Non-average price impact in order-driven markets

Bellani, Claudio; Brigo, Damiano; Pakkanen, Mikko S.; Sánchez-Betancourt, Leandro

doi:10.2139/ssrn.4012167

Cited by 1 publication

(4 citation statements)

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“…Finally, the online optimization algorithm can be used to train SDE models (including point process models) of limit order books (Bellani et al., 2021; Kumar, 2021; Lu and Abergel, 2018; Morariu‐Patrichi & Pakkanen, 2022; Shi & Cartlidge, 2022). Order books involve large numbers of high‐frequency events (

\sim 10^5 - 10^6

events per day per stock) and high‐dimensional dynamics (many price levels, each with limit order submissions and cancellations, as well as market orders, hidden orders, and transactions).…”

Section: Numerical Performance Of the Online Algorithmmentioning

confidence: 99%

“…Recent examples of such model frameworks for the simulation of the order books include Morariu‐Patrichi and Pakkanen (2022), Bellani et al. (2021), Shi and Cartlidge (2022), Lu and Abergel (2018), Kumar (2021). Morariu‐Patrichi and Pakkanen (2022), Bellani et al.…”

Section: Numerical Performance Of the Online Algorithmmentioning

confidence: 99%

“…Morariu‐Patrichi and Pakkanen (2022), Bellani et al. (2021), Shi and Cartlidge (2022), Lu and Abergel (2018), and Kumar (2021) develop stochastic point process models to model the event‐by‐event dynamics in order books.…”

Section: Numerical Performance Of the Online Algorithmmentioning

confidence: 99%

“…Multidimensional point process models can model the dynamics of the entire order book (e.g., limit order submissions, cancellations, market orders, hidden orders, and transactions) Morariu‐Patrichi and Pakkanen (2022), Bellani et al. (2021).…”

Section: Numerical Performance Of the Online Algorithmmentioning

confidence: 99%

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Continuous‐time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

Wang,

Sirignano

2023

Mathematical Finance

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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 using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multidimensional Ornstein–Uhlenbeck process) and present its 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 (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high‐dimensional optimal control of SDEs and training stochastic point process models of limit order book events.

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\sim 10^5 - 10^6

events per day per stock) and high‐dimensional dynamics (many price levels, each with limit order submissions and cancellations, as well as market orders, hidden orders, and transactions).…”

Section: Numerical Performance Of the Online Algorithmmentioning

confidence: 99%