부산대학교 도서관

Let $\alpha$ belong to $\left]-1,0\right[,$ $\gamma$ to $\left]0,\infty\right[$ and $\beta$ to $\left]\alpha,\gamma\right[.$ Suppose $p_{\alpha},\ p_{\beta}$ and $p_{\gamma}$ are positive probabilities that sum to one. Let $\Omega=\left\{\alpha,\beta,\gamma\right\},$ $P=\left\{p_{\alpha},p_{\beta},p_{\gamma}\right\},$ ${\Omega}_{\Bbb N}={\Omega}^{\Bbb N}$ and $P_{\Bbb N}=P^{\Bbb N},$ the product measure on ${\Omega}_{\Bbb N}.$ Let $\left\{X_n,\ n\in \Bbb N\right\}$ be a sequence of i.i.d.\ random variables generating $P_{\Bbb N}.$ Consider then the sequence ${\scr{S}}=\left\{S_n,\ n\in\Bbb N\right\}$ defined by $S_0,$ positive and deterministic, and $S_n=\left(1+X_n\right)S_{n-1},$ $ n\geq 1.$ $Q$ is a martingale measure for ${\scr{S}}$ if it is equivalent to $P_{\Bbb N}$ and ${\scr{S}}$ is a martingale with respect to $Q.$ $M\left(P_{\Bbb N}\right)$ is the family of martingale measures for ${\scr{S}}$ for which $\left\{X_n,\ n\in \Bbb N\right\}$ remains i.i.d. $I\left(Q,P_{\Bbb N}\right)$ is the Kullback-Leibler number defined by $$I\left(Q,P_{\Bbb N}\right)= \cases E_{P_{\Bbb N}}\left[\frac{dQ}{dP_{\Bbb N}}\ln\left[\frac{dQ}{dP_{\Bbb N}}\right]\right]&\text{if}\ Q\ll P_{\Bbb N},\\ \infty &\text{otherwise}.\endcases$$ The minimal entropy martingale measure is that element $Q^{\star}\in M\left(P_{\Bbb N}\right)$ for which $$I\left(Q^{\star},P_{\Bbb N}\right)=\min_{Q\in M\left(P_{\Bbb N}\right)}I\left(Q,P_{\Bbb N}\right).$$ One learns in this paper that $Q^{\star}$ exists, is unique and may be computed. An explicit expression for $\frac{dQ^{\star}}{dP_{\Bbb N}},$ in terms of ${\scr{S}},$ is also provided. Similar results are obtained for the reverse minimal entropy martingale measure in which $\left[\frac{dQ}{dP_{\Bbb N}}\ln\left[\frac{dQ}{dP_{\Bbb N}}\right]\right]$ is replaced by $-\ln\left[\frac{dQ}{dP_{\Bbb N}}\right]$, and the minimal $\varphi$-divergence distance martingale measure in which $\left[\frac{dQ}{dP_{\Bbb N}}\ln\left[\frac{dQ}{dP_{\Bbb N}}\right]\right]$ is replaced by $\varphi\left(\frac{dQ}{dP_{\Bbb N}}\right)$. In the latter case some constraints on $\varphi$ must be imposed.