"Monte Carlo Methods in Finance, Statistics and Biostatistics"

这项研究是关于蒙特卡罗方法在金融与统计问题中的应用。我们经常希望估计随机过程的函数的期望值，该随机过程的分布Q是不可分解的或难以模拟的。我们调查了“随机重要性抽样”的使用情况。我们从另一个更简单的分布P进行模拟，然后使用(随机)重要性抽样(IS)权重W对观测进行加权，这弥补了错误分布被用于进行模拟的事实。该估计器是具有随机权重的加权平均值。W估计两个度量下的观测的相对可能性，并且可以选择仅采用整数值或重新缩放、丢弃权重为0的观测。由于随机过程的概率(Radon Nikodym导数)很难精确计算，因此有许多潜在的应用，但它们的无偏估计是容易处理的。例如，如果过程是扩散或跳跃扩散过程，则评估可能性需要评估过程的积分，即在区间内的每个时间点对其进行模拟。然而，很容易得到一个只在有限多个点处对过程进行抽样的无偏估计。使用这种估计器的模拟将被用于评估风险或评估期权价格。我们也可以仅使用分布的特征函数来获得我们的抽样权重。我们将这种方法应用于随机波动率模型的分析，这是对金融学中不适合的Black-Scholes模型的一个重要改进。我们将考虑其他已知原木现货价格特征函数的问题，如有随机利率和不有随机利率的仿射随机波动率模型，时变的Levy过程，以及其他复杂模型。使用基于模拟的随机重要性样本权重允许采用更广泛的模型类别，或容纳不完整的观测数据，因为它允许在简单的度量P下进行分配。这项工作还将促进统计学和生物统计学中连续时间纵向模型中更复杂的缺失数据问题的分析。

This research concerns the applications of Monte Carlo methods to problems in Finance and Statistics. We frequently wish to estimate the expected value of a function of a stochastic process, whose distribution Q is intractible or difficult to simulate from. We investigate the use of "randomized importance sampling". We simulate from an alternate simpler distribution P and then weight the observations using (random) importance sampling (IS) weights W, which compensate for the fact that the wrong distribution was used to conduct the simulation. The estimator is a weighted average with random weights W. W estimates the relative likelihood of the observations under the two measures and can be chosen to take only integer values or rescaled, observations with weight 0 discarded. There are emany potential applications since for stochastic processes, likelihoods ("Radon Nikodym derivatives") are difficult to compute exactly, but unbiased estimators of them are tractable. For example if the process is a diffusion or jump-diffusion process, evaluating the likelihood requires evaluating an integral of the process, i.e. simulating it at every time point in an interval. However it is easy to obtain an unbiased estimator which samples the process only at finitely many points. Simulations using such estimators will be used to assess risk or evalute option prices. We can also obtain our sampling weights using only the characteristic function of a distribution. We apply this methodology to the analysis of stochastic volatilitiy models, which provide a critical improvement over the ill-fitting Black-Scholes model in finance. We will consider other problems in which the characteristic function of the log-spot price is known, such as affine stochastic volatility models with and without stochastic interest rates, time-changed Levy processes, and other complex models. Using simulation-based randomized importance sample weights permits adoption of broader classes of models, or accommodating incomplete observed data, since it allows imputation to be carried out under the simple measure P. This work will also facilitate the analysis of more complex missing data problems in continuous-time longitudinal models in statistics and biostatistics.