Minimization with respect to entropy in the problem of finding a martingale measure (Q2780498)

Let \((\Omega, F,P)\) be the probability space and \(h_1,h_2,\dots, h_N\) be real-valued random variables having a continuous joint distribution with everywhere non-zero density \(f_0(x_1,\dots,x_N)> 0\). Let \(S_n= S_0 e^{h_1+\cdots+ h_n}\) be the price of a stock at time \(n= 1,2,\dots, N\), \(B_0\equiv 1\) be the price of a bond, \({\mathcal I}_n= \sigma(h_1,\dots, h_n)\), \(n=1,\dots, N\), and denote by \({\mathcal P}(P)\) the set of all probability measures equivalent to the measure \(P\) and such that \((S_n/B_n)\) is a martingale with respect to the filtration \(({\mathcal I}_n)_n\). The author determines the density \(f\) of the joint distribution of \(h_1,\dots, h_N\) corresponding to a measure \(\widetilde P\in{\mathcal P}(P)\) that satisfies the conditions \(E_{\widetilde P}G_i\) \((S_1,\dots, S_N)= \int G_i f dx_1\cdots dx_N= C_i\), \(i=1,\dots,M\), where the \(G_i\) are given functions to be interpreted as discounted payment functions of certain options and the \(C_i\) are the given values of their prices at time \(0\).