Information uncertainty related to marked random times and optimal investment

Probability, Uncertainty and Quantitative Risk (2018) 3:3 DOI 10.1186/s41546-018-0029-8 RESEARCH Probability, Uncertainty and Quantitative Risk Open Access Information uncertainty related to marked random times and optimal investment Ying Jiao · Idris Kharroubi Received: 19 January 2017 / Accepted: 18 April 2018 / © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Abstract We study an optimal investment problem under default risk where related information such as loss or recovery at default is considered as an exogenous random mark added at default time. Two types of agents who have different levels of information are considered. We first make precise the insider’s information flow by using the theory of enlargement of filtrations and then obtain explicit logarithmic utility maximization results to compare optimal wealth for the insider and the ordinary agent. Keywords Information uncertainty · Marked random times · Enlargement of filtrations · Utility maximization MSC 60G20; 91G40; 93E20 1 Introduction The optimization problem in presence of uncertainty on a random time is an important subject in finance and insurance, notably for risk and asset management when it concerns a default event or a catastrophic occurrence. Another related source of risk is the information associated with the random time concerning resulting payments, the price impact, the loss given default or the recovery rate, etc. Measuring Y. Jiao Université Claude Bernard - Lyon 1, Institut de Science Financière et d’Assurances, 50 Avenue Tony Garnier, 69007 Lyon, France e-mail: I. Kharroubi () Sorbonne Université, Sorbonne Paris Cité, CNRS, Laboratoire de Probabilités Statistique et Modélisation, LPSM, F-75005 Paris, France e-mail: Page 2 of 24 Y. Jiao, I. Kharroubi these random quantities is, in general, difficult since the relevant information on the underlying firm is often not accessible to investors on the market. For example, in the credit risk analysis, modelling the recovery rate is a subtle task (see, e.g. Duffie and Singleton (2003) Section 6, Bakshi et al. (2006), and Guo et al. (2009)). In this paper, we study the optimal investment problem with a random time and consider the information revealed at the random time as an exogenous factor of risk. We suppose that all investors on the market can observe the arrival of the random time, such as the occurrence of a default event. However, for the associated information, such as the recovery rate, there are two types of investors: the first is an informed insider and the second is an ordinary investor. For example, the insider has private information on the loss or recovery value of a distressed firm at the default time, but the ordinary investor must wait for the legitimate procedure to be finished to know the result. Both investors aim at maximizing the expected utility from the terminal wealth and each of them will determine their investment strategy based on the corresponding information set. Following Amendinger et al. (1998, 2003), we will compare the optimization results and deduce the additional gain of the insider. Let the financial market be described by a probability space (, A, P) equipped with a reference filtration F = (Ft )t≥0 which satisfies the usual conditions. In the literature, the theory of enlargements of filtrations provides essential tools for the modelling of different information flows. In general, the observation of a random time, in particular a default time, is modelled by the progressive enlargement of filtration, as proposed by Elliott et al. (2000) and Bielecki and Rutkowski (2002). The knowledge of insider information is usually studied by using the initial enlargement of filtration as in Amendinger et al. (1998, 2003) and Grorud and Pontier (1998). In this paper, we suppose that the filtration F represents the market information known by all investors, including the default information. Let τ be an F-stopping time which represents the default time. The information flow associated with τ is modelled by a random variable G on (, A) valued in a measurable space (E, E ). In the classic setting of insider information, G is added to F at the initial time t = 0, while in our model, the information is added punctually at the random time τ . Therefore, we need to specify the corresponding filtration. Let the insider’s filtration G = (Gt )t≥0 be a punctual enlargement of F by adding the information of G at the random time τ . In other words, G is the smallest filtration which contains F and such that the random variable G is Gτ -measurable. This provides a new type of enlargement of filtrations which is an extension of the classical initial enlargement. In the literature, other generalizations of enlargement have also been considered such as in Kchia et al. (2013) and Kchia and Protter (2015) where the authors study extensions of progressive enlargement. We shall make precise the adapted and predictable processes in the filtration G that we define in order to describe investment strategy and wealth processes. As usual in the asymmetric information literature, we suppose the hypothesis that the F-conditional law of G is equivalent and hence admits a positive density with respect to its probability law. By adapting arguments in Föllmer and Imkeller (1993) and in Grorud and Pontier (1998), we deduce the insider martingale measure Q which plays an important role in the study of (semi)martingale processes in the filtration G. Our main mathematical result is to give the decomposition formula of an Probability, Uncertainty and Quantitative Risk (2018) 3:3 Page 3 of 24 F-martingale as a semimartingale in G, which gives a positive answer to the Jacod’s (H’)-hypothesis. In the optimization problem with random default times, it is often supposed that the random time satisfies the intensity hypothesis (e.g., Lim and Quenez (2011) and Kharroubi et al. (2013)) or the density hypothesis (e.g., Blanchet-Scalliet et al. (2008), Jeanblanc et al. (2015), and Jiao et al. (2013)), so that it is a totally inaccessible stopping time in the market filtration. In particular, in Jiao et al. (2013), we consider marked random times where the random mark represents the loss at default and we suppose that the vector of default time and mark admits a conditional density. In this paper, the random time τ we consider does not necessarily satisfy the intensity nor the density hypothesis: it is a general stopping time in F and may also contain a predictable part. Following the approach of Amendinger et al. (1998), we obtain the optimal strategy and wealth for (...truncated)