重要抽樣法是一種主要的變異數縮減技術之一。藉由適當選擇新的機率測度，可以有效降低估計的標準誤差。然而，選擇新的機率測度的方式不只一種，所以我選擇最小熵測度作為新測度。此種方法稱為相對熵重要抽樣法。
本篇論文將會討論四種Lévy 隨機過程下的相對熵重要抽樣法。最後會討論模型的參數估計。

Importance sampling the approach of variance reduction is one of the most important methods to estimate the probability of rare events. By properly changing of measure, this method can reduce the variance of new estimator. However, there are several ways of choosing the appropriate probability measure to implement importance sampling. The “Minimal Entropy Measure” is chosen as suitable probability measure, which is entropy-based importance sampling.
This article takes the Lévy Processes as examples, such as jump diffusion process, variance gamma process and normal inverse Gaussian process. Moreover, our method works on stochastic volatility jump model. We compare the numerical results with basic Monte Carlo and other distance function to demonstrate that this method is effective. Finally, we estimate the parameters by method of moment on examples of Lévy Process, and briefly describe estimation of parameters of stochastic volatility jump model by Markov chain Monte Carlo method.

Abstract i
摘要 ii
Contents iii
Chapter 1 Introduction 1
Chapter 2 Entropy-based Importance Sampling 3
2.1 Change of Probability Measure 3
2.2 Principle of Importance Sampling 4
2.3 Selection of an Equivalent Probability Measure P ̃ 5
Chapter 3 Lévy Processes 8
3.1 Introduction of Lévy process 8
3.2 Examples of Lévy Processes 9
Chapter 4 Estimation of Default Probability under Lévy Processes 13
4.1 Variance Gamma Process 13
4.2 Normal Inverse Gaussian Process 18
4.3 Compound Poisson Process 21
4.4 Jump Diffusion Process 27
Chapter 5 Estimation of Default Probability beyond Lévy processes 31
5.1 Heston model 31
5.2 Stochastic Volatility Jump Model 32
Chapter 6 Estimation of Parameters 35
6.1.1 Method of Moment 35
6.1.2 Simulation Studies 36
6.1.3 Empirical Studies 38
6.2 MCMC for Processes beyond Lévy 41
Chapter 7 Conclusion 42
Reference 43

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