Forthcoming events in this series


Thu, 14 Nov 2019

16:00 - 17:00
L4

Viscosity solutions for controlled McKean-Vlasov jump-diffusions

Matteo Burzoni
((Oxford University))
Abstract

We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean-Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution. Based on a joint work with V. Ignazio, M. Reppen and H. M. Soner

Thu, 07 Nov 2019

16:00 - 17:00
L4

Sensitivity Analysis of the Utility Maximization Problem with Respect to Model Perturbations

Oleksii Mostovyi
(University of Connecticut)
Abstract

First, we will give a brief overview of the asymptotic analysis results in the context of optimal investment. Then, we will focus on the sensitivity of the expected utility maximization problem in a continuous semimartingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled by a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and construct trading strategies that match the indirect utility function up to the second order. If a risk-tolerance wealth process exists, using it as numeraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita–Watanabe decomposition. Then we discuss possible extensions and special situations, in particular, the power utility case and models that admit closed-form solutions. The central part of this talk is based on the joint work with Mihai Sirbu.

Thu, 31 Oct 2019

16:00 - 17:00
L4

On a mean-field optimal control problem.

Vardan Voskanyan
(Centro de Matemática da Universidade de Coimbra)
Abstract

In this talk we consider a mean field optimal control problem with an aggregation-diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton-Jacobi and a Fokker-Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. We will discuss the existence and regularity of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker-Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.

Thu, 17 Oct 2019

14:00 - 15:00
L5

Deep Learning: Asymptotics and Financial Applications

Justin Sirignano
(University of Illinois)
Abstract

Deep learning has revolutionized image, text, and speech recognition. Motivated by this success, there is growing interest in developing deep learning methods for financial applications. We will present some of our recent results in this area, including deep learning models of high-frequency data. In the second part of the talk, we prove a law of large numbers for single-layer neural networks trained with stochastic gradient descent. We show that, depending upon the normalization of the parameters, the law of large numbers either satisfies a deterministic partial differential equation or a random ordinary differential equation. Using similar analysis, a law of large numbers can also be established for reinforcement learning (e.g., Q-learning) with neural networks. The limit equations in each of these cases are discussed (e.g., whether a unique stationary point and global convergence can be proven).  

Thu, 10 Oct 2019

16:00 - 17:00
L4

Universal Approximation with Deep Narrow Networks

Patrick Kidger
(University of Oxford)
Abstract

The classical Universal Approximation Theorem certifies that the universal approximation property holds for the class of neural networks of arbitrary width. Here we consider the natural `dual' theorem for width-bounded networks of arbitrary depth, for a broad class of activation functions. In particular we show that such a result holds for polynomial activation functions, making this genuinely different to the classical case. We will then discuss some natural extensions of this result, e.g. for nowhere differentiable activation functions, or for noncompact domains.
 

Thu, 20 Jun 2019

13:00 - 14:00
L3

Spectral methods for certain inverse problems on graphs and time series data

Mihai Cucuringu
(Statistics Oxford University)
Further Information

We study problems that share an important common feature: they can all be solved by exploiting the spectrum of their corresponding graph Laplacian. We first consider a classic problem in data analysis and machine learning, of establishing a statistical ranking of a set of items given a set of inconsistent and incomplete pairwise comparisons. We formulate the above problem of ranking with incomplete noisy information as an instance of the group synchronization problem over the group SO(2) of planar rotations, whose least-squares solution can be approximated by either a spectral or a semidefinite programming relaxation, and consider an application to detecting leaders and laggers in financial multivariate time series data. An instance of the group synchronization problem over Z_2 with anchor information is broadly applicable to settings where one has available a sparse signal such as positive or negative news sentiment for a subset of nodes, and would like to understand how the available measurements propagate to the remaining nodes of the network. We also present a simple spectral approach to the well-studied constrained clustering problem, which captures constrained clustering as a generalized eigenvalue problem with graph Laplacians. This line of work extends to the setting of clustering signed networks and correlation clustering, where the edge weights between the nodes of the graph may take either positive or negative values, for which we provide theoretical guarantees in the setting of a signed stochastic block model and numerical experiments for financial correlation matrices. Finally, we discuss a spectral clustering algorithm for directed graphs based on a complex-valued representation of the adjacency matrix, motivated by the application of extracting cluster-based lead-lag relationships in time series data.
 

