英文标题：
《Approximation methods for piecewise deterministic Markov processes and
  their costs》
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作者：
Peter Kritzer, Gunther Leobacher, Michaela Sz\\\"olgyenyi, Stefan
  Thonhauser
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最新提交年份：
2019
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英文摘要：
  In this paper, we analyse piecewise deterministic Markov processes, as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of piecewise deterministic Markov processes. In this context, one is interested in computing certain quantities of interest such as the probability of ruin of an insurance company, or the insurance company\'s value, defined as the expected discounted future dividend payments until the time of ruin. Instead of explicitly solving the integro-(partial) differential equation related to the quantity of interest considered (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. On the analytical side, we prove a convergence result for our PDMP approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and provide a comparative study of deterministic and Monte Carlo integration.
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中文摘要：
本文分析了Davis（1984）提出的分段确定马尔可夫过程。保险数学中的许多模型都可以用分段确定性马尔可夫过程的一般概念来表示。在这种情况下，人们感兴趣的是计算一定数量的利息，例如保险公司破产的概率，或保险公司的价值，定义为直到破产时的预期贴现未来股息支付。我们没有显式求解与所考虑的关注量相关的积分（偏）微分方程（这种方法只能在少数特殊情况下使用），而是以允许我们应用确定性数值积分算法（如准蒙特卡罗规则）的方式来调整问题；这与应用蒙特卡罗等随机积分算法形成对比。为此，我们将一个一般的代价泛函重新表述为一个特定积分算子的不动点，这允许对泛函进行迭代逼近。此外，我们还介绍了一种应用于所涉及的被积函数的平滑技术，以使用确定性容积规则的误差界。在分析方面，我们证明了我们的PDMP近似的收敛结果，这是一个独立的有趣的结果，因为它证明了过程级别上的相位类型近似。我们举例说明了风险理论中的平滑技术，并对确定性积分和蒙特卡罗积分进行了比较研究。
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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