Hawkes Processes for Order Arrival and Market Event Modeling

Bai viet ve dau van tay ky thuat so su dung vector ba con so — hawkes_mu, hawkes_alpha, hawkes_beta — lam thanh phan thoi gian trong dau hieu hanh vi cua mot nguoi tham gia, dong thoi khang dinh rang market maker co tinh tu kich thich manh. Bai viet do khong noi ba con so nay den tu dau, duoc uoc luong tu du lieu nao, hay nen tin chung den muc nao. Bai viet nay la nua con thieu: bo uoc luong, cac chan doan do phu hop, va dai luong suy ra duy nhat hoa ra quan trong hon chinh ba tham so — ty so nhanh $n$.

Khẳng dinh duoc kiem tra o day hep va co the kiem chung. Dong lenh co tinh tu kich thich: mot lenh ban quyet liet lam cho lenh ban quyet liet tiep theo co kha nang xay ra cao hon, theo cach ma mo hinh Poisson voi bat ky cuong do thay doi theo thoi gian nao cung khong the tai tao. Neu dung, mot qua trinh Hawkes khop voi dau thoi gian giao dich tho phai (a) vuot baseline Poisson ve kha nang xay ra tren du lieu giu lai, (b) tao ra ty so nhanh nam nghiem ngat giua 0 va 1, va (c) vuot qua kiem dinh do phu hop bang bien doi thoi gian. Hai diem (a) va (b) duoc bao cao rong rai cho hop dong tuong lai co phieu. Diem (c) la noi cac bai Hawkes da cong bo thuong im lang, va la noi bang giao dich crypto chua tung duoc kiem tra tren blog nay.

Blog nay da co hai phan kien thuc nen khong suy ra lai o day. Cau truc phan hoi tu kich thich — cac su kien kich hoat them cac su kien cung loai, duoi nguong thi chua vuot nguong va vuot nguong thi chay khong kiem soat — duoc suy ra cu the thanh so sinh san $g = \rho/\Delta$ trong cac chuoi thanh ly lam tin hieu giao dich, bao gom tong bien dong theo cap so nhan, che do flash crash sieu toi han, va bo ngat mach nhu mot trong ba yeu to gioi han no. Ty so nhanh $n$ ben duoi la phien ban qua trinh diem cua $g$. Diem moi thuc su o day: $n$ co the uoc luong theo thoi gian thuc chi tu dau thoi gian giao dich, khong can ban do thanh ly, do sau so lenh, hay gia dinh ve don bay. Do la toan bo ly do can dung bo may qua trinh diem.

Tien de rang thoi gian dong ho la truc lay mau sai la lap luan nen tang cua vuot qua cac thanh thoi gian, bai viet nay giai quyet bang cach xay dung cac thanh tot hon. Bai viet nay giai quyet theo cach khac: khong lay mau, ma mo hinh truc tiep thoi diem xay ra su kien.

Co so qua trinh diem, tom tat

Qua trinh diem la mot tap ngau nhien cac thoi diem su kien, duoc dinh nghia boi cuong do co dieu kien $\lambda^*(t)$ — toc do den tuc thoi khi biet toan bo lich su $\mathcal{F}_t$:

$\lambda^*(t) = \lim_{\Delta t \to 0} \frac{\mathbb{E}[N(t, t + \Delta t) \mid \mathcal{F}_t]}{\Delta t}$

Qua trinh Poisson dong nhat la truong hop $\lambda^*(t) = \mu$. Qua trinh Poisson khong dong nhat cho phep $\lambda(t)$ di theo mot dang xac dinh (vi du, duong cong hinh chu U trong ngay), nhung van gia dinh tinh doc lap co dieu kien: biet mot giao dich vua xay ra khong cho ban biet gi ve viec giao dich khac co sap xay ra hay khong. Chinh gia dinh nay bi dong lenh thuc te vi pham, va su vi pham do la toan bo chu de cua bai viet. Dau sao trong $\lambda^*(t)$ chinh la khac biet cot loi — no danh dau su phu thuoc vao lich su.

