SDES的参数估计是一个研究水平的领域,因而相当重要。整本书都是关于这个话题的。请随意查看这些内容以了解更多详细信息。
但这里有一个简单的方法来解决这个问题。首先,请注意,GBM的对数是一个仿射变换的维纳过程(即线性ITO漂移扩散过程)。所以
d ln(s_t)=(mu-sigma^2/2)d t+sigma db_t
因此,我们可以估计出日志过程参数,并将其转换为与原始过程相匹配的参数。退房
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,请
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例如。
这是一个脚本,它以两种简单的方式来实现这个漂移(只是想看看不同之处),并且只有一种用于传播(抱歉)。测井过程的漂移估计为:
(X_T - X_0) / T
以及通过增量MLE(参见代码)。扩散参数是以偏态的方式估计的,其定义是无穷小的方差。
import numpy as np
np.random.seed(9713)
# Parameters
mu = 1.5
sigma = 0.9
x0 = 1.0
n = 1000
dt = 0.05
# Times
T = dt*n
ts = np.linspace(dt, T, n)
# Geometric Brownian motion generator
def gbm(mu, sigma, x0, n, dt):
step = np.exp( (mu - sigma**2 / 2) * dt ) * np.exp( sigma * np.random.normal(0, np.sqrt(dt), (1, n)))
return x0 * step.cumprod()
# Estimate mu just from the series end-points
# Note this is for a linear drift-diffusion process, i.e. the log of GBM
def simple_estimate_mu(series):
return (series[-1] - x0) / T
# Use all the increments combined (maximum likelihood estimator)
# Note this is for a linear drift-diffusion process, i.e. the log of GBM
def incremental_estimate_mu(series):
total = (1.0 / dt) * (ts**2).sum()
return (1.0 / total) * (1.0 / dt) * ( ts * series ).sum()
# This just estimates the sigma by its definition as the infinitesimal variance (simple Monte Carlo)
# Note this is for a linear drift-diffusion process, i.e. the log of GBM
# One can do better than this of course (MLE?)
def estimate_sigma(series):
return np.sqrt( ( np.diff(series)**2 ).sum() / (n * dt) )
# Estimator helper
all_estimates0 = lambda s: (simple_estimate_mu(s), incremental_estimate_mu(s), estimate_sigma(s))
# Since log-GBM is a linear Ito drift-diffusion process (scaled Wiener process with drift), we
# take the log of the realizations, compute mu and sigma, and then translate the mu and sigma
# to that of the GBM (instead of the log-GBM). (For sigma, nothing is required in this simple case).
def gbm_drift(log_mu, log_sigma):
return log_mu + 0.5 * log_sigma**2
# Translates all the estimates from the log-series
def all_estimates(es):
lmu1, lmu2, sigma = all_estimates0(es)
return gbm_drift(lmu1, sigma), gbm_drift(lmu2, sigma), sigma
print('Real Mu:', mu)
print('Real Sigma:', sigma)
### Using one series ###
series = gbm(mu, sigma, x0, n, dt)
log_series = np.log(series)
print('Using 1 series: mu1 = %.2f, mu2 = %.2f, sigma = %.2f' % all_estimates(log_series) )
### Using K series ###
K = 10000
s = [ np.log(gbm(mu, sigma, x0, n, dt)) for i in range(K) ]
e = np.array( [ all_estimates(si) for si in s ] )
avgs = np.mean(e, axis=0)
print('Using %d series: mu1 = %.2f, mu2 = %.2f, sigma = %.2f' % (K, avgs[0], avgs[1], avgs[2]) )
输出:
Real Mu: 1.5
Real Sigma: 0.9
Using 1 series: mu1 = 1.56, mu2 = 1.54, sigma = 0.96
Using 10000 series: mu1 = 1.51, mu2 = 1.53, sigma = 0.93
