Log-optimal and numéraire portfolios for market models stopped at a random time

This paper focuses on numéraire and log-optimal portfolios when a market model ( S , F , P ) $(S,\mathbb{F},P)$ – specified by its assets’ price S $S$ , its flow of information F $\mathbb{F}$ and a probability measure P $P$ – is stopped at a random time τ $\tau $ . The flow of information that incorporates both F $\mathbb{F}$ and τ $\tau $ , denoted by G $\mathbb{G}$ , is the progressive enlargement of F $\mathbb{F}$ with τ $\tau $ . For the resulting stopped model ( S τ , G , P ) $(S^{\tau},\mathbb{G},P)$ , we study the two portfolios in different manners and describe their computations in terms of F $\mathbb{F}$ -observable parameters of the pair ( S , τ ) $(S, \tau )$ . In particular, we single out the types of risks induced by τ $\tau $ that really affect the numéraire portfolio, and address the following questions: 1) What are the conditions on τ $\tau $ (preferably in terms of information-theoretic concepts such as entropy) that guarantee the existence of the log-optimal portfolio for ( S τ , G , P ) $(S^{\tau},\mathbb{G},P)$ when that for ( S , F , P ) $(S,\mathbb{F},P)$ already exists? 2) What are the factors that fully determine the increment in maximal expected logarithmic utility from terminal wealth for the models ( S τ , G , P ) $(S^{\tau},\mathbb{G},P)$ and ( S , F , P ) $(S,\mathbb{F},P)$ , and how can one quantify them?