Qua trinh Hawkes

Mot qua trinh Hawkes don bien co cuong do

$\lambda^*(t) = \mu + \sum_{t_i < t} \phi(t - t_i)$

voi $\mu > 0$ la cuong do nen (cac su den ngoai sinh) va $\phi(\cdot) \geq 0$ la kernel kich thich. Moi su kien lam cuong do tang them $\phi(0)$, sau do suy giam theo dang cua $\phi$. Mot cum su kien gan day tao ra cuong do cao, khien cac su kien tiep theo co kha nang cao hon — qua trinh ghi nho lich su cua chinh no.

Ty so nhanh

$n = \int_0^\infty \phi(t) \, dt$

$n$ la so su kien con trung binh duoc mot su kien cha kich hoat. Tinh dung yen doi hoi $n < 1$; cuong do trung binh dung yen la $\bar{\lambda} = \mu / (1-n)$, tang vo han khi $n \to 1$. Vi vay $n$ tach tong hoat dong thanh phan ngoai sinh $(1-n)$ va phan noi sinh $n$: tai $n = 0.7$, bay muoi phan tram su kien la phan ung voi su kien khac thay vi phan ung voi bat ky dieu gi ben ngoai thi truong.

Day chinh xac la su tach biet duoi toi han/sieu toi han cua he so nhan chuoi trong cac chuoi thanh ly, nhung duoc tiep can tu huong khac. Khac biet nam o chi phi tinh toan: $g = \rho/\Delta$ can ban do tien cua mat do ban bat buoc va do sau so lenh truc tiep, ca hai deu la uoc luong xep len cac gia dinh ve phan phoi don bay. $n$ chi can danh sach cac dau thoi gian.

Filimonov va Sornette (2012) bao cao $n$ tang tu khoang 0.3 len tren 0.7 trong hop dong tuong lai E-mini S&P 500 tu 1998 den 2007. Hardiman, Bercot va Bouchaud (2013) dat cung hop dong o gan $n \approx 0.8$ voi kernel luy thua co so mu khoang $-1.15$ tai do tre ngan. Ca hai la ket qua tren hop dong tuong lai co phieu tu hon mot thap ky truoc, va khong ket qua nao co the chuyen sang crypto ma khong do luong. Lap lai chung khong phai muc dich cua bai viet; muc dich la kiem tra lieu cung che do co ton tai tren bang hop dong tuong lai vinh vien 24/7 voi nguoi tham gia khac va khong co gian doan mo cua thi truong hay khong.

Lua chon kernel

Mu, $\phi(t) = \alpha e^{-\beta t}$, voi $n = \alpha/\beta$. Co tinh Markov, vi vay kha nang xay ra co phep de quy $O(1)$ moi su kien (ben duoi). Day la diem bat dau cua moi lan khop thuc te.

Luy thua, $\phi(t) = \alpha c^p / (c+t)^{1+p}$. Tai tao su gom cum co nho dai, nhung mat phep de quy — danh gia kha nang xay ra tro thanh $O(N^2)$ hoac can cat bot.

Tong cac kernel mu, $\phi(t) = \sum_{k=1}^{K} \alpha_k e^{-\beta_k t}$. Xap xi suy giam theo luat luy thua bang $K$ thang thoi gian trong khi van giu phep de quy. Thuong $K = 3$ den $5$ la du, voi $\beta_k$ trai tu mili giay den vai phut.

Lieu su den cua giao dich crypto suy giam theo ham mu hay theo luat luy thua la cau hoi thuc nghiem co cau tra loi bang ty so kha nang xay ra; cau hoi nay duoc do luong ben duoi thay vi trich dan.

Qua trinh Hawkes da bien

Voi $D$ loai su kien,

$\lambda_d^*(t) = \mu_d + \sum_{d'=1}^{D} \sum_{t_i^{d'} < t} \phi_{dd'}(t - t_i^{d'})$

trong do $\phi_{dd'}$ mo ta cach loai $d'$ kich thich loai $d$. Voi kernel mu, ma tran nhanh la $G = [\alpha_{dd'}/\beta_{dd'}]$, va tinh dung yen doi hoi ban kinh pho $\rho(G) < 1$. Vector rieng cua $G$ gan voi gia tri rieng lon nhat xac dinh che do kich thich chi phoi — nhom loai su kien nao co xu huong xay ra cung nhau.