Thu, 23 May 2019

13:00 - 14:00
L4

Monotone Solutions to the Moral Hazard Problem

Hanqing Jin
((Oxford University))
Abstract

We investigate monotone solutions of the moral hazard problems without the monotone likelihood ratio property. The solutions are explicitly characterised by a concave envelope relaxation approach for a two-action model in which the principal is risk neutral or exhibits constant absolute risk aversion.  

Thu, 09 May 2019

13:00 - 14:00
L4

Talks by Dphil students

Theerawat Bhudisaksang & Yufei Zhang (DPhil students)
Abstract

Theerawat Bhudisaksang
----------------------

Adaptive robust control with statistical learning

We extend the adaptive robust methodology introduced in Bielecki et al. and propose a continuous-time version of their approach. Bielecki et al. consider a model in which the distribution of the underlying (observable) process depends on unknown parameters and the agent uses observations of the process to estimate the parameter values. The model is made robust to misspecification because the agent employs a set of ambiguity measures that contains measures where the parameter are inside a confidence region of their estimator. In our extension, we construct the set of ambiguity measures such that each probability measure in the set has a semimartingale characterisation lies in a restricted set. Finally, we prove the dynamic programming principle of the adaptive robust control in continuous time problem using measurable selection theorems, and we show that the value function can be characterised as the solution of a non-linear partial differential equation.

Yufei Zhang
-----------

A neural network based policy iteration algorithm with global convergence of values and controls for stochastic games on domains

In this talk, we propose a class of neural network based numerical schemes for solving semi-linear Hamilton-Jacobi-Bellman-Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit a policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the H^2-norm, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to HJBI boundary value problems corresponding to controlled diffusion processes with oblique boundary reflection. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method. 
 

Thu, 02 May 2019

13:00 - 14:00
L4

A class of stochastic games and moving free boundary problems

Renyuan Xu
(Berkeley)
Abstract

Stochastic control problems are closely related to free boundary problems, where both the underlying fully nonlinear PDEs and the boundaries separating the action and waiting regions are integral parts of the problems. In this talk, we will propose a class of stochastic N-player games and show how the free boundary problems involve moving boundaries due to the additional game nature. We will provide explicit Nash equilibria by solving a sequence of Skorokhod problems. For the special cases of resource allocation problems, we will show how players change their strategies based on different network structures between players and resources. We will also talk about the insights from a sharing economy perspective. This talk is based on a joint work with Xin Guo (UC Berkeley) and Wenpin Tang (UCLA).

Thu, 07 Mar 2019

13:00 - 14:00
L4

Optimal execution with rough path signatures

Imanol Perez
((Oxford University))
Further Information


 

Abstract

We consider a well-studied optimal execution problem under little assumptions on the underlying midprice process. We do so by using signatures from rough path theory, that allows converting the original problem into a more computationally tractable problem. We include a few numerical experiments where we show that our methodology is able to retrieve the theoretical optimal execution speed for several problems studied in the literature, as well as some cases not included in the literatture. We also study some estensions of our framework to other settings.
 

Thu, 14 Feb 2019

13:00 - 14:00
L4

Pathwise functional portfolio generation and optimal transport

Micheal Monoyios
((Oxford University))
Further Information

We make precise a remarkable connection, first observed by Pal and Wong (2016) and further analysed in the doctoral thesis of Vervuurt (2016), between functionally generated investments and optimal transport, in a model-free discrete-time financial market. A functionally generated portfolio (FGP) computes the investment in each stock through the prism of the super-differential of the logarithm of a concave function (the generating function of the FGP) of the market weight vector. Such portfolios have been shown to outperform the market under suitable conditions. Here, in our pathwise discrete-time scenario, we equate the convex-analytic cyclical monotonicity property characterising super-differentials, with a $c$-cyclical monotonicity property of the unique Monge solution of an appropriately constructed optimal transport problem with cost function $c$, which transfers the market portfolio distribution to the FGP distribution. Using the super-differential characterisation of functional investments, we construct optimal transport problems for both traditional (multiplicative) FGPs, and an ``additive'' modification introduced by Karatzas and Ruf (2017), featuring the same cost function in both cases, which characterise the functional investment. In the multiplicative case, the construction differs from Pal and Wong (2016) and Vervuurt (2016), who used a ``multiplicative'' cyclical monotonicity property, as opposed to the classical cyclical monotonicity property used here.
  
