Beyond Brownian Motion

Lévy processes. Processes with independent, identically distributed increments are called Lévy processes. The Lévy–Itô decomposition splits every one of them into a drift, a Brownian part and a (compensated) sum of jumps, so this section's model is a typical member of the family.

Itô's formula with jumps. For $dX = \sigma\,dW + dJ$: $$f(X_t) = f(X_0) + \int_0^t f'(X_{s})\,\sigma\,dW_s + \tfrac12\int_0^t f''(X_s)\,\sigma^2\,ds + \sum_{s \le t} \big[f(X_s) - f(X_{s-})\big].$$ The continuous part follows Chapter 4. Each jump adds its full change in $f$, not a Taylor approximation.

In finance. Merton's jump-diffusion model (1976) adds normally distributed log-price jumps to Black–Scholes. The hedging argument of section 6.1 then breaks down, because a jump can't be hedged with the stock alone.

References: R. Merton, "Option pricing when underlying stock returns are discontinuous", J. Financial Economics 3 (1976), 125–144. R. Cont & P. Tankov, Financial Modelling with Jump Processes (Chapman & Hall/CRC, 2004). Shreve, Stochastic Calculus for Finance II, ch. 11.

import numpy as np
rng = np.random.default_rng(23)
lam, share, paths = 2.0, 0.8, 200_000                # 2 jumps per unit time carry 80% of the variance
sigma, delta = np.sqrt(1 - share), np.sqrt(share / lam)
n = rng.poisson(lam, paths)                          # number of jumps by t = 1, per path
X1 = sigma * rng.normal(size=paths) + delta * np.sqrt(n) * rng.normal(size=paths)   # sum of n N(0, delta^2) jumps
print(f"variance {X1.var():.3f} (1)   kurtosis {np.mean(X1 ** 4) / X1.var() ** 2:.2f} (theory {3 + 3 * share ** 2 / lam:.2f}; bell curve 3)")
print(f"share of |X1| > 3: {np.mean(np.abs(X1) > 3):.4f}   (bell curve 0.0027)")

Output when we ran it:

variance 1.006 (1)   kurtosis 3.97 (theory 3.96; bell curve 3)
share of |X1| > 3: 0.0079   (bell curve 0.0027)

Optimality. For linear signal and observation equations with Gaussian noise, the conditional distribution of $X_t$ given the observations up to $t$ is exactly normal, with mean $m_t$ and variance $P_t$. No other estimator built from the same data has a smaller mean squared error.

The innovation is Brownian motion. The scaled innovations $\tfrac1r(dY_t - m_t\,dt)$ form a Brownian motion. The filter extracts everything predictable from the data, and what remains is pure, independent noise.

Beyond linear models. For nonlinear models the conditional distribution is no longer normal. It solves a stochastic PDE (the Kushner–Stratonovich or Zakai equation), and in practice it is approximated by the extended Kalman filter or by particle filters.

References: R. Kalman, "A new approach to linear filtering and prediction problems", J. Basic Engineering 82 (1960), 35–45. R. Kalman & R. Bucy, "New results in linear filtering and prediction theory", J. Basic Engineering 83 (1961), 95–108. Øksendal, Stochastic Differential Equations, ch. 6.

import numpy as np
rng = np.random.default_rng(24)
theta, sigma, r, T, steps = 1.0, 1.0, 0.5, 10.0, 2000
dt = T / steps
x = m = 0.0
P = sigma ** 2 / (2 * theta)                         # no information yet
err = []
for _ in range(steps):
    dY = x * dt + r * np.sqrt(dt) * rng.normal()     # noisy observation of the hidden signal
    m += -theta * m * dt + (P / r ** 2) * (dY - m * dt)    # Kalman-Bucy update of the estimate
    P += (-2 * theta * P + sigma ** 2 - P ** 2 / r ** 2) * dt   # Riccati equation for its variance
    x += -theta * x * dt + sigma * np.sqrt(dt) * rng.normal()   # the hidden signal moves on
    err.append(m - x)
p_inf = r ** 2 * (np.sqrt(theta ** 2 + sigma ** 2 / r ** 2) - theta)
print(f"filter error spread {np.std(err):.3f}   theory sqrt(P_inf) = {np.sqrt(p_inf):.3f}   no-data guess {np.sqrt(0.5):.3f}")

Output when we ran it:

filter error spread 0.506   theory sqrt(P_inf) = 0.556   no-data guess 0.707