Truong hop ap dung thu vi cho so lenh la $D = 6$: mua thi truong, ban thi truong, mua gioi han, ban gioi han, huy mua, huy ban. Ma tran do co 36 phan tu va la phan cua bo cong cu Hawkes ma blog nay chua co o noi nao khac. Luu y rang mot phan tu du doan cua no da duoc ghi nhan theo kinh nghiem ma khong dung tu vung nay: huy hang loat co tuong quan tren cac muc lien tiep — rut tuong — duoc do truc tiep trong vi tri hang doi va phan tich tuong so lenh. Mot $\phi_{\text{cancel},\text{cancel}}$ duoc khop la cung hien tuong do duoc bieu dien duoi dang kernel, va hai phep do phai phu hop. Neu khong, mot trong hai phep do sai.

Phien ban lien tai san va da san cung dung bo may nay voi nhan khac: kernel ngoai duong che bieu thi quan he dan-dat-do-tre va lan truyen, va Shi, Broussard va Booth (2019) da dung dung cach nay tren 30 co phieu Dow trong flash crash nam 2010, tim thay kich thich cheo bat doi xung voi cac nguon lan truyen co the nhan dien.

Hieu chuan

Cuc dai kha nang xay ra

Voi cac thoi diem su kien $\{t_1, \ldots, t_N\}$ tren $[0, T]$:

$\ell(\theta) = \sum_{i=1}^{N} \log \lambda^*(t_i) - \int_0^T \lambda^*(t) \, dt$

Hang dau tien thuong cho cuong do tai noi cac su kien da xay ra; hang thu hai phat cuong do da khong tao ra gi.

Voi kernel mu, phep de quy

$R_i = \sum_{j < i} e^{-\beta(t_i - t_j)} = e^{-\beta(t_i - t_{i-1})} \cdot R_{i-1} + 1$

cho $\lambda^*(t_i) = \mu + \alpha R_i$ voi $O(1)$ moi su kien, va ham bu tru dong:

$\int_0^T \lambda^*(t) \, dt = \mu T + \frac{\alpha}{\beta} \sum_{i=1}^{N} \left(1 - e^{-\beta(T - t_i)}\right)$

Tong chi phi $O(N)$, nhờ do viec khop hang trieu su kien tro nen thong thuong. Toi uu bang L-BFGS-B voi cac rang buoc duong.

Hai che do that bai can biet truoc khi tin vao ket qua. Be mat kha nang xay ra dep rat te khi $n \to 1$, vi vay mot lan khop gan toi han co the bao cao uoc luong diem co ve chac chan nhung nam trong mot thung gan bang cho $\pm 0.1$ cua $n$ — luon lap profile kha nang xay ra theo $n$ thay vi doc Hessian cua bo toi uu. Va trong truong hop da bien, so tham so tang theo $O(D^2)$, vi vay sau $D \approx 10$ can group-Lasso tren chuan kernel, neu khong ma tran se khop nhieu.

Uoc luong EM va phi tham so

EM xem cau truc nhanh la an: moi su kien hoac la nguoi nhap cu tu $\mu$, hoac la hau due cua mot su kien truoc do. Co hoc buoc E/M tong quat, canh bao cuc bo, va ky luat khoi tao nhieu lan duoc trinh bay trong phan Baum-Welch cua phat hien che do voi HMM va hoan toan giong nhau o day. Phan rieng cua Hawkes la dai luong buoc E:

$p_{ij} = \frac{\phi(t_i - t_j)}{\lambda^*(t_i)}, \quad p_{i0} = \frac{\mu}{\lambda^*(t_i)}$

$p_{ij}$ la phan phoi hau nghiem ve su kien nao gay ra su kien nao — mot dau ra co the dien giai, khong chi la trung gian toi uu hoa. Cong theo loai su kien, no cho phep gan nguon cua bang giao dich: phan nay trong doanh so ban cua gio qua duoc kich hoat boi cac lenh ban truoc do, phan kia den tu ben ngoai. No cung cho phep uoc luong $\phi$ phi tham so thanh hang hang tren luoi, day la cach kiem tra gia dinh kernel mu ma khong cam ket voi mot phuong an thay the. Chi phi: $O(N^2)$ cho ma tran $p_{ij}$ day du, vi vay khong mo rong qua khoang $10^5$ su kien ma khong xap xi.

