Girsanov Log-Ratio for Drift-Tilted Diffusions

Computes the Radon–Nikodym derivative (log form) associated with a Girsanov change of measure for an SDE:

Usage

girsanov_logratio(theta_path, Winc, dt)

Arguments

theta_path

Numeric vector of drift tilts \(\theta_t\).

Winc

Numeric vector of Brownian increments \(\Delta W_t\).

dt

Time step size.

Value

A numeric log-likelihood ratio.

Details

\(dX_t = b(X_t)\, dt + dW_t\)

tilted by an alternative drift:

\(dX_t = (b(X_t) + \theta_t)\, dt + dW_t\).

The log-likelihood ratio is:

\(\log \frac{dQ}{dP} = \sum_t \left( \theta_t W_{t} - \frac{1}{2}\theta_t^2\, dt \right)\).

This function returns the log-ratio for a given path of drift tilts theta_path, Brownian increments Winc, and time step dt.