关于Skorokhod嵌入问题，迄今为止的研究绝大多数都假设相应的停止过程一致可积。当X是一个连续鞅时，这个假设等价于位势秩条件：起始分布的位势不小于目标分布的位势。但是一些简单的例子说明，即使没有这个条件，仍可构造恰当的Skorokhod嵌入。为此我们挑选一类特殊的Skorokhod嵌入：Root型嵌入来进行系统性的研究。由之前的工作，我们对这类嵌入的结构性质有了相当深的了解。在此基础上，我们会研究，对于任意的起始分布和目标分布，在没有位势秩条件的情况下，是否存在Root型的嵌入，以及如何构造；进一步，它是否满足给定的标准(一致可积不成立，我们考虑的是Monroe最小停时条件)。注意到，一致可积条件下Root型嵌入的最优性在这里不成立，我们会研究其他适用的最优性；最后我们研究其在金融中的应用：关于变差期权的定价问题。我们还研究将类似结果对其他类型嵌入(Rost型)是否适用。

Up to now, almost all research about Skorokhod embedding problem assumes that the corresponding stopped processes are uniformly integrable. When the underlying process is a continuous martingale, this assumption is equivalent to the potential ordering condition: the potential of initial distribution is not less than the potential of target distribution. However, some simple examples tells us, even when this condition fails, one can still find suitable Skorokhod embedding. We pick a class of special embedding, Root's embedding, as the research object, to study this problem systematically. Thanks to our early work, we already have deep insight in Root's embedding: its construction, its properties. Now, given that the potential ordering condition fails, we study the existence of Root's embedding, and how to construct it; moreover, when given the existence result, does the Root's embedding satisfy some special criterion (since the uniform integrability fails, we consider the minimality of Monroe's sense)? Another question is about the optimality. One can find that the optimality of classical Root's embedding (minimal residual expectation) obviously fails as the uniform integrability fails. So we will explore some other optimality of Root's embedding under the general setting. At last, we will study its application in mathematical finance, particularly, the (model-independent, or robust) pricing problem of variance options. We will also study if the results obtained for Root's embedding are available for other types of embeddings, for example, Rost's embedding.