Hidden semi Markov models; stochastic control.

除了信号处理、语音处理和其他领域，隐马尔可夫模型最成功的应用之一是基因组和蛋白质测序。我与L·阿贡和J·摩尔合著的《隐马尔可夫模型：估计与控制》一书于1995年由斯普林格出版社出版，1997年再版。2008年印制了修订版和扩充版。我还发表了其他几篇关于隐马尔可夫模型的文章。然而，当离散序列建模时，例如在离散时间或在生物应用中，如果隐藏过程是马尔可夫链，则其在任何状态下的(随机)占用时间都是几何分布的随机变量。对于生物应用，似乎应该考虑更一般的占用时间。这导致我们考虑半马尔可夫模型，其中跳跃到新状态的概率不仅取决于当前状态，而且取决于链已经处于该状态的时间长度。我们将考虑“隐藏的”半马尔可夫模型，即半马尔可夫链不是直接观察到的，而是调制第二个观察到的过程的情况。我们相信，这些将更好地模拟蛋白质和其他‘GNOME’测序中显示的行为。我们将为这些模型发展在我们的书和以前的工作中找到的结果。第二行的研究将使用我们最近关于倒向随机微分方程的结果来研究部分可观测的随机控制问题。伴随过程用倒向随机微分方程来描述；对于部分观测问题，这是一个倒向随机偏微分方程。最初的工作将考虑部分观察到的马尔可夫链的控制问题

In addition to signal processing, speech processing and other areas, one of the most successful applications of hidden Markov models has been to genome and protein sequencing. My book with L Aggoun and J Moore on "Hidden Markov Models: Estimation and Control" was published in 1995 by Springer Verlag and reprinted in 1997. A revised and expanded edition was printed in 2008. I have published several other contributions to hidden Markov models. However, when modelling discrete sequences, for example in discrete time or in biological applications, if the hidden process is a Markov chain its (random) occupation time in any state is a geometrically distributed random variable. For biological applications it appears that more general occupation times should be considered. This leads us to consider semi-Markov models where the probability of jumping to a new state depends not only on the present state but how long the chain has been in that state. We shall consider 'hidden' semi-Markov models, that is situations where the semi-Markov chain is not observed directly but modulates a second, observed, process. We believe these will better model behaviour exhibited in protein and other 'gnome' sequencing. We shall develop for these models the results found in our book and previous works. A second line of research will use our recent results on backward stochastic differential equations to investigate partially observed stochastic control problems. The adjoint process is described by a backward stochastic differential equation; for partially observed problems this is a backward stochastic partial differential equation. Initial work will consider this problem for the control of a partially observed Markov chain