by Gimle Labs
Categorical Semantics Compilation Approximation Theory Stochastic Calculus

Stochastic Calculus

Mathematical foundations of stochastic calculus in Asgard. The Runtime page covers the API; this page covers the underlying mathematics.

Stochastic Processes

A one-dimensional stochastic process $X_t$ is defined by a stochastic differential equation (SDE):

$$dX_t = a(t, X_t),dt + b(t, X_t),dW_t$$

where:

Equivalently, in integral form:

$$X_t = X_0 + \int_0^t a(s, X_s),ds + \int_0^t b(s, X_s),dW_s$$

The first integral is the standard Riemann integral. The definition of the stochastic integral $\int f,dW_t$ determines which stochastic calculus we are working in. There are two viable choices: Itô and Stratonovich.

Itô Calculus

The Itô integral evaluates the integrand at the left endpoint of each partition interval. This makes it non-anticipating (the integrand does not "look ahead" to future values of $W_t$), which gives the integral the martingale property: the conditional expectation of future values equals the current value.

Itô Lemma

The Itô Lemma is the chain rule of stochastic calculus. For a twice differentiable function $f$ and an $m$-dimensional continuous semimartingale $X = \langle X^1, \ldots, X^m \rangle$:

$$f(X_t) = f(X_0) + \sum_{i=1}^{m} \int_0^t \frac{\partial f}{\partial x_i}(X_s),dX_s^i + \frac{1}{2}\sum_{i,j=1}^{m} \int_0^t \frac{\partial^2 f}{\partial x_i \partial x_j}(X_s),d[X^i, X^j]_s$$

where $[X^i, X^j]_s$ is the quadratic covariation of processes $X^i$ and $X^j$.

The extra second-order term $\frac{1}{2}\sum \frac{\partial^2 f}{\partial x_i \partial x_j},d[X^i, X^j]$ is what distinguishes Itô calculus from classical calculus. It arises because Brownian motion paths have non-zero quadratic variation ($[W, W]_t = t$).

For the common scalar case $dX_t = a,dt + b,dW_t$ and a function $g(t, X_t)$:

$$dg = \frac{\partial g}{\partial t},dt + \frac{\partial g}{\partial x},dX_t + \frac{1}{2}\frac{\partial^2 g}{\partial x^2},b^2,dt$$

Stratonovich Calculus

The Stratonovich integral evaluates the integrand at the midpoint of each partition interval. The key advantage: it preserves the classical chain rule from ordinary calculus. If $f$ is a smooth function and $X_t$ satisfies an SDE, then:

$$df(X_t) = f'(X_t),dX_t \quad \text{(Stratonovich)}$$

with no correction term. This makes Stratonovich calculus the natural choice when modelling continuous physical systems where the classical rules of calculus should hold.

However, the Stratonovich integral is not a martingale and does not satisfy the Fokker-Planck equation directly.

Itô-Stratonovich Conversion

The two integrals are related by a correction term. For a function $h$ of Brownian motion:

$$\int_0^T h(W_t) \circ dW_t = \int_0^T h(W_t),dW_t + \frac{1}{2}\int_0^T h'(W_t),dt$$

where $\circ,dW_t$ denotes the Stratonovich integral and $dW_t$ denotes the Itô integral.

For SDEs, the conversion works through a drift correction. An SDE in Stratonovich form:

$$dX_t = a(t, X_t),dt + b(t, X_t) \circ dW_t$$

is equivalent to the Itô SDE:

$$dX_t = \bar{a}(t, X_t),dt + b(t, X_t),dW_t$$

where the corrected drift is:

$$\bar{a}(t, X) = a(t, X) + \frac{1}{2},b(t, X),\frac{\partial b}{\partial X}(t, X)$$

The correction term $\frac{1}{2}b\frac{\partial b}{\partial X}$ is called the noise-induced drift or Stratonovich correction. It vanishes when the diffusion coefficient $b$ is constant (additive noise), in which case both calculi agree.

When to Use Which

Criterion Itô Stratonovich
Chain rule Itô Lemma (extra term) Classical chain rule
Martingale Yes No
Fokker-Planck Directly applicable Requires conversion
Moment equations Clean expressions More complex
Physical systems Less natural Natural (classical rules)
Discrete models Better fit Less natural
Numerical schemes Euler-Maruyama is $O(\sqrt{\Delta t})$ strong Euler-Heun needed
Convergence Standard discretization Brownian bridge convergence

Rules of thumb:

Since the two formulations are always interconvertible via the drift correction formula, the choice is one of convenience rather than correctness.

Taylor Expansions

Both calculi have stochastic generalizations of the Taylor expansion, used in higher-order numerical schemes.

The Itô-Taylor expansion of a function $f(X_t)$ involves iterated stochastic integrals $I_\alpha$ indexed by multi-indices $\alpha$. Truncating to order $m$ gives schemes of increasing accuracy:

The Stratonovich-Taylor expansion has similar structure but uses Stratonovich iterated integrals, which satisfy simpler algebraic identities due to the classical chain rule.

Connection to Asgard

Asgard implements both calculi through StochasticCalculus (parameters) and StochasticCircuitExecutor (execution):

from gimle.asgard.circuit.circuit import Circuit
from gimle.asgard.runtime.stochastic_circuit_executor import (
    StochasticCircuitExecutor,
)
from gimle.asgard.runtime.stream_evaluator import StochasticCalculus

# Define the calculus (drift, diffusion, Monte Carlo settings)
calculus = StochasticCalculus(
    drift=0.0,
    diffusion=1.0,
    n_paths=1000,
    dt=0.01,
)

# Compile and execute a circuit
circuit = Circuit.from_string("composition(var(noise), register(t))")
executor = StochasticCircuitExecutor(circuit)
paths = executor.execute(
    input_streams=[noise_stream],
    stochastic_dims={"t": calculus},
    t_start=0.0,
    t_end=2.0,
)

The stochastic circuit executor:

  1. Generates Brownian paths via jax.random with configurable seed
  2. Applies Euler-Maruyama (Itô) or Euler-Heun (Stratonovich) time-stepping
  3. Computes ensemble statistics (mean, variance, quantiles) across paths
  4. Handles Itô-Stratonovich conversion internally when needed

Circuit operations like register and deregister adapt their behaviour based on the calculus: under stochastic semantics, register implements stochastic integration rather than deterministic coefficient shifting.

Next Steps