Seminari: Time inconsistent control - Bond markets and absence of arbitrage beyond the existence of a bank account

Venerdì 7 Marzo 2014, ore 9:00 - Aula 1BC45 - Tomas Bjork, Thorsten Schmidt



Venerdì 7 Marzo 2014, in Aula 1BC45 si terranno i seminari:

ore 9:00
Tomas Bjork
"Time inconsistent control" (joint with Agatha Murgoci)

We develop a theory for stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for Nash subgame perfect equilibrium points.
For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of non-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. Most known examples of time inconsistency in the literature are easily seen to be special cases of the present theory. We also prove that for every time inconsistent problem, there exists an associated time consistent problem such that the optimal control and the optimal value function for the consistent problem coincides with the equilibrium control and value function respectively for the time inconsistent problem. We also study some concrete examples.

Ore 10:00
Thorsten Schmidt
"Bond markets and absence of arbitrage beyond the existence of a bank account" (joint with Irene Klein and Josef Teichmann)

We investigate default-free bond markets where the standard relationship between a possibly existing bank account process and the term structure of bond prices is broken, i.e. the bank account process is not a valid numeraire. We argue that this feature is not the exception but rather the rule in bond markets when starting with, e.g., terminal bonds as numeraires.
Our setting are general cad lag processes as bond prices, where we employ directly methods from large financial markets. Moreover, we do not restrict price process to be semimartingales, which allows for example to consider markets driven by fractional Brownian motion. In the core of the article we relate the appropriate no arbitrage assumptions (NAFL), i.e. no asymptotic free lunch, to the existence of an equivalent local martingale measure with respect to the terminal bond as numeraire, and no arbitrage opportunities of the first kind (NAA1) to the existence of a supermartingale deflator, respectively. In all settings we obtain existence of a generalized bank account as a limit of convex combinations of roll-over bonds.
Additionally we provide an alternative definition of the concept of a numeraire, leading to a possibly interesting connection to bubbles. If we can construct a bank account process through roll-overs, we can relate the impossibility of taking the bank account as numeraire to liquidity effects. Here we enter endogenously the arena of multiple yield curves.

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