This thesis aims to study two important problems in mathematical finance— dynamic coherent risk measures and dynamic mean-variance optimizations. It contains six chapters, where the first one is an exordium, the 2nd and 3rd ones discuss coherent and dynamic coherent risk measures, and the last three chapters study the dynamic mean-variance optimization problems.We first generalize the discussion of coherent risk measures from L^{∞} space to the more generally Banach space L^{p}, and prove that a || · ||_{p}-norm lower continuous coherent risk measure defined on L^{P} is necessarily and sufficiently defined by a collect of q-square integrable probability measures where q is the dual index of p. In convenience of giving an axiomatic definition of a dynamic coherent risk measure, we propose a new concept of risk measure the (?)-coherent risk measure where (?) is all of the information available at r time and r may be a fixed time or a stopping time like random time. We obtain that a (?)-coherent risk measure satisfying Fatou property should be defined by a special collect (?) of probability measures absolutely continuous with respect to the objective probability measure and coincident with it when restricting in (?) and if it satisfies more the relevant property, it then can be defined by the collect (?)^{e}_{}of all the equivalent ( with respect to the objective probability measure ) probability measures belonging to (?). We also illustrate that both (?) and (?) are (?)-convex set, i.e., satisfy the (?)-convexity property. We define a collect with time set (?) (discrete, or continuous) as the index set of (?)-coherent risk measures as a dynamic coherent risk measure and by this we give an axiomatic definition for dynamic coherent risk measures. We prove that a dynamic coherent risk measure satisfying Fatou property, the relevant property and the time-consistent property is necessarily and sufficiently defined directly by a multiplicatively stable set ( m-stable for short ) of probability measures. Furthermore, we generalize successfully all of the main theorems concerning the representations of... |