Stochastics Seminar

Academic Year 2018-2019                                                                                               Previous Semesters: AY2017-2018  Spring 2017

Fall 2018                                                                                     

 Date/Time/Location:  August 30, 2018 11:00-11:50 a.m. @ ECOT 226
 Speaker:  Nils Detering (Department of Statistics and Applied Probability, University of California at Santa Barbara)
 Title:   Managing Default Contagion In Inhomogeneous Financial Networks
 Abstract: The aim of this paper is to quantify and manage systemic risk caused by default contagion in the interbank market. Our results allow us to determine the impact of local shocks to the entire system and the wider economy. As a central application, we characterize resilient and non-resilient cases. In particular, for the prominent case where the network has a degree sequence without second moment, we show that a small number of initially defaulted banks can trigger a substantial default cascade. Paralleling regulatory discussions, we determine minimal capital requirements for financial institutions sufficient to make the network resilient to small shocks. It is joint work with Thilo Meyer-Brandis, Konstantinos Panagiotou and Daniel Ritter (all University of Munich)

Date/Time/Location:  September 20, 2018 11:00-11:50 a.m. @ ECOT 226
 Speaker:  Matteo Basei (Department of Industrial Engineering and Operations Research, University of California at Berkeley)
 Title:   Nonzero-sum stochastic differential games with impulse controls
 Abstract: We consider a general class of nonzero-sum impulsive games with N players. By means of a suitable system of quasi-variational  inequalities, we provide a verification theorem for the equilibrium  strategies and the value functions of the game. In particular, we focus on the regularity conditions required by the theorem. We then  present some practical applications. Finally, we focus on some ongoing  extensions and generalizations.

Date/Time/Location:  October 18, 2018 11:00-11:50 a.m. @ ECOT 226
 Speaker:  Ruimeng Hu (Department of Statistics, Columbia University)
 Title:   Optimal Portfolio under Fractional Stochastic Environments
 Abstract: Rough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non-Markovian) fractional stochastic environment (for all Hurst index H \in (0,1)). We rigorously establish a first-order approximation of the optimal value, where the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein-Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Similar results are also obtained under fast mean-reverting fractional stochastic environment. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.

Date/Time/Location:  October 25, 2018 11:00-11:50 a.m. @ ECOT 226
 Speaker:  Yerkin Kitapbayev (Sloan School of Management, MIT)
 Title:   American option pricing under stochastic volatility models via Picard iterations
 Abstract: This paper studies the valuation of American options for a general one-factor stochastic volatility model. Using the local time-space calculus on surfaces we derive an early exercise premium representation for the option price, parametrized by the optimal exercise surface. The exercise surface is the unique solution to an integral equation of Volterra type. The paper proposes a new numerical scheme to solve the integral equation based on the Picard iterations method. The method is flexible and can handle a wide class of non-affine models. Performance is illustrated for the Black-Scholes, Heston and 3/2 models. The approach provides fast convergence, simple implementation and good runtime/RMSE tradeoff and can be extended to other multi-dimensional stopping problems  (joint work with J. Detemple and L. Zhang).

Date/Time/Location:  November 8, 2018 11:00-11:50 a.m. @ ECOT 226
 Speaker:  Joshua Aurand (Department of Applied Mathematics, University of Colorado Boulder)
 Title:   Epstein-Zin Utility Maximization over Random Horizon
 Abstract: This talk focuses on solving the consumption-investment problem for an agent with stochastic differential utility of Epstein-Zin type. In contrast to prior literature, our time horizon is random, taken from the class of stopping times in the augmented Brownian filtration. Parameter specification is empirically relevant, with both relative risk-aversion and the elasticity of intertemporal substitution (EIS) in excess of one. The theory of BSDE (backward stochastic differential equations) is used to establish existence and uniqueness of the continuation value process, while martingale method is employed to derive optimal strategies. Relevant examples in decision theory will be discussed (Joint work with Yu-Jui Huang).

Date/Time/Location:  November 29, 2018 11:00-11:50 a.m. @ ECOT 226
 Speaker:  Saeed Khalili (Department of Mathematics, University of Colorado Boulder)
 Title:   Optimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints
 Abstract: This paper investigates optimal consumption in the stochastic Ramsey problem with the Cobb-Douglas production function. Contrary to prior studies, we allow for general consumption processes, without any a priori boundedness constraint. A non-standard stochastic differential equation, with neither Lipschitz continuity nor linear growth, specifies the dynamics of the controlled state process. A mixture of probabilistic arguments are used to construct the state process, and establish its non-explosiveness and strict positivity. This leads to the optimality of a feedback consumption process, defined in terms of the value function and the state process. Based on additional viscosity solutions techniques, we characterize the value function as the unique classical solution to a nonlinear elliptic equation, among an appropriate class of functions. This characterization involves a condition on the limiting behavior of the value function at the origin, which is the key to dealing with unbounded consumptions. Finally, relaxing the boundedness constraint is shown to increase, strictly, the expected utility at all wealth levels (Joint work with Yu-Jui Huang).

 Date/Time/Location:  December 13, 2018 11:00-11:50 a.m. @ ECOT 226
 Speaker:  Tien Khai Nguyen (Department of Mathematics, North Carolina State University)
 Title:   A Stochastic Model of Optimal Debt Management and Bankruptcy
 Abstract: Consider a problem of optimal debt management which is modeled as a non-cooperative game between a borrower and a pool of risk-neutral lenders. Since the debtor may go bankrupt, lenders charge a higher interest rate to offset the possible loss of part of their investment. In this talk, I will present results on existence and properties of optimal strategies, both in a deterministic and in a stochastic framework.