# Stochastic Analysis and Stochastic Finance Seminar

## Risk measures for processes and BSDEs

Speaker(s):
Irina Penner (HU Berlin)
Date:
Thursday, June 20, 2013 - 5:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin

In the talk we analize risk assessment for cash flows in continuous time using the notion of convex risk measures for processes. By combining a decomposition result for optional measures, and a dual representation of a convex risk measure for bounded cadlag processes, we show that this framework provides a systematic approach to the both issues of model ambiguity, and uncertainty about the time value of money. We also provide examples of such risk measures in terms of BSDEs, that depend on the whole path of a process.

## Martingale Optimal Transport and Robust Hedging in Continuous Time

Speaker(s):
Yan Dolinsky (Hebrew University of Jerusalem)
Date:
Thursday, June 20, 2013 - 4:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin

The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity.

## Asymptotic Independence of Three Statistics of the Maximal Increments of Random Walks and Lévy Processes

Speaker(s):
Aleksandar Mijatovic (Imperial College London)
Date:
Thursday, June 6, 2013 - 5:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041

Let $H(x) = \inf\{n:\, \exists\, k x\}$ be the first epoch that an increment of the size larger than $x>0$ of a random walk $S$ occurs and consider the path functionals: $R_n = \max_{m\in\{0, \ldots, n\}}\{S_{n} - S_m\}, R_n^* = \max_{m,k\in\{0, \ldots, n\}, m\leq k} \{S_{k}-S_m\}$ and $O_x=R_{H(x)}-x.$ The main result states that, under Cram\'{e}r's condition on the step-size distribution of $S$, the statistics $R_n$, $R_n^* -y$ and $O_{x+y}$ are asymptotically independent as $\min\{n,y,x\}\uparrow\infty$.

## A Mathematical Treatment of Bank Monitoring Incentives

Speaker(s):
Dylan Possamai (École Polytechnique in Paris)
Date:
Thursday, June 6, 2013 - 4:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041

In this paper, we take up the analysis of a principal/agent model with moral hazard introduced in by Pagès, with optimal contracting between competitive investors and an impatient bank monitoring a pool of long-term loans subject to Markovian contagion. We provide here a comprehensive mathematical formulation of the model and show using martingale arguments in the spirit of Sannikov, how the maximization problem with implicit constraints faced by investors can be reduced to a classic stochastic control problem.

## Probabilistic models for interacting agents facing binary decisions

Speaker(s):
Marco Tolotti (Università Ca' Foscari Venezia)
Date:
Thursday, May 23, 2013 - 5:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041 im Erdgeschoß

In this talk I present a class of stochastic models useful to represent the dynamics of a system of many interacting agents facing binary decisions. At each time, agents update their choice in order to maximize their payoff, depending on their action, on the state of the system and on a random noise. In particular, I put attention on two different mechanisms for the updating: a “sequential updating” scheme and a “parallel updating” one. The former is more similar in spirit to classical statistical-mechanics, whereas the latter mimics a non-cooperative game played by the agents.

## The Joint Dynamics of Labor Income, Stock Prices, and House Prices and the Implications for Household Decisions

Speaker(s):
Holger Kraft (Goethe Universität Frankfurt am Main)
Date:
Thursday, May 23, 2013 - 4:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041 im Erdgeschoß

We show that in the 1953-2010 U.S. data stock prices, house prices, and labor income have time-varying expected growth rates. These drift rates are correlated with shocks to stock prices, house prices, and income so that these quantities tend to move together in the longer run in spite of low contemporaneous correlations. Incorporating these long-run relations into a life-cycle household optimization problem affects the optimal consumption, housing, and investment decisions.

## Equilibrium considerations in a financial market with interacting investors

Speaker(s):
Gonçalo Dos Reis (Technische Universität Berlin)
Date:
Thursday, April 25, 2013 - 5:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041 im Erdgeschoß

While trading on a financial market, the agents we consider take the performance of their peers into account. By maximizing individual utility subject to investment constraints, the agents may ruin each other even unintentionally so that no equilibrium can exist. However, when the agents are willing to waive little expected utility, an approximated equilibrium can be established. The study of the associated backward stochastic differential equation (BSDE) reveals the mathematical reason for the absence of an equilibrium.

## On the Reflected Backward Stochastic Partial Differential Equations

Speaker(s):
Jinniao Qiu (Humboldt-Universität zu Berlin)
Date:
Thursday, April 25, 2013 - 4:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041 im Erdgeschoß

In this talk, we first introduce the backward SPDEs and the quasi-linear reflected backward SPDEs. Basing on the classical parabolic capacity and potential theory, we associate the reflected backward SPDE to a variational problem, and present the well-posedness of the quasi-linear reflected backward SPDEs. Some related results, which include the comparison principle for solutions of RBSPDEs, as well as the connections with reflected backward stochastic differential equations and the optimal stopping problems, are also addressed.

## Dynamics of Contract Design with Screening

Speaker(s):
Jaksa Cvitanic (EDHEC Business School, Nice)
Date:
Thursday, April 11, 2013 - 5:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041 im Erdgeschoß

We analyze a novel principal-agent problem of moral hazard and adverse selection in continuous time. The constant private shock revealed at time zero when the agent selects the contract has a long-term impact on the optimal contract. The latter is based not only on the continuation value of the agent who truthfully reports, but also contingent upon the continuation value of the agent who misreports, called temptation value. The good agent is retired when the temptation value of the bad agent becomes large, because then it is expensive to motivate the good agent.

## On arbitrages arising with honest times

Speaker(s):
Monique Jeanblanc (Université d'Evry Val d'Essone)
Date:
Thursday, April 11, 2013 - 4:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041 im Erdgeschoß

In the context of a general continuous financial market model, we study whether the additional information associated with an \emph{honest time} $\tau$ gives rise to arbitrage profits. By relying on the theory of progressive enlargement of filtrations, we explicitly show that arbitrage profits can never be realized strictly before $\tau$, while classical arbitrage opportunities can be realized exactly at $\tau$ and stronger arbitrages of the first kind always exist after $\tau$.