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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!

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  • $\begingroup$ You would simulate the process, (the SDE) and find the values of the parameters that best fit the data. You could calibrate both parameters individually using MLE, and memory, the MLE of the poisson process is just the sample mean. But I'm assuming you would want to calibrate all the parameters simultaneously, then you would need to simulate as said above or use some type of Bayesian sampling method $\endgroup$
    – THATS MY QUANT MY QUANTITATIVE
    Commented Oct 20 at 13:00
  • $\begingroup$ What does $\eta$ represent, why not use a constant? Seems overly complicated. What parameters are you estimating exactly? Also, how do you define order flow and how often is it sampled? $\endgroup$
    – Freelunch
    Commented Oct 23 at 15:23
  • $\begingroup$ lets say $\(\eta\)$ represent the volume of each agent that arrives. since the volume varies between agent to agent, the model doesn't consider it as a constant... I want to estimate the distribution F from which $\(\eta\)$ is sampled and, $(\kappa) and the parameter of the poisson distribution. I define order flow as the total volume of sell (buy). I don't understand what do you need by how often, since it is discrete anyway. $\endgroup$
    – Quant master
    Commented Oct 23 at 20:50
  • $\begingroup$ That doesn't make sense, total volume is monotonically increasing so there is no mean reversion. $\endgroup$
    – Freelunch
    Commented Oct 24 at 7:07
  • $\begingroup$ I mean the total volume in a given time, like the total volume in t=5 or in t=8.0643... $\endgroup$
    – Quant master
    Commented Oct 24 at 12:45

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