for the logarithmic decay rate of transition probabilities in a jump process as the jump intensity tends to zero.
• To establish a connection between the generator of a jump process and the associated Hamilton-Jacobi-Bellman equation through Legendre duality.
• To provide a semi-group-theoretic translation of exponential martingale techniques used in stochastic analysis for Poisson-driven processes.

Proposed method

• The paper introduces a time-rescaled generator $ L^h $ associated with a Lévy measure $ \mu(x,dz) $, modeling jump dynamics.
• It defines the Hamiltonian $ H(x,\xi) $ as the integral of $ \exp[\langle z,\xi \rangle] - 1 - \langle z,\xi \rangle $ with respect to the Lévy measure.
• The Legendre transform $ L(x,\alpha) $ of $ H(x,\xi) $ is used to define the action functional $ S(\phi) = \int_0^1 L(\phi(t), \dot\phi(t)) dt $, which governs the large deviation rate.
• A modified generator $ \overline{L}^h $ is constructed on $ \mathbb{R}^d \times \mathbb{R} $ to model exponential martingales, ensuring $ \overline{L}^h \exp[\langle C,x \rangle - y] = 0 $.
• The proof uses a semi-group representation and Hölder's inequality to control the growth of transition probabilities, relying on Gronwall's lemma to bound the exponential moment.
• Polygonal paths are used to discretize the path space, and the action is approximated via piecewise linear control, with the Legendre transform used to bound the deviation from the true action.

Experimental results

Research questions

• RQ1 How can Wentzel-Freidlin estimates for jump processes be reformulated in the language of semi-group theory?
• RQ2 What is the upper bound for the logarithmic decay rate of transition probabilities in a jump process as the jump intensity $ h \to 0 $?
• RQ3 How does the Legendre transform of the Hamiltonian relate to the large deviation rate function for jump processes?
• RQ4 Can exponential martingale techniques from stochastic analysis be adapted to the semi-group framework for pure jump processes?
• RQ5 What conditions on the Lévy measure and generator ensure the existence and regularity of the large deviation rate function?

Key findings

• The upper bound for the logarithmic decay of transition probabilities is given by $ \overline{\lim}_{h \to 0} h \log P_1^h[1_O](x) \leq -\inf_{y \in O} l(x,y) $, where $ l(x,y) $ is the infimum of the action functional over paths from $ x $ to $ y $.
• The action functional $ S(\phi) = \int_0^1 L(\phi(t), \dot\phi(t)) dt $ is finite and continuous under the hypotheses, ensuring the existence of a well-defined rate function.
• The Legendre transform $ L(x,\alpha) $ is uniformly convex in $ \alpha $ with a Hessian bounded below by $ mI_d $ for bounded $ \alpha $, ensuring regularity of the rate function.
• The proof establishes that the probability of paths deviating from the optimal path decays exponentially fast, with the decay rate controlled by the action functional.
• The use of the modified generator $ \overline{L}^h $ and exponential martingale structure allows control of the semi-group norm, leading to uniform bounds in $ h $ via Gronwall's inequality.
• The result holds under minimal regularity assumptions: continuity of the Lévy density $ g(x,z) $ for $ z \neq 0 $, and uniform boundedness of the second absolute moment of the Lévy measure.