insight - Algorithms and Data Structures - # Exact Simulation of First Passage Time of Tempered Stable Subordinator

Fast and Exact Simulation of the First Passage Time of a Tempered Stable Subordinator Across a Non-Increasing Function

Q: How can the algorithms developed in this paper be extended to handle more general non-increasing functions b(t) beyond absolutely continuous ones

The algorithms developed in the paper can be extended to handle more general non-increasing functions beyond absolutely continuous ones by considering different types of functions that are still non-increasing. One approach could be to explore piecewise non-increasing functions, where the function b(t) is defined by different segments with varying properties. By adapting the algorithms to handle these piecewise functions, the simulation process can account for different behaviors and characteristics of the function over different intervals. This extension would involve modifying the sampling and simulation steps to accommodate the changing nature of the non-increasing function.

Q: What are the potential applications of the exact simulation algorithm beyond solving fractional partial differential equations and pricing barrier options, as discussed in the paper

The exact simulation algorithm developed in the paper has various potential applications beyond solving fractional partial differential equations and pricing barrier options. Some of these applications include: Risk Management: The algorithm can be used in risk management to simulate the first-passage time of a subordinator across certain risk thresholds, helping in assessing and managing risks effectively. Queueing Theory: In queueing theory, the simulation of first-passage events can aid in analyzing the waiting times and workloads in queueing systems, leading to improvements in system performance and efficiency. Biomedical Modeling: The algorithm can be applied in biomedical modeling to study processes where the first passage of a subordinator across certain levels has significance, such as in physiological systems or disease progression models. Environmental Modeling: In environmental modeling, the simulation of first-passage events can be used to study the crossing of critical thresholds in environmental processes, helping in predicting and managing environmental risks. These applications demonstrate the versatility and utility of the exact simulation algorithm in various fields where the first-passage time of a subordinator plays a crucial role.

Q: Can the techniques used in this paper be adapted to develop efficient exact simulation algorithms for other classes of stochastic processes, such as general Lévy processes or diffusions

The techniques used in this paper can be adapted to develop efficient exact simulation algorithms for other classes of stochastic processes, such as general Lévy processes or diffusions, by considering the specific characteristics and properties of these processes. The key steps involved in the algorithms, such as sampling from specific densities, numerical inversion methods, and accept-reject sampling, can be tailored to suit the requirements of the particular stochastic process under consideration. For general Lévy processes, the algorithms can be modified to handle the specific jump characteristics and drift properties of these processes. By adjusting the sampling methods and simulation steps to align with the Lévy process dynamics, an efficient exact simulation algorithm can be developed. Similarly, for diffusions, the techniques used in the paper can be adapted to account for the continuous nature of diffusion processes and the specific diffusion coefficients involved. By incorporating the relevant diffusion properties into the simulation algorithms, accurate and efficient simulations of first-passage events for diffusions can be achieved. Overall, the principles and methodologies presented in the paper can serve as a foundation for developing exact simulation algorithms for a wide range of stochastic processes beyond tempered stable subordinators.

Core Concepts

This paper develops a fast and exact algorithm for simulating the first passage time, undershoot, and overshoot of a tempered stable subordinator over an arbitrary non-increasing absolutely continuous function. The algorithm has finite exponential moments and its expected running time is bounded explicitly in terms of the model parameters.

Abstract

The paper presents an algorithm called TSFFP-Alg that can efficiently and exactly simulate the first passage time (τb), undershoot (Sτb-), and overshoot (Sτb) of a tempered stable subordinator S over an arbitrary non-increasing absolutely continuous function b(t).

The key steps are:

TSFFP-Alg reduces the problem to the stable case (q=0) using a rejection sampling approach based on the Esscher transform.
The stable case is handled by SFP-Alg, which first samples the crossing time τb conditional on τb ≤ t*, then determines if the crossing occurred via a jump or by creeping.
The most technically challenging part is the simulation of the undershoot Sτb- in the jump case, which is achieved by SU-Alg. SU-Alg samples from the undershoot density by breaking up the state space and using tailored accept-reject algorithms on each subinterval.

