Estimating the Drift, Diffusion, and Lévy Measure of a Lévy Process
Summary
A Lévy process is characterized by a drift vector, a diffusion matrix, and a Lévy measure, which together describe its continuous variation and jumps. The document distinguishes finite-dimensional estimation of drift and diffusion parameters from the potentially nonparametric estimation of the Lévy measure. It asks for practical computational references and implementations, especially in R or Python, noting that a technical textbook and a chapter in a specialized volume are the resources the author has found so far.
The material frames an estimation problem but provides no estimator, implementation, or empirical comparison. It is useful as an introduction to the parameter structure and the challenge of estimating jump behavior, while offering no guidance on sampling assumptions, observation frequency, or methods for selecting among estimation approaches.
Key ideas
- A Lévy process is specified by drift, diffusion, and a Lévy measure.
- Drift and diffusion are finite-dimensional parameters that can be approached parametrically.
- The Lévy measure may require nonparametric estimation.
- The document identifies a need for practical computational methods but does not provide an estimator or implementation.
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# Resource recommendations: Levy process estimation using programming languages
# Resource recommendations: Levy process estimation using programming languages
Perhaps this type of question is not very suitable for this forum, but I'll try to make my question a little useful.
I'm studying stochastic processes, more precisely, Levy processes. A Levy process $X=(X_t)_{t\geq 0}$ is characterized by its characteristic function given by: $$\log \varphi_{X_t}(u)= t\left(ib'u+ \frac{u'a u}{2} + \int_{\mathbb R^d - \{0\}} \left[ e^{i u'x} - 1 - i u'x \mathbf{1}_{(|x|\leq 1)} \right] \nu(dx)\right)$$ More precisely, the whole process is characterized by the triple $(a,b, \nu)$.
A good but extremely technical book is Lévy Processes and Stochastic Calculus. However, I am interested in estimating the triple $(a,b, \nu)$. Note that the drift $b\in\mathbb R^d$ and the matrix $a$ are finite dimensional parameters and, consequently, could be estimated with parametric techniques. On the other hand, the Levy measure $\nu$ might require non-parametric techniques.
So I'm looking for bibliographical references that help me in the computational part and implementation of the estimation of $(a,b, \nu)$.
The first chapter of book Lévy Matters IV looks like a good book to me, but it's the only decent reference I've found. Also, maybe it's not as suitable for someone who doesn't have a lot of experience in the subject. So I would like to have suggestions for books with a more practical approach and, perhaps, using some language (R, python or some other).
Can you help me?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.