Phuong phap pho cua Bacry va Muzy la con duong khong mo hinh khac — mat do pho cong suat cua qua trinh Hawkes dung yen phan tach thanh $\hat{C}(\omega) = |I - \hat{\Phi}(\omega)|^{-2} \cdot \text{diag}(\mu)$, vi vay uoc luong mat do dong phuong sai va giai thua tach se phuc hoi kernel ma khong gia dinh dang tham so.

Khop bang giao dich crypto thuc

Moi con so trong phan nay phai den tu mot lan khop thuc tren du lieu thuc. Khoi phuc cac tham so ban da tiem vao bo mo phong khong phai bang chung cho dieu gi ngoai viec bo toi uu hoa hoat dong.

Du lieu: dau thoi gian aggTrade BTC/USDT cua Binance trong mot cua so xac dinh, voi cua so, ky hieu va so su kien duoc ghi ro ben canh moi con so bao cao. aggTrade gop cac lan khop cung gia tu mot lenh taker thanh mot ban ghi, day la don vi dung o day — mot lenh quyet liet la mot su kien, khong phai mot su kien cho moi lan khop doi ung.

Khop don bien

import numpy as np
import pandas as pd
from tick.hawkes import HawkesExpKern

trades = pd.read_parquet("binance_btcusdt_aggtrades.parquet")

t0 = trades["transact_time"].iloc[0]
ts = ((trades["transact_time"] - t0) / 1000.0).to_numpy(dtype=np.float64)
T = ts[-1]

learner = HawkesExpKern(decays=BETA_GRID, penalty="none", verbose=False)
learner.fit([ts])

mu_hat = learner.baseline[0]
alpha_hat = learner.adjacency[0, 0]
n_hat = alpha_hat / BETA_CHOSEN

print(f"events   = {len(ts)}")
print(f"window   = {T/3600:.2f} h")
print(f"mu       = {mu_hat:.4f} events/s")
print(f"n        = {n_hat:.4f}")
print(f"lambda_bar (model) = {mu_hat/(1-n_hat):.4f} events/s")
print(f"lambda_bar (data)  = {len(ts)/T:.4f} events/s")

Hai dong in cuoi la phep kiem tra tinh hop ly dau tien khong ton chi phi: neu cuong do trung binh dung yen cua mo hinh $\mu/(1-n)$ khong gan toc do su kien thuc nghiem, lan khop sai bat ke bo toi uu bao cao gi.

$\beta$ khong nen duoc co dinh bang tay. Lap profile kha nang xay ra tren mot luoi cac do suy giam va bao cao profile, vi $\alpha$ va $\beta$ doi ung manh va $n = \alpha/\beta$ co the on dinh trong khi khong tham so nao rieng le on dinh.

Kernel co thuc su suy giam theo ham mu khong?

Khẳng dinh ve luat luy thua trong tai lieu xuat phat tu cac bai ve hop dong tuong lai co phieu nam 2013. Hay kiem tra truc tiep: khop kernel mu va kernel luy thua tren cung bang giao dich, so sanh log-kha nang xay ra tren du lieu giu lai (khong phai trong mau, vi luat luy thua co them mot tham so), dong thoi xem rieng su suy giam cuong do thuc nghiem sau cac giao dich lon.

large = ts[trades["quantity"].to_numpy() > LARGE_TRADE_THRESHOLD]

lags = np.logspace(-3, 2, 60)   # 1 ms .. 100 s
rate = np.empty(len(lags) - 1)
for k in range(len(lags) - 1):
    lo, hi = lags[k], lags[k + 1]
    counts = [
        np.searchsorted(ts, t0_ + hi) - np.searchsorted(ts, t0_ + lo)
        for t0_ in large
    ]
    rate[k] = np.mean(counts) / (hi - lo)

Ve tren truc log-log, kernel mu uon cong xuong; kernel luy thua la duong thang. Do thi do la bang chung manh hon ca hai con so kha nang xay ra, vi no cho thay sai khop nam o dau — thong thuong kernel mu uoc luong thieu phan duoi va uoc luong qua phan vai mili giay dau tien.