The authors provide a detailed complexity analysis, showing that the expected running time of TSFFP-Alg has finite exponential moments and grows at most cubically in the stability parameter 1/α as it approaches 0 or 1. The running time is also linear in the tempering parameter q and the initial value of the function b(0).

Numerical experiments demonstrate the good agreement between the theoretical complexity bounds and the observed performance of the implemented algorithms.

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Stats

The paper does not contain any explicit numerical data or statistics to support the key claims. The analysis focuses on the theoretical complexity of the algorithms.

Quotes

"The expected running time grows at most cubically in the stability parameter (as it approaches either 0 or 1) and is linear in the tempering parameter and the initial value of the function."
"The running time of our algorithm has finite exponential moments and provide bounds on its expected running time with explicit dependence on the characteristics of the process and the initial value of the function."

Key Insights Distilled From

Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function

by Jorg... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2303.11964.pdf

Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function

Deeper Inquiries

How can the algorithms developed in this paper be extended to handle more general non-increasing functions b(t) beyond absolutely continuous ones

The algorithms developed in the paper can be extended to handle more general non-increasing functions beyond absolutely continuous ones by considering different types of functions that are still non-increasing. One approach could be to explore piecewise non-increasing functions, where the function b(t) is defined by different segments with varying properties. By adapting the algorithms to handle these piecewise functions, the simulation process can account for different behaviors and characteristics of the function over different intervals. This extension would involve modifying the sampling and simulation steps to accommodate the changing nature of the non-increasing function.

What are the potential applications of the exact simulation algorithm beyond solving fractional partial differential equations and pricing barrier options, as discussed in the paper

The exact simulation algorithm developed in the paper has various potential applications beyond solving fractional partial differential equations and pricing barrier options. Some of these applications include:

Risk Management: The algorithm can be used in risk management to simulate the first-passage time of a subordinator across certain risk thresholds, helping in assessing and managing risks effectively.
Queueing Theory: In queueing theory, the simulation of first-passage events can aid in analyzing the waiting times and workloads in queueing systems, leading to improvements in system performance and efficiency.
Biomedical Modeling: The algorithm can be applied in biomedical modeling to study processes where the first passage of a subordinator across certain levels has significance, such as in physiological systems or disease progression models.
Environmental Modeling: In environmental modeling, the simulation of first-passage events can be used to study the crossing of critical thresholds in environmental processes, helping in predicting and managing environmental risks.

These applications demonstrate the versatility and utility of the exact simulation algorithm in various fields where the first-passage time of a subordinator plays a crucial role.

Can the techniques used in this paper be adapted to develop efficient exact simulation algorithms for other classes of stochastic processes, such as general Lévy processes or diffusions

The techniques used in this paper can be adapted to develop efficient exact simulation algorithms for other classes of stochastic processes, such as general Lévy processes or diffusions, by considering the specific characteristics and properties of these processes. The key steps involved in the algorithms, such as sampling from specific densities, numerical inversion methods, and accept-reject sampling, can be tailored to suit the requirements of the particular stochastic process under consideration.
For general Lévy processes, the algorithms can be modified to handle the specific jump characteristics and drift properties of these processes. By adjusting the sampling methods and simulation steps to align with the Lévy process dynamics, an efficient exact simulation algorithm can be developed.
Similarly, for diffusions, the techniques used in the paper can be adapted to account for the continuous nature of diffusion processes and the specific diffusion coefficients involved. By incorporating the relevant diffusion properties into the simulation algorithms, accurate and efficient simulations of first-passage events for diffusions can be achieved.
Overall, the principles and methodologies presented in the paper can serve as a foundation for developing exact simulation algorithms for a wide range of stochastic processes beyond tempered stable subordinators.