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Quantitative Finance · المسرد

ما معنى Multivariate Hawkes process؟

Definition 7.7 Quantitative Methods · الفصل 7 — Point Processes and Hawkes Processes

A multivariate Hawkes process has components N(1),…,N(d)N^{(1)}, \dots, N^{(d)} with intensities λt(i)=μi+∑j∑tk(j)<tgij(t−tk(j))\lambda^{(i)}_t = \mu_i + \sum_j\sum_{t^{(j)}_k < t}g_{ij}(t - t^{(j)}_k); the branching matrix is G=(∥gij∥1)G = (\lVert g_{ij}\rVert_1), and a stationary version exists when its spectral radius is below one.

Left: one minute of a simulated trading day with = 0.1, = 0.7 and = 1 per second (branching ratio 0.7): the intensity jumps by at each event (ticks) and decays toward the baseline (dashed). Right: variance-to-mean ratio of the counts in bins of length , from the covariance density and in the simulated day, against one for a Poisson process (dashed), rising to 1/(1 - 0.7)2 = 11.1. Data: the chapter’s tutorial, seeded.
Figure 7.2. Left: one minute of a simulated trading day with μ=0.1\mu = 0.1, α=0.7\alpha = 0.7 and β=1\beta = 1 per second (branching ratio 0.7): the intensity jumps by α\alpha at each event (ticks) and decays toward the baseline (dashed). Right: variance-to-mean ratio of the counts in bins of length τ\tau, from the covariance density and in the simulated day, against one for a Poisson process (dashed), rising to 1/(1−0.7)2=11.11/(1 - 0.7)^2 = 11.1. Data: the chapter’s tutorial, seeded.

أمثلة

Example 7.8 (Buys and sells that excite each other)

Let buys and sells each have baseline 0.10.1 per second and exponential kernels with β=1\beta = 1, a buy raising the buy intensity by 0.20.2 and the sell intensity by 0.40.4, and symmetrically. The branching matrix G=(0.20.40.40.2)G = \bigl(\begin{smallmatrix}0.2 & 0.4\\0.4 & 0.2\end{smallmatrix}\bigr) has eigenvalues 0.60.6 and −0.2-0.2, so the process is stable; taking expectations as in Proposition 7.5, the mean intensities solve λˉ=μ+Gλˉ\bar\lambda = \mu + G\bar\lambda, giving λˉ=(I2−G)−1μ=(0.25,0.25)\bar\lambda = (I_2 - G)^{-1}\mu = (0.25, 0.25) per second. Cross-excitation dominates: of the triggered activity on each side, two thirds comes from the other side’s trades.

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