Modeling Buy and Sell Order Flows with a Jump Ornstein-Uhlenbeck Process
Summary
The document poses a parameter-estimation problem for positive and negative trading flows modeled as independent jump Ornstein-Uhlenbeck processes. Each flow decays toward zero at a mean-reversion rate, while Poisson events introduce jumps whose sizes are drawn from an independent, identically distributed sequence. The model also distinguishes these flow processes from the Brownian motion driving the mid-price.
The author has observed time series of buy and sell flows and asks how to fit them to the specified dynamics, including both the mean-reversion and jump components. No estimation procedure, fitted parameters, dataset, or empirical results are provided, so the document does not resolve the question. It serves as a clearly stated modeling setup and research problem; practical estimation would require additional choices, such as how to identify jumps and specify or estimate their size distribution.
Key ideas
- The proposed model represents buy and sell order flows as separate mean-reverting jump processes.
- Mean reversion is governed by a decay parameter, while jump arrivals follow Poisson processes.
- Jump sizes are modeled as independent draws from a distribution.
- The document states an estimation question but gives no fitting method or empirical findings.
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# Estimating Parameters for a Jump Ornstein-Uhlenbeck Process from Positive and Negative Order Flow Time Series
# Estimating Parameters for a Jump Ornstein-Uhlenbeck Process from Positive and Negative Order Flow Time Series
Question: Estimating Parameters for a Jump Ornstein-Uhlenbeck Process from Positive and Negative Order Flow Time Series
I’m working with a model of buy and sell order flows that are described as stochastic mean-reverting processes with jumps, following a Jump Ornstein-Uhlenbeck (OU) process. The setup of the problem is as follows:
μ + represents the positive (buy) order flow. μ − represents the negative (sell) order flow. The dynamics of the order flows are governed by the following SDE:
$$ d\mu_t^{\pm} = -\kappa \mu_t^{\pm} dt + \eta_{1+N_{t^-}^{\pm}} dN_t^{\pm}, $$
$ N^\pm_t $ are independent Poisson processes with intensity $\lambda$.$$$$ $\{\eta^\pm_1, \eta^\pm_2, ...\}$ are i.i.d random variables, with distribution function F - representing jumps in trading volume. All are independent of $N^\pm_t$ and of $W_t$ (the Brownian motion which drives the mid-price).
My Question: I have time series data for both the positive and negative order flows, μ + and μ − , and I’m trying to estimate the parameters for this Jump OU process. Specifically:
How can I match the order flow data to the given OU process form? I want to ensure that both the mean-reversion (Ornstein-Uhlenbeck) behavior and the jump dynamics are accurately captured.
Thanks in advance!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.