| Martingale and local martingale are basic concepts in stochastic analysis.Martingale property and local martingale property are defined by conditional expectation,so they are related to the probability measure in the probability space.This book report focuses on the following questions:can a local martingale find an equivalent probability measure to make it a martingale under the new probability measure.The answer to this question is yes in discrete time,but it needs local martingales to satisfy some properties in continuous time to become martingales.And this problem is related to the basic theorem of assets in financial mathematics,which will also be introduced in this paper.By collecting and consulting literature,this paper summarizes the properties of local martingales transformed into martingales in discrete time and continuous time.The first chapter introduces the work of this paper.The second chapter will introduce some basic knowledge.The third chapter introduces that in discrete time,local martingales can be transformed into martingales by measure transformation.The fourth chapter will introduce that the local martingale of continuous time becomes martingale through probability measure transformation under certain properties. |