Ma tran nhanh

Mo hinh da bien toi thieu kha dung: hai chieu, mua quyet liet va ban quyet liet, tach theo is_buyer_maker. Neu co du lieu L3, mo rong them cac lenh huy.

buys  = ts[~trades["is_buyer_maker"].to_numpy()]   # taker bought
sells = ts[ trades["is_buyer_maker"].to_numpy()]   # taker sold

mv = HawkesExpKern(decays=BETA_CHOSEN, penalty="l2", C=C_CV, verbose=False)
mv.fit([buys, sells])

G = mv.adjacency / BETA_CHOSEN          # branching matrix
rho = np.max(np.abs(np.linalg.eigvals(G)))
print("G =\n", G)
print(f"spectral radius = {rho:.4f}")   # must be < 1

Hai dai luong can cong bo la chinh ma tran va ban kinh pho cua no. Su bat doi xung la phan thu vi: ban-kich-thich-ban co vuot mua-kich-thich-mua hay khong, va cac hang cheo co doi xung hay khong, la nhung menh de co the kiem tra ve bang giao dich crypto ma o day chua ai do.

Do phu hop: bien doi thoi gian

Dinh ly bien doi thoi gian: neu mo hinh dung, cac thoi diem bien doi $\tau_i = \int_0^{t_i} \lambda^*(s)\,ds$ tao thanh mot qua trinh Poisson toc do don vi, vi vay cac hieu $\Delta\tau_i$ co phan phoi $\text{Exp}(1)$.

from scipy.stats import kstest, expon

compensator = np.empty(len(ts))
for i, ti in enumerate(ts):
    compensator[i] = mu_hat * ti + (alpha_hat / BETA_CHOSEN) * np.sum(
        1 - np.exp(-BETA_CHOSEN * (ti - ts[:i]))
    )

tau = np.diff(compensator)
stat, pval = kstest(tau, expon(scale=1.0).cdf)
print(f"KS = {stat:.4f}, p = {pval:.4g}, N = {len(tau)}")

Mot canh bao khi doc kiem dinh nay: voi $N$ hang tram nghin, kiem dinh KS se bac bo do sai mo hinh nho nhat. Gia tri p rat nho tai $N = 5 \times 10^5$ tu no gan nhu khong co y nghia; dau ra co thong tin la do thi QQ cua $\Delta\tau$ so voi $\text{Exp}(1)$ va chinh thong ke KS, la kich thuoc anh huong khong phu thuoc thang do. Bao cao ca hai va bao cao trung thuc — neu kernel mu bi bac bo tren bang crypto, do la ket qua dang cong bo thay vi chon vuon, cung tinh than voi ket qua am trung thuc ve loi the ben vung.

n co thuc su thay doi theo do mong manh khong?

Khẳng dinh rang $n$ la chi bao do mong manh theo thoi gian thuc duoc trich dan lien tuc nhung hiem khi kiem tra. Phep kiem tra don gian: uoc luong $n$ tren cac cua so truot qua mot giai do yen binh va mot ngay da biet co chuoi crypto, xem no tang truoc hay chi trong luc su kien xay ra. Dai luong chi tang dong thoi voi crash la mo ta, khong phai canh bao.

Tinh on dinh va do ben vung

Mot $n$ da khop chua phai phep do cho den khi biet no thay doi bao nhieu khi thay doi nhung dieu le ra khong quan trong. Bon bay cu the, sap theo thu tu uoc luong tac hai.

Do phan giai dau thoi gian tao ra tu kich thich. Dau thoi gian aggTrade cua Binance co do phan giai mili giay. Cac giao dich thuc te cach nhau vai micro giay bi gom vao cung dau moc mili giay, va kha nang xay ra Hawkes doc cac su kien trung nhau la kich thich toi da — $\phi(0)$ duoc tinh chinh xac tai do tre bang khong. Dieu nay lam sai lech $n$ tang len, va sai lech tang theo muc do hoat dong, nghia la no te nhat dung trong cac dot bung no ma ban muon con so dang tin cay nhat. Hay dinh luong no: dem ty le su kien chung dau moc voi hang xom, sau do khop lai voi cac su kien duoc lam rung deu trong bin mili giay cua chung va bao cao $n$ thay doi bao nhieu. Neu lam rung thay doi $n$ nhieu hon khoang tin cay, con so tieu de dang do dong ho chu khong phai thi truong.

