Abstract: In this talk, we show how to use the Markov diffusion generator framework introduced by Ledoux in order to obtain limit theorems for the approximation of invariant measures of diffusions, where a sufficient condition for convergence is given in terms of a finite linear combination of moments. We will illustrate these results for the Pearson class of probability measures, and recover classical results as particular cases.

pdf Presentation from Conference (1.38 MB)

Title: Adaptive Robust Hedging Under Model Uncertainty

Abstract. We propose a new methodology, called adaptive robust control, for solving a discrete-time Markovian control problem subject to Knightian uncertainty. We apply the general framework to a financial hedging problem where the uncertainty comes from the fact that the true law of the underlying model is only known to belong to a certain family of probability laws. We provide a learning algorithm that reduces the model uncertainty through progressive learning about the unknow system. One of the pillars in the proposed methodology is the recursive construction of the confidence sets for the unknown parameter. This allows, in particular, to establish the corresponding Bellman system of equations.

document Presentation from Conference (3.40 MB)

Title: Rate Control under Heavy Traffic with Strategic Servers (a mean-field game analysis.

Joint with Erhan Bayraktar and Amarjit Budhiraja.

Abstract: We consider a large queueing system that consists of many strategic servers that are weakly interacting. Each server processes jobs from its unique critically loaded buffer and controls the rate of arrivals and departures associated with its queue to minimize its expected cost. The rates and the cost functions in addition to depending on the control action, can depend, in a symmetric fashion, on the size of the individual queue and the empirical measure of the states of all queues in the system. In order to determine an approximate Nash equilibrium for this finite player game we construct a Lasry-Lions type mean-field game (MFG) for certain reflected diffusions that governs the limiting behavior. Under conditions, we establish the convergence of the Nash-equilibrium value for the finite size queuing system to the value of the MFG. In general closed form solutions of such MFG are not available and thus numerical methods are needed. We use the Markov chain approximation method to construct approximations for the solution of the MFG and establish convergence of the numerical scheme.

pdf Presentation from Conference (878 KB)

Title: Asset pricing under optimal contracts.

Joint with Hao Xing.

Abstract: We consider the problem of finding equilibrium asset prices in a financial market in which a portfolio manager (Agent) invests on behalf of an investor (Principal), who compensates the manager with an optimal contract. We extend a model from Buffa, Vayanos and Woolley (2014), BVW (2014), by allowing general contracts. We find that the optimal contract rewards Agent for taking specific risk of individual assets and not only the systematic risk of the index by using the quadratic variation of the deviation between the portfolio return and the return of an index portfolio.

Similarly to BVW (2014), we find that, in equilibrium, the stocks in large supply have high risk premia, while the stocks in low supply have low risk premia, and this effect is stronger as agency friction increases. However, by using our risk-incentive optimal contract, the sensitivity of the price distortion to agency frictions is of an order of magnitude smaller compared to the price distortion in BVW (2014), where only contracts linear in portfolio value and index are allowed.

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Title: Eigenvalue problems arising in models with small transaction costs

Abstract: When studying option pricing or portfolio optimization in the limit of small transaction costs, one typically encounters a PDE eigenvalue problem for the correction to the limiting quantity of interest. We will present some recent results and a few open problems for a prototype of these eigenvalue problems.

pdf Presentation from Conference (1.20 MB)

Title: Optimal rebalancing frequency of functionally generated portfolios

Abstract: Functionally generated portfolios, such as constant weighted portfolios, have been shown to outperform the market capitalization index over long time horizons in a diverse volatile market. In actual applications time is discrete and one encounters the problem of determining how frequently to rebalance such a portfolio. We will show that the optimal frequency is determined by the angle at which geodesics in a certain geometry on the unit simplex intersect. The geometry is also intimately connected to an interesting Monge-Kantorovich optimal transport problem that is of independent interest.

Title: Dynamic Programming for multivariate Problems

Abstract: Multivariate risk measures appear naturally in markets with transaction costs or when measuring the systemic risk of a network of banks.

Recent research suggests that time consistency of these multivariate risk measures can be interpreted as a set-valued Bellman principle. And a more general structure emerges that might also be important for other applications and is interesting in itself. In this talk I will show that this set-valued Bellman principle holds also for the dynamic mean-risk portfolio optimization problem. In most of the literature, the Markowitz problem is scalarized and it is well known that this scalarized problem does not satisfy the (scalar) Bellman principle. However, when we do not scalarize the problem, but leave it in its original form as a vector optimization problem, the upper images, whose boundary is the efficient frontier, recurse backwards in time under very mild assumptions. I will present conditions under which this recursion can be exploited directly to compute a solution in the spirit of dynamic programming and will state some open problems and challenges for the general case.

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Title: Superprocesses, singular control problems in finance, and path-dependent PDEs

Joint with Eyal Neuman and Alexander Kalinin

Abstract: We consider a class of stochastic control problems that are connected with the problem of finding adaptive mean-variance-optimal portfolio liquidation strategies in the Almgren-Chriss framework or in currency target zone models. We give closed-form solutions to these control problems in terms of the log-Laplace transforms of certain J-functionals of (catalytic) Dawson-Watanabe superprocesses. In the case of historical superprocesses, these log-Laplace transforms are mild solutions to a semilinear path-dependent PDE, and we show that they solve this path-dependent PDE also in the viscosity sense.

Title: Applications of the analysis on Wiener space to statistical inference

Joint with Luis A. Barboza, Khalifa es-Sebaiy, and Leo Neufcourt.

Abstract: Toolsfrom the analysis on Wiener space, including Wiener chaos calculus and the Malliavin calculus, were promoted historically to help develop the theory of stochastic analysis and its applications to other parts of probability and analysis. They are becoming increasingly helpful as sharp tools for the quantitative analysis of asymptotic questions. In this talk, after a brief introduction to these tools and their connections to normal approximations, we will discuss some applications to the generalized method of moments and to extensions of least-squares ideas for parametric estimation of general Gaussian sequences and processes. Illustration of our general results and practical performance are presented in the case of Ornstein-Uhlenbeck processes driven by fractional Gaussian noise. Some of the results are accompanied by demonstrably sharp normal asymptotics. We may mention some intriguing implications for partial observation situations.

pdf Presentation from Conference (596 KB)

Title: Some Results of Time-Inconsistent Optimal Control Problems

Abstract: Time-inconsistency exists in optimal control problems. An optimal control found for a given initial time and an initial state usually will not stay optimal afterwards. The main reasons causing such a situation are people’s time-preferences and risk-preferences. One way to treat the problem is to seek equilibrium strategies, instead of optimal controls. In this talk, some recent results concerning on the problem will be presented.

pdf Presentation from Conference (3.43 MB)

Title: A Martingale Approach for Fractional Brownian Motions and Related Path Dependent PDEs

Joint with Frederi Viens

Abstract: Empirical studies show that the volatilities could be rough, which typically go beyond the semimartinagle framework and the fractional Brownian Motion (fBM) becomes a natural tool. Compared with BM, fBM has two features: (i) non-Markoivan; (ii) non-semimartingale (when the Hurst parameter $H< {1\over 2}$). We shall show that the recent development of path dependent PDEs provides a convenient tool to extend the standard literature of pricing/hedging derivatives to an fBM framework.