**Date and Time:** 03/16/20, 4-5pm

**Room:** SH306

**Abstract: **This talk starts with an optimal inventory control problem using a long-term average criterion. In absence of ordering, the inventory process is modeled by a one-dimensional diffusion on some interval of (-∞, ∞) with general drift and diffusion coefficients and boundary points that are consistent with the notion that demands tend to reduce the inventory level. Orders instantaneously increase the inventory level and incur both positive fixed and level dependent costs. In addition, state-dependent holding/backorder costs are incurred continuously. Using average expected occupation and ordering measures and weak convergence arguments, weak conditions are given for the optimality of the (s_{*}, S_{*}) ordering policy in the general class of admissible policies. The second part of the talk is devoted to the construction of the probability measure on a single path space for an admissible intervention policy subject to an uncertain impulse mechanism. An important feature of this construction is that the intervention times are stopping times with respect to the natural filtration. In addition, when the intervention policy results in deterministic distributions for each impulse, the paths between interventions are independent and, moreover, if the same distribution is used for each impulse, then the cycles following the initial cycle are identically distributed. We also identify a class of impulse policies under which the resulting controlled process is Markov. The decision to use an (s, S) ordering policy in inventory management provides an example of an impulse policy for which the process is Markov and has i.i.d.~cycles so a benefit of the constructed model is that one is allowed to use classical renewal arguments.

This is a joint work with Kurt Helmes (Humboldt University of Berlin) and Richard Stockbridge

(University of Wisconsin-Milwaukee).

]]>**Date and Time:** 04/06/2020

**Room:** SH306

**Abstract: **We consider the case in which a trader has anticipative information about the value of a stock assumed to be driven by a Brownian motion. We show the usage of two different ways to model the problem; the first one with forward integration and the second one with Skorohod integration. Both of them are generalizations of Itô’s integration, which we cannot apply in this scope because, in principle, we do not have integrands adapted to the natural filtration of the Brownian motion unless we use suitable restrictions. We offer a simulation of the models in the real stock market when the price updates are daily. We also discuss how to get anticipative information in the scope of high-frequency trading (HFT). In this case, latency is the mean factor, and we can exploit it in macroeconomic announcements.

https://www.wpi.edu/news/wpi-mathematician-receives-prestigious-teaching-award

]]>https://www.wpi.edu/news/wpi-researcher-uses-math-help-army-protect-soldiers-chemical-attack

]]>**Date and Time:** 03/06/20, 11am-12

**Room:** SH203

**Abstract: **The complexity of large population multi-agent dynamical systems, such as occur in economics, communication systems, and environmental and transportation systems, makes centralized control infeasible and classical game theoretic solutions intractable.

In this talk we first present the Mean Field Game (MFG) theory of large population systems. Going to the infinite population limit, individual agent feedback strategies exist which yield Nash equilibria. These are given by the MFG equations consisting of (i) a Hamilton-Jacobi-Bellman equation generating the Nash values and the best response control actions, and (ii) a McKean-Vlasov-Fokker-Planck–Kolmogorov equation for the probability distribution of the states of the population, otherwise known as the mean field.

Next we shall introduce Graphon Mean Field Game and Control theory. Very large scale networks linking dynamical agents are now ubiquitous, with examples being given by electrical power grids and social media networks. In this setting, the emergence of the graphon theory of infinite networks has enabled the formulation of the Graphon Mean Field Game equations. Just as for MFG theory, it is the simplicity of the infinite population GMFG strategies which permits their application to otherwise intractable problems involving large populations and networks.

**Date and Time:** 02/24/20, 4-5pm

**Room:** SH306

**Abstract: **In this talk, we consider equilibria in the presence of asymmetric information and misinformed agents (noise traders). We establish existence of two equilibria. First, a full communication one where the informed agents’ signal is disclosed to the market, and static policies are optimal. Second, a partial communication one where the signal disclosed is affine in the informed and noise traders’ signals. Here, information asymmetry creates a demand for a dark pool with endogenous participation where private information trades can be implemented. Markets are endogenously complete and equilibrium prices have a three factor structure. Results are valid for multiple dimensions; constant absolute risk averse investors; fundamental processes following a general diffusion; and non-linear terminal payoffs. Asset price dynamics and public information flows are endogenous, and are established using multiple filtration enlargements, in conjunction with predictable representation theorems for random analytic maps. Rational expectations equilibria are special cases of the general results. This is joint work with J. Detemple and M. Rindisbacher of Boston University.

**Date and Time:** 03/23/2020, 4-5pm

**Abstract:** A common challenge faced by many institutional and retail investors is to effectively control risk exposure to various market factors. There is a great variety of indices designed to provide different types of exposures across sectors and asset classes. Some of these indices can be difficult or impossible to trade directly, but investors can trade the associated financial derivatives if they are available in the market. For example, the CBOE Volatility Index (VIX), often referred to as the fear index, is not directly tradable, but investors can gain exposure to the index and potentially hedge against market turmoil by trading futures and options written on VIX.

We develop a methodology for index tracking and risk exposure control using financial derivatives. Under a continuous-time diffusion framework for price evolution, we present a pathwise approach to construct dynamic portfolios of derivatives in order to gain exposure to an index and/or market factors that may be not directly tradable. A general tracking condition is established to relate the portfolio drift to the desired exposure coefficients under any given model. We derive the slippage process that reveals how the portfolio return deviates from the targeted return. In our multi-factor setting, the portfolio’s realized slippage depends not only on the realized variance of the index, but also the realized covariance among the index and factors. Trading strategies are implemented under a number of models using different derivatives, such as futures and options.

]]>**Date and Time:** 11/05/2019, 4-5pm

**Abstract:** Stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of stochastic gradient with momentum where a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates towards a global minimum. Many works reported its empirical success in practice for solving stochastic non-convex optimization problems, in particular it has been observed to outperform overdamped Langevin Monte Carlo-based methods such as stochastic gradient Langevin dynamics (SGLD) in many applications. Although asymptotic global convergence properties of SGHMC are well known, its finite-time performance is not well-understood. In this work, we study two variants of SGHMC based on two alternative discretizations of the underdamped Langevin diffusion. We provide finite-time performance bounds for the global convergence of both SGHMC variants for solving stochastic non-convex optimization problems with explicit constants. Our results lead to non-asymptotic guarantees for both population and empirical risk minimization problems. For a fixed target accuracy level, on a class of non-convex problems, we obtain complexity bounds for SGHMC that can be tighter than those for SGLD. These results show that acceleration with momentum is possible in the context of global non-convex optimization. This is based on the joint work with Xuefeng Gao and Mert Gurbuzbalaban.

**Date and Time:** 10/29/19, 4-5pm

**Abstract:**In this talk, we present limit theorems related to integrals of functions taken at the values of Brownian motion. These limit theorems involve regenerative processes and the local time of the Brownian motion.