Itô isometry


In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals. One of its main applications is to enable the computation of variances for random variables that are given as Itô integrals.
Let denote the canonical real-valued Wiener process defined up to time, and let be a stochastic process that is adapted to the natural filtration of the Wiener process. Then
where denotes expectation with respect to classical Wiener measure.
In other words, the Itô integral, as a function from the space of square-integrable adapted processes to the space of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products
and
As a consequence, the Itô integral respects these inner products as well, i.e. we can write
for .

Numerical Simulation

The Itô isometry can be illustrated through numerical simulation using Monte Carlo methods. Such simulations help verify the theoretical relationship between the expected value of squared stochastic integrals and the expected value of integrated squared processes. A typical Monte Carlo experiment involves generating numerous sample paths of Brownian motion and computing both sides of the isometry equation for different choices of the integrand process. The simulation approximates the continuous-time stochastic integral using a discrete-time summation:
where represents the Brownian increments over small time intervals. The isometry can be demonstrated using various processes on the interval :
  1. Constant process:
  2. * Analytical value:,
  3. Linear deterministic process:
  4. * Analytical value:,
  5. Trigonometric deterministic process:
  6. * Analytical value:,
  7. Path-dependent stochastic process:
  8. * Analytical value:,
  9. Compensated Poisson :
  10. * Analytical value:.

  1. =
  2. Convergence test for five processes
  3. – 4 Brownian‑based Itô integrals
  4. – 1 compensated‑Poisson martingale integral
  5. Output:
  6. • nicely formatted error table
  7. • log–log convergence plot
  8. – each Brownian curve in its own colour
  9. – Poisson curve orange & dashed

  10. =
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from IPython.display import display
np.random.seed # reproducible demo
  1. ---------------------- global settings ---------------------
T = 1.0 # horizon
lam = 3.0 # Poisson intensity λ
M_paths = 20_000 # Monte‑Carlo paths
N_list = # mesh refinements
  1. dataframe to hold absolute errors
index = pd.Index
err_table = pd.DataFrame
  1. -------------------- main simulation loop ------------------
for N in N_list:
dt = T / N
sqrt_dt = np.sqrt
t_left = np.linspace # left endpoints
# --- Brownian increments & paths ------------------------
dW = np.random.normal
W = np.zeros)
W = np.cumsum
# --- Compensated Poisson increments --------------------
dN = np.random.poisson
dM = dN - lam * dt
# helper for deterministic X_t
lhs_det = lambda X_grid, inc: np.mean.sum
# 1) X_t ≡ 1 with Brownian integrator
X1 = np.ones_like
err_table.loc = abs
# 2) X_t = t
Xt = t_left
err_table.loc = abs
# 3) X_t = sin
Xs = np.sin
err_table.loc = abs
# 4) X_t = W_t
lhs_W = np.mean.sum
err_table.loc = abs
# 5) compensated Poisson with X_t ≡ 1
err_table.loc = abs
  1. ---------------------- show table --------------------------
display.set_caption
  1. ---------------------- plotting ----------------------------
colour_map =
markers =
styles =
plt.figure
for col in err_table.columns:
plt.plot
plt.xscale
plt.yscale
plt.xlabel
plt.ylabel
plt.title"
plt.grid
plt.legend
plt.tight_layout
plt.show

N1tsinWtPoisson-1
503.202e-035.356e-033.551e-031.497e-053.625e-02
1009.887e-033.152e-037.880e-038.480e-034.595e-02
2001.290e-026.827e-033.569e-036.790e-032.950e-02
5003.515e-033.692e-045.956e-036.902e-031.780e-02
10005.203e-035.764e-032.894e-042.334e-033.605e-02
20008.342e-034.412e-032.358e-031.208e-023.330e-02
50005.890e-031.908e-036.048e-031.243e-028.335e-02

A Monte Carlo simulation with 20,000 sample paths and 1,000 time steps produces results that closely match the theoretical values predicted by the Itô isometry. As shown in the table above, the absolute errors between the simulated left-hand side and the analytical right-hand side are typically on the order of or smaller, confirming the validity of the isometry relationship.
These simulations serve as empirical evidence for the Itô isometry and provide insight into how the relationship holds across different types of processes, both deterministic and stochastic. The close agreement between theoretical and simulated values demonstrates the robustness of the isometry as a fundamental property of stochastic calculus.

Generalization to Martingales

The Itô isometry extends beyond the standard Wiener process to a broader class of stochastic processes, particularly martingales, providing a powerful framework for computing variances of stochastic integrals. A stochastic process is a martingale with respect to the filtration if:
  • for all
Martingales can be further classified as:
A local martingale is a process for which there exists a sequence of stopping times with such that is a martingale for each.

The Itô Isometry for Martingales

The Itô integral is defined for a broader class of integrators beyond Brownian motion. For a predictable process and an appropriate integrator, the Itô integral is defined as the limit in of simple predictable processes approximating :
where each is -measurable.
The Itô isometry holds when the integrator is one of:
For these cases, the isometry takes the form:
where denotes the quadratic variation process of.