**Oxford-Man Institute**

**Machine Learning and Quantitative Finance Workshop**

**Wednesday 16 ^{th} February 2022**

**At Eagle House and online**

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**Schedule of talks:**

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3.00pm – 3.05pm Welcome & Introduction to the workshop (Álvaro Cartea, OMI Director)

##### 3.05pm – 3.50pm Justin Sirignano, Associate Professor of Mathematics, University of Oxford

**Title: **Online Stochastic Optimization of SDEs

**Abstract: **We develop a new continuous-time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model’s parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of our online algorithm for linear SDE models and present numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations, which are then used to analyze the parameter fluctuations in the algorithm.

3.50pm – 4.00pm Questions and discussion

##### 4.00pm – 4.45pm Dr. Ryan Donnelly, Lecturer in Financial Mathematics, King’s College London

**Title**: Optimal Execution with Exploratory Trading

**Abstract**: An agent wishes to liquidate a block of shares subject to price impact effects, but also desires to explore regions of the state and control space that would be avoided

according to the optimal strategy of the model. This is accomplished by incorporating unpredictable randomness in the trading strategy at each point in time, and offering a reward in the form of Shannon’s differential entropy of the distribution of this random component. We propose a framework in discrete time with captures this objective and solve for the optimal distribution of trades. At each point in time the optimal trade distribution is Gaussian with parameters that are given in terms of the solution to a backwards stochastic difference equation. The solution to this backwards stochastic difference equation can be approximated by a continuous time analogue, which can be solved in closed form. Using this approximation, we demonstrate the relation between this discrete time model and other pieces of literature which work in continuous time.

4.45pm – 4.55pm Questions and discussion** **

##### 4.55pm – 5.40pm Professor Olivier Guéant, Professor of Applied Mathematics at Université Paris 1 Panthéon Sorbonne

**Title: **Risk budgeting: new results and stochastic optimization algorithms

**Abstract: **In recent years, interest has grown for portfolio construction methods that do not rely on expected returns. Among risk-based methods, the most popular ones are minimum variance, maximum diversification, and risk budgeting (especially equal risk contribution, aka. ERC). Risk budgeting is particularly attracting because of its versatility: based on the Euler decomposition of positively homogenous functions, it can be used with a large range of risk « measures », from volatility to expected shortfall/CVaR and beyond. The goal of this talk is to present new mathematical results about the construction of risk budgeting portfolios for a very wide spectrum of risk « measures » and to show that, in many cases, stochastic optimization techniques can be used to build risk budgeting portfolios.

5.40pm – 5.50pm Questions and discussion