---
title: Girsanov Transform for Sub-Diffusions
url: https://www.emergentmind.com/topics/girsanov-transform-for-sub-diffusions
type: topic
---

# Girsanov Transform for Sub-Diffusions

The Girsanov transform for sub-diffusions provides a rigorous stochastic change-of-measure framework for time-changed Brownian motion models driven by an inverse \(\alpha\)-stable subordinator. This construction yields a geometric sub-diffusive model for asset prices that generalizes the classical Black-Scholes setup, capturing anomalous diffusion phenomena observed in financial time series, notably extended waiting periods and periods of inactivity. The framework results in a time-fractional partial differential equation (PDE) for option pricing and introduces several technical novelties in both theory and explicit solutions [2511.10371].

## 1. Mathematical Structure of Sub-Diffusive Processes

Let \(S^*(u)\), \(u \geq 0\), denote a strictly increasing \(\alpha\)-stable subordinator with Laplace exponent \(\E[e^{-\lambda S^*(u)}] = e^{-u \lambda^\alpha}\) for \(0 < \alpha < 1\). The inverse subordinator \(E(t) = \inf\{u > 0 : S^*(u) > t\}\) serves as the random operational time, inducing sub-diffusive dynamics. The sojourn (waiting time) property of \(E(t)\) produces temporal heterogeneities characteristic of sub-diffusions.

Defining \(B(s)\) as a standard Brownian motion independent of \(S^*\), the time-changed process \(X(t) = B(E(t))\) is a continuous martingale with quadratic variation \(\langle X \rangle_t = E(t)\), progressing more slowly than its classical Brownian counterpart. The sub-diffusive geometric asset price is modeled as
\[
S(t) = S(0) \exp\big(\mu E(t) + \sigma B(E(t))\big),
\]
where \(\mu\) is drift and \(\sigma\) is volatility. In differential form,
\[
dS(t) = S(t)\big(\mu\, dE(t) + \sigma\, dB(E(t))\big),
\]
interpreted as a stochastic differential equation under the filtration \(\mathcal{F}_t = \sigma\{X(s), E(s): s \leq t\}\).

## 2. Stochastic Differential Equations Under the Physical Measure

Under \(\mathbb{P}\), the stock dynamics can be expressed as
\[
dS(t) = S(t)\left(\mu\, dE(t) + \sigma\, dM_t\right),
\]
where \(M_t := B(E(t))\) is an \(\mathcal{F}_t\)-martingale with quadratic variation \(E(t)\). Integration with respect to \(dB(E(s))\) reflects the non-Markovian operational time, leading to sub-diffusive sample paths with intermittency.

## 3. Girsanov Transform Adapted to Sub-Diffusions

The Girsanov theorem is extended to this time-changed setting as follows. For an \(\mathcal{F}_t\)-adapted process \(\theta:[0,T]\times\Omega \to \mathbb{R}\) satisfying the "fractional Novikov condition"
\[
\E \exp\left\{\frac{1}{2}\int_0^T \theta(s)^2\,dE(s)\right\} < \infty,
\]
define the Radon-Nikodym density process
\[
L_t = \exp\left\{-\int_0^t \theta(s)\, dB(E(s)) - \frac{1}{2}\int_0^t \theta(s)^2\, dE(s)\right\},\quad 0 \leq t \leq T.
\]
\(L_t\) is a true \(\mathbb{P}\)-martingale by Novikov's criterion, giving rise to the equivalent measure \(\mathbb{Q}\) via \(d\mathbb{Q}/d\mathbb{P}|_{\mathcal{F}_t} = L_t\).

Under \(\mathbb{Q}\), the process
\[
Z(t) = B(E(t)) + \int_0^t \theta(s)\, dE(s)
\]
becomes a time-changed Brownian motion with variance process \(E(t)\). Equivalently, setting \(\widetilde{B}(s) = B(s) + \int_0^s \theta(\tau)\, d\tau\), one has \(Z(t) = \widetilde{B}(E(t))\).

## 4. Risk-Neutral Dynamics and Equivalent Martingale Measure

To align the drift with the risk-free rate \(r\), the choice \(\theta(t) = \frac{\mu - r}{\sigma}\) removes the physical drift, resulting in
\[
dS(t) = S(t)\big(r\,dE(t) + \sigma\,d\widetilde{B}(E(t))\big).
\]
In this measure, the discounted process \(e^{-r E(t)}S(t)\) is a \(\mathbb{Q}\)-martingale. The sub-diffusion equivalent martingale measure (EMM) thus exists, ensuring arbitrage-freedom in the sense of "no free lunch with vanishing risk." However, except for the deterministic case (\(\alpha=1\)), the model remains incomplete since there are infinitely many EMMs, analogous to classical models driven by processes with jumps or stochastic volatility.

## 5. Time-Fractional Black-Scholes PDE

For a European contingent claim with payoff \(\psi(S(T))\), the price process \(V(S, t)\) under risk-neutral valuation is
\[
V(S, t) = \E^\mathbb{Q}\big[e^{-r (E(T)-E(t))}\,\psi(S(T))\,|\,\mathcal{F}_t\big].
\]
The associated pricing equation is a time-fractional PDE:
\[
\frac{\partial V}{\partial t}
+ r S \frac{\partial V}{\partial S}
+ \frac{1}{2} \sigma^2 S^2 D_t^{1-\alpha}\frac{\partial^2 V}{\partial S^2}
- r V = 0, \quad 0 < t < T,\, S > 0,
\]
where \(D_t^{1-\alpha}\) denotes the Caputo fractional derivative of order \(1-\alpha\), inheriting memory effects from the underlying sub-diffusive operational time.

## 6. Explicit Solution for European Call Options

For a European call (\(\psi(x) = (x-K)^