[Paper Review] Penalising symmetric stable Lévy paths

Kouji Yano, Yuko Yano+1 more|ArXiv.org|Jul 27, 2008

Stochastic processes and financial applications9 references8 citations

TL;DR

This paper develops a penalisation theory for symmetric stable Lévy processes of index $1 < \alpha \leq 2$, introducing a universal $\sigma$-finite measure $\mathscr{P}$ that unifies limit theorems analogous to those in the Brownian motion case ($\alpha = 2$). By constructing a martingale generator $M_{t,x}(F)$ under $\mathscr{P}$, the authors characterize the limiting normalized measures for weight functionals involving local time and exponential occupation time, proving almost-sure convergence and a decomposition of supermartingales that generalizes the Brownian penalisation framework.

ABSTRACT

Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable Lévy process of index $ 1 &lt; α\le 2 $. The first kind is a function of the local time at the origin, and the second kind is the exponential of an occupation time integral. Special emphasis is put on the role played by a stable Lévy counterpart of the universal $ σ$-finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which $ α=2 $.

Motivation & Objective

• To extend the penalisation framework from Brownian motion ($\alpha = 2$) to general symmetric stable Lévy processes with index $1 < \alpha \leq 2$.
• To introduce a universal $\sigma$-finite measure $\mathscr{P}$ as an analogue of the Brownian bridge measure $\mathscr{W}$, unifying limit theorems across different weight functionals.
• To characterize the limiting normalized measures for weight functionals based on local time and exponential occupation time using the measure $\mathscr{P}$.

Proposed method

• Introduce a new $\sigma$-finite measure $\mathscr{P} = \frac{\Gamma(1/\alpha)}{\alpha\pi}\int_0^\infty \frac{du}{u^{1/\alpha}} Q^{(u)} \bullet P^h_0$, where $Q^{(u)}$ is the bridge law and $P^h_0$ is the $h$-path process for $|x|^{\alpha-1}$.
• Define the martingale generator $M_{t,x}(F)$ as the $P_x$-conditional expectation of $F$ under $\mathscr{P}_x$, ensuring $M_{t,x}(F) \to 0$ $P_0$-a.s. and $M_{t,x}(F)/(1+h(X_t)) \to F$ $\mathscr{P}$-a.s.
• Use Itô’s excursion measure $n$ and the local time $L_t$ to disintegrate the stable process and derive the meander distribution $M^{(t)}$ via normalization of excursion measure on $\{R > t\}$.
• Prove that for any non-negative $\mathscr{P}_x$-integrable functional $F$, the process $M_{t,x}(F)$ is a non-negative $P_x$-martingale with $P_x$-a.s. convergence to zero and normalized convergence to $F$ under $\mathscr{P}$.
• Establish a decomposition of non-negative $P_0$-supermartingales into three components: a martingale part $M_{t,x}(F)$, a uniformly integrable martingale part $P_0[N_\infty|\mathcal{F}_t]$, and a residual part $\xi_t$ vanishing a.s. and in normalized form.
• Apply the convergence $M_{t,x}(F)/(1+h(X_t)) \to F$ $\mathscr{P}$-a.s. to prove that the limit measure $P^\Gamma_x$ in penalisation problems is given by $P^\Gamma_x = \Gamma_\infty \cdot \mathscr{P}_x / \mathscr{P}_x[\Gamma_\infty]$, thus characterizing the limit.

Experimental results

Research questions

• RQ1 Can the penalisation problem for symmetric stable Lévy processes be unified via a universal $\sigma$-finite measure analogous to $\mathscr{W}$ in the Brownian case?
• RQ2 Does the normalized measure $\Gamma_t \cdot P_x / P_x[\Gamma_t]$ converge almost surely along $\mathcal{F}_s$ as $t \to \infty$ for weight functionals $\Gamma_t$ based on local time or exponential occupation time?
• RQ3 Can the limit measure $P^\Gamma_x$ be characterized via a martingale generator $M_{t,x}(F)$ under the universal measure $\mathscr{P}_x$?
• RQ4 Is there a decomposition of any non-negative $P_0$-supermartingale into components corresponding to $\mathscr{P}$-integrable functionals, $P_0$-integrable terminal values, and a vanishing residual part?

Key findings

• The universal $\sigma$-finite measure $\mathscr{P}$ is constructed as $\frac{\Gamma(1/\alpha)}{\alpha\pi}\int_0^\infty \frac{du}{u^{1/\alpha}} Q^{(u)} \bullet P^h_0$, providing a stable Lévy counterpart to the Brownian bridge measure $\mathscr{W}$.
• For any non-negative $\mathscr{P}_x$-integrable functional $F$, the process $M_{t,x}(F)$ is a non-negative $P_x$-martingale such that $M_{t,x}(F)/(1+h(X_t)) \to F$ $\mathscr{P}$-almost surely as $t \to \infty$, establishing a key convergence property.
• The limit measure $P^\Gamma_x$ in penalisation problems is characterized as $P^\Gamma_x = \Gamma_\infty \cdot \mathscr{P}_x / \mathscr{P}_x[\Gamma_\infty]$, where $\Gamma_\infty$ is the almost sure limit of $\Gamma_t / \mu(t)$ under $\mathscr{P}_x$, proving the existence and form of the limit.
• Any non-negative $P_0$-supermartingale $N_t$ admits a unique decomposition $N_t = M_t(F) + P_0[N_\infty|\mathcal{F}_t] + \xi_t$, where $M_t(F)$ is a martingale with $M_t(F)/(1+h(X_t)) \to F$ $\mathscr{P}$-a.s., $P_0[N_\infty|\mathcal{F}_t]$ is uniformly integrable, and $\xi_t/(1+h(X_t)) \to 0$ $\mathscr{P}$-a.s.
• The martingale generator $M_{t,x}(F)$ satisfies $P^\Gamma_x|_{\mathcal{F}_t} = \frac{M_{t,x}(\Gamma_\infty)}{\mathscr{P}_x[\Gamma_\infty]} \cdot P_x|_{\mathcal{F}_t}$, providing a pathwise characterization of the limiting measure.
• The convergence $M_{t,x}(F)/(1+h(X_t)) \to F$ $\mathscr{P}$-a.s. is used to prove that $P^\Gamma_x$ is the unique limit of normalized measures $\Gamma_t \cdot P_x / P_x[\Gamma_t]$, thus solving the penalisation problem in full generality.