We establish uniqueness of the solution to the relevant optimal transport problem, elevating the connection observed by Pal and Wong (2016) to an exact equivalence between optimal transport and functional generation. We explore ramifications, including pathwise discrete-time master equations for the evolution of the relative wealth of the investment when using the market portfolio as numeraire. We take the pathwise continuous time limit, assuming continuous paths which admit well-defined quadratic variation, to establish model-free continuous-time master equations for both types of functionally generated investment, providing an alternative derivation to the recent proof of Schied et al (2018) of the master equation for multiplicative FGPs, as well as an extension to the case of additive functionally generated trading strategies.

Thu, 24 Jan 2019

13:00 - 14:00
L4

Talks by Dphil students

Tanut Treetanthiploet and Julien Vaes (Dphil students)
Abstract

Tanut Treetanthiploet
---------------------
Exploration vs Exploitation under Statistical Uncertainty

The exploration vs Exploitation trade-off can be quantified and studied through the notion of statistical uncertainty using the theory of nonlinear expectations. The dynamic allocation problem of multi-armed bandits will be discussed. In the case of a finite state space in discrete time, we can describe the value function in terms of the solution to a discrete BSDE and obtain a similar notion to the Bellman equation. We also give an approximation scheme to evaluate decisions in the simple setting.


Julien Vaes
-----------
Optimal Execution Strategy Under Price and Volume Uncertainty

In the seminal paper on optimal execution of portfolio transactions, Almgren and Chriss define the optimal trading strategy to liquidate a fixed volume of a single security under price uncertainty. Yet there exist situations, such as in the power market, in which the volume to be traded can only be estimated and becomes more accurate when approaching a specified delivery time. To meet the need of efficient strategies in these situations, we have developed  a model that accounts for volume uncertainty and show that a risk-averse trader has benefit in delaying their trades. We show that the optimal strategy is a trade-off between early and late trades to balance risk associated to both price and volume. With the incorporation of a risk term for the volume to trade, the static optimal strategies obtained with our model avoid the explosion in the algorithmic complexity associated to dynamic programming solutions while yielding to competitive performance.

 

Thu, 29 Nov 2018

13:00 - 14:00
L4

OPTIMAL CONTROL UNDER CONTROLLED-LOSS CONSTRAINTS VIA REACHABILITY APPROACH AND COMPACTIFICATION

Geraldine Bouveret
(Smith School of Enterprise and the Environment)
Abstract

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton-Jacobi-Bellman equation usually calls for additional strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then apply a level-set approach to describe the reachable set. With this approach, the state constraints can be managed through an exact penalization technique. However, a new set of state and control variables enters the definition of this stochastic target problem. In particular, those controls are unbounded. A “compactification” of the problem is then performed. (joint work with Athena Picarelli)
 

Thu, 08 Nov 2018

13:00 - 14:00
L4

Talks by graduate students

Donovan Platt and Yufei Zhang (DPhil students)
((Oxford University))
Abstract

Donovan Platt
-------------
Economic Agent-Based Model Calibration

Interest in agent-based models of financial markets and the wider economy has increased consistently over the last few decades, in no small part due to their ability to reproduce a number of empirically-observed stylised facts that are not easily recovered by more traditional modelling approaches. Nevertheless, the agent-based modelling paradigm faces mounting criticism, focused particularly on the rigour of current validation and calibration practices, most of which remain qualitative and stylised fact-driven. While the literature on quantitative and data-driven approaches has seen significant expansion in recent years, most studies have focused on the introduction of new calibration methods that are neither benchmarked against existing alternatives nor rigorously tested in terms of the quality of the estimates they produce. We therefore compare a number of prominent ABM calibration methods, both established and novel, through a series of computational experiments in an attempt to determine the respective strengths and weaknesses of each approach and the overall quality of the resultant parameter estimates. We find that Bayesian estimation, though less popular in the literature, consistently outperforms frequentist, objective function-based approaches and results in reasonable parameter estimates in many contexts. Despite this, we also find that agent-based model calibration techniques require further development in order to definitively calibrate large-scale models.

Yufei Zhang
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A penalty scheme and policy iteration for stochastic hybrid control problems with nonlinear expectations

We propose a penalty method for mixed optimal stopping and control problems where the objective is evaluated
by a nonlinear expectation. The solution and free boundary of an associated HJB variational inequality are constructed from a sequence
of penalized equations, for which the penalization error is estimated. The penalized equation is then discretized by a class of semi-implicit
monotone approximations. We further propose an efficient iterative algorithm with local superlinear convergence for solving the discrete
equation. Numerical experiments are presented for an optimal investment problem under ambiguity to demonstrate the effectiveness of
the new schemes.  Finally, we extend the penalty schemes to solve stochastic hybrid control problems involving impulse controls.

