The Martingale Approach Sample Clauses

The Martingale Approach. For a fixed the time horizon T ∈ [0, ∞], St is a stochastic process on a probability space (Ω, F, P) modelling the stock price in the market. As mentioned in previous sections, S = St denotes a locally bounded semimartingale. A probability measure equivalent to the original measure Q P is an equivalent local martingale measure for S such that S is a local martingale in measure Q. We denoted Mα(S) as the family of absolutely continuous local martingale measure and it follows that Mα(S) /= ∅. For a utility function defined above, we define the convex dual V (y) be V (y) := sup U (x) − xy , the Xxxxxxxx transform of −U (−x), where x is in the domain of utility function U and y is nonnegative. Thus, we have the relation Ur = (−V r)—1. By the definition of V (y), U (x) ≤ V (y) + xy holds for any x runs through the domain of U and nonnegative y. Hence, for any y > 0 and for any equivalent martingale measure Q, we have the duality upper bound E[U (Xπ)] ≤ E V ydQ(y) + yE [X ] T dP dP ≤ E V ydQ(y) + yx Q(y) T holds for any strategy π that its wealth process Xπ is a supermartingale under the equivalent martingale measure Q. We assume that such equivalent martingale measure always exists. Suppose there exists a strategy πˆt and a martingale measure Qˆ satisfying yˆdQˆ dP = Ur(Xπˆ) and EQˆ [Xˆπˆ] = x. (2.13) Then πˆ maximises the expected utility E U (Xπ) over any possible sets of wealth processes that are supermartingale under measure Qˆ. Then, the primal optimisation problem (2.12) can turn into problem of finding a solution of v(y) = inf Q∈Mα(S) E V ydQ . Consider the original optimisation problem u(x) = sup E U x + ∫ πtdSt , for all admissible πt, the optimal solution πˆ exists and is unique. Denote y = ur(x), there exists a positive yˆ such that the dual problem E V yˆdQ is minimised by a so-called dual minimiser Qˆ. Relation (2.13) can be described as ∫ − x + πˆtdSt = V r yˆdQˆ dP and yˆdQˆ = U dP r x + ∫ πˆtdSt 0