학술논문
The minimal entropy and minimal $\phi$-divergence distance martingale measures for the trinomial scheme.
Document Type
Journal
Author
Glonti, O. (GE-TBIL-PB) AMS Author Profile; Jamburia, L. (GE-TBIL-PB) AMS Author Profile; Kapanadze, N. (GE-TBIL-PB) AMS Author Profile; Khechinashvili, Z. (GE-TBIL-PB) AMS Author Profile
Source
Subject
60 Probability theory and stochastic processes -- 60G Stochastic processes
60G42Martingales with discrete parameter
91Game theory, economics, social and behavioral sciences -- 91B Mathematical economics
91B28Finance, portfolios, investment
94Information and communication, circuits -- 94A Communication, information
94A17Measures of information, entropy
60G42
91
91B28
94
94A17
Language
English
Georgian
Georgian
ISSN
15120074
Abstract
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 probabilitiesthat sum to one. Let $\Omega=\left\{\alpha,\beta,\gamma\right\},$ $P=\left\{p_{\alpha},p_{\beta},p_{\gamma}\right\},$${\Omega}_{\Bbb N}={\Omega}^{\Bbb N}$ and $P_{\Bbb N}=P^{\Bbb N},$ theproduct measure on ${\Omega}_{\Bbb N}.$Let $\left\{X_n,\ n\in \Bbb N\right\}$ be a sequence of i.i.d. randomvariables generating $P_{\Bbb N}.$Consider then the sequence${\scr{S}}=\left\{S_n,\ n\in\Bbb N\right\}$ defined by $S_0,$ positive and deterministic, and$S_n=\left(1+X_n\right)S_{n-1},$ $ n\geq 1.$ $Q$ is a martingale measure for ${\scr{S}}$if it is equivalent to $P_{\Bbb N}$and ${\scr{S}}$ is a martingale with respect to $Q.$ $M\left(P_{\Bbb N}\right)$is the family of martingale measures for ${\scr{S}}$ for which$\left\{X_n,\ n\in \Bbb N\right\}$ remains i.i.d.$I\left(Q,P_{\Bbb N}\right)$ is the Kullback-Leibler number defined by$$I\left(Q,P_{\Bbb N}\right)=\cases E_{P_{\Bbb N}}\left[\frac{dQ}{dP_{\Bbb N}}\ln\left[\frac{dQ}{dP_{\BbbN}}\right]\right]&\text{if}\ Q\ll P_{\Bbb N},\\\infty &\text{otherwise}.\endcases$$The minimal entropy martingale measure is that element $Q^{\star}\inM\left(P_{\Bbb N}\right)$ for which$$I\left(Q^{\star},P_{\Bbb N}\right)=\min_{Q\in M\left(P_{\BbbN}\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 expressionfor $\frac{dQ^{\star}}{dP_{\Bbb N}},$ in terms of ${\scr{S}},$ is also provided. Similar results areobtained for the reverse minimal entropymartingale 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 measurein 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.