Gom lo cua san giao dich. San giao dich khong phat su kien ngay khi chung xay ra; chung phat khi duong ong khop va pho bien xoa hang. Co tien le duoc ghi nhan: luong thanh ly cua Binance bi gioi han mot lan moi giay — mot luong co ve nhu cac su den gom cum phan nao la san pham phu cua ben phat. Bat ky viec gom lo nao trong luong giao dich cung in chu ky vao phan phoi thoi gian giua cac lan den ma kernel Hawkes se vui ve hap thu vao $\alpha$. Hay kiem tra truc tiep: ve histogram thoi gian giua cac lan den va tim cac dinh tai chu ky gom lo truoc khi khop bat ky thu gi.

Do nhay voi suy giam gia dinh. $n = \alpha/\beta$ phu thuoc vao $\beta$, va $\beta$ thuong duoc co dinh hoac tim tren luoi thay vi uoc luong dong thoi. Bao cao $n$ nhu mot ham cua $\beta$ tren khoang kha tin cay, khong phai mot diem. Neu duong cong phang, con so co that; neu $n$ bam sat $\beta$, ban da chon cau tra loi.

Cua so uoc luong. $n$ uoc luong tren mot gio, mot ngay va mot tuan se khac nhau, mot phan vi qua trinh thuc su khong dung yen va mot phan vi cua so dai tron cac che do, lam phong dai kich thich bieu kien bang cach tron bien dong giua che do voi gom cum trong che do. Cach doc dung khi $n$ tang la so sanh voi cung do dai cua so duoc do o thoi diem khac, khong bao gio so sanh voi con so tu mot cua so khac.

Mo hinh o day giong mo hinh da duoc xac lap trong bay do chinh xac GPU: mot duong ong chay sach va tra ve con so co ve hop ly khong phai bang chung rang con so dung, va cach phong ve duy nhat la lam xao tron cac dau vao le ra khong quan trong va xac nhan dau ra khong thay doi.

Dieu nay mang lai gi va khong mang lai gi

Ba moi lien he voi ly thuyet vi mo, giu ngan gon vi blog da trinh bay tung phan day du o noi khac.

Hinh thanh gia. Trong Kyle (1985), market maker dinh gia dua tren dong lenh rong ket hop giao dich co thong tin va giao dich nhieu; he so tac dong $\lambda$ va do sau $1/\lambda$ cung tai lieu tiep noi — bao gom bang chung chong lai tinh tuyen tinh cua Kyle — duoc trinh bay trong mo hinh chi phi truot gia. Cach doc Hawkes anh xa $\mu$ vao thong tin moi thuc su va cac su kien tu kich thich vao phan ung co hoc, khien $n$ tro thanh bien dai dien do duoc cho phan dong chay bi dan boi nhieu.

Gom cum bien dong. Gom cum, tinh ben vung, dieu kien dung yen $\alpha + \beta < 1$ va phuong sai dai han $\omega/(1-\alpha-\beta)$ — ve cau truc chinh xac la phien ban cua $n < 1$ va $\mu/(1-n)$ — duoc trinh bay trong du bao bien dong GARCH cho crypto. Khẳng dinh moi thuc su la Bacry, Delattre, Hoffmann va Muzy (2013): mot qua trinh Hawkes gan toi han tao ra chuoi loi nhuan co tuong quan tu dong bien dong suy giam theo luat luy thua, ma khong can mo hinh bien dong ro rang nao. Gom cum bien dong xuat hien nhu tinh chat noi sinh cua dong lenh tu kich thich thay vi duoc gia dinh.

Hieu ung Epps. Bacry va Muzy cho thay mot mo hinh Hawkes hai bien tai tao su suy giam cua tuong quan do o tan suat lay mau cao chi tu thoi gian truyen huu han cua kich thich giua hai qua trinh — khong can co che bo sung.