Thu, 18 Oct 2018

13:00 - 14:00
L4

Dynamic clearing and contagion in an Eisenberg-Noe framework

Zachary Feinstein
((Washington University in St. Louis))
Abstract

We will consider an extension of the Eisenberg-Noe model of financial contagion to allow for time dynamics in both discrete and continuous time. Mathematical results on existence and uniqueness of firm wealths under discrete and continuous-time will be provided. The financial implications of time dynamics will be considered, with focus on how the dynamic clearing solutions differ from those of the static Eisenberg-Noe model.
 

Fri, 01 Jun 2018

13:00 - 14:00
L6

Multilevel Monte Carlo for Estimating Risk Measures

Mike Giles
Abstract

Joint work with Abdul-Lateef Haji-Ali

This talk will discuss efficient numerical methods for estimating the probability of a large portfolio loss, and associated risk measures such as VaR and CVaR. These involve nested expectations, and following Bujok, Hambly & Reisinger (2015) we use the number of samples for the inner conditional expectation as the key approximation parameter in the Multilevel Monte Carlo formulation. The main difference in this case is the indicator function in the definition of the probability. Here we build on previous work by Gordy & Juneja (2010) who analyse the use of a fixed number of inner samples, and Broadie, Du & Moallemi (2011) who develop and analyse an adaptive algorithm. I will present the algorithm, outline the main theoretical results and give the numerical results for a representative model problem. I will also discuss the extension to real portfolios with a large number of options based on multiple underlying assets.

Fri, 18 May 2018

13:00 - 14:00
L6

A probabilistic approach to non-parametric local volatility

Martin Tegner
Abstract

The local volatility model is a celebrated model widely used for pricing and hedging financial derivatives. While the model’s main appeal is its capability of reproducing any given surface of observed option prices—it provides a perfect fit—the essential component of the model is a latent function which can only be unambiguously determined in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested involving steps of interpolation and extrapolation, most often of parametric form and with point-estimates as result. We seek to look at the calibration problem in a probabilistic framework with a nonparametric approach based on Gaussian process priors. This immediately gives a way of encoding prior believes about the local volatility function, and a hypothesis model which is highly flexible whilst being prone to overfitting. Besides providing a method for calibrating a (range of) point-estimate, we seek to draw posterior inference on the distribution over local volatility to better understand the uncertainty attached with the calibration. Further, we seek to understand dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this gives us means of inferring predictive distributions not only locally, but also for entire surfaces forward in time.

Fri, 04 May 2018

13:00 - 14:00
L6

Talks by Phd Students

Leandro Sánchez Betancourt and Jasdeep Kalsi
Abstract

Leandro Sánchez Betancourt
--------------------------
The Cost of Latency: Improving Fill Ratios in Foreign Exchange Markets

Latency is the time delay between an exchange streaming market data to a trader, the trader processing information and deciding to trade, and the exchange receiving the order from the trader.  Liquidity takers  face  a  moving target problem as a consequence of their latency in the marketplace -- they send marketable orders that aim at a price and quantity they observed in the LOB, but by the time their order was processed by the Exchange, prices (and/or quantities) may have worsened, so the  order  cannot  be  filled. If liquidity taking orders can walk the limit order book (LOB), then orders that arrive late may still be filled at worse prices. In this paper we show how to optimally choose the discretion of liquidity taking orders to walk the LOB. The optimal strategy balances the tradeoff between the costs of walking the LOB and targeting  a desired percentage of filled orders over a period of time.  We employ a proprietary data set of foreign exchange trades to analyze the performance of the strategy. Finally, we show the relationship between latency and the percentage of filled orders, and showcase the optimal strategy as an alternative investment to reduce latency.

Jasdeep Kalsi
-------------
An SPDE model for the Limit Order Book

I will introduce a microscopic model for the Limit Order Book in a static setting i.e. in between price movements. Here, order flow at different price levels is given by Poisson processes which depend on the relative price and the depth of the book. I will discuss how reflected SPDEs can be obtained as scaling limits of such models. This motivates an SPDE with reflection and a moving boundary as a model for the dynamic Order Book. An outline for how to prove existence and uniqueness for the equation will be presented, as well as some simple simulations of the model.