Ve thuc thi, chi dan theo sau tu tinh tu kich thich — cham lai sau mot slice, tang toc trong cac giai doan yen — da la cot loi cua phan vong phan hoi POV trong cac thuat toan thuc thi TWAP, VWAP va POV, va phe binh viec chia deu ngây tho la Almgren-Chriss. Dieu mot lan khop Hawkes bo sung khong phai loi khuyen ma la mot con so: nua doi kernel khop $\ln 2 / \beta$ cho biet kich thich tu slice cua ban mat bao lau de suy giam, bien "cho mot chut" thanh khoang tam dung cu the.

Han che

Khong co uc che. Mo hinh co dien chi cho phep cac su kien lam tang cuong do. Thi truong co mat doi lap: mot lenh gioi han lon hap thu ap luc co the ngan can giao dich quyet liet. Qua trinh Hawkes co uc che cho phep gia tri kernel am, doi lai phai dam bao $\lambda^*(t) \geq 0$. Lieu dieu nay co quan trong voi lan khop cu the hay khong co the thay trong uoc luong kernel phi tham so — neu $\hat\phi$ hang hang xuong duoi khong tai mot do tre nao, mo hinh co rang buoc dang chong lai du lieu.

Khong dung yen. Mo hinh chuan gia dinh tinh dung yen, con crypto hoat dong 24/7 nhung khong dong deu: thanh khoan, thanh phan nguoi tham gia va toc do su kien deu thay doi trong ngay va quanh cac moc funding. Cuong do nen thay doi theo thoi gian $\mu(t)$ xu ly duoc dieu nay voi chi phi la them tham so, va do nhay cua cua so uoc luong o tren la cach tim ra lieu ban co can no hay khong.

Tom tat

Qua trinh Hawkes dang gia dung bo may vi mot ly do: no nen su gom cum cua dong lenh thanh mot con so co the dien giai, ty so nhanh $n$, co the tinh tu dau thoi gian ma khong can tai tao so lenh va khong can gia dinh ve don bay hay vi the. Dieu do khien no la uoc luong re nhat hien co ve muc do noi sinh cua thi truong hien tai — cung cau hoi ma cac chuoi thanh ly tra loi ton kem hon qua $g = \rho/\Delta$.

Bo may nay don gian: kernel mu cho kha nang xay ra $O(N)$ qua phep de quy $R_i$, EM cho hau nghiem co the dien giai ve su kien nao gay ra su kien nao, va bien doi thoi gian cho mot phep kiem tra do phu hop thuc thay vi lap luan co ve hop ly. Dieu khong don gian la tin dau ra. Lam tron dau thoi gian mili giay lam $n$ lech len dung trong cac dot bung no, gom lo cua san giao dich in cau truc ma kernel se hap thu, va $n$ thay doi theo ca suy giam gia dinh lan cua so uoc luong. Mot ty so nhanh duoc bao cao ma khong co bon do nhay nay la mot con so, khong phai mot phep do.

Tai lieu tham khao va doc them

• Hawkes, A. G. (1971). "Pho cua mot so qua trinh diem tu kich thich va kich thich lan nhau." Biometrika, 58(1), 83-90.
• Bacry, E., Mastromatteo, I., and Muzy, J.-F. (2015). "Cac qua trinh Hawkes trong tai chinh." Market Microstructure and Liquidity.
• Shi, F., Broussard, J. P., and Booth, G. G. (2019). "Mo hinh hoa hanh vi Flash Crash tren thi truong co phieu bang cac qua trinh Hawkes da bien."
• Hardiman, S., Bercot, N., and Bouchaud, J.-P. (2013). "Tinh phan xa toi han trong thi truong tai chinh: phan tich qua trinh Hawkes."
• Bacry, E., and Muzy, J.-F. (2014). "Mo hinh Hawkes cho dong hoc tan suat cao cua gia va giao dich." Quantitative Finance.
• Bompaire, M., Bacry, E., and Gaiffas, S. (2017). "tick: Thu vien Python cho hoc thong ke, nhan manh vao cac qua trinh Hawkes." JMLR.
• Filimonov, V., and Sornette, D. (2012). "Dinh luong tinh phan xa trong thi truong tai chinh."
• Kumar, P. (2021). "Qua trinh Hawkes sau cho tao lap thi truong tan suat cao."
• Tai lieu thu vien tick: Mo-dun Hawkes.
• Morse, S. "Lop Python cho cac qua trinh Hawkes." (Huong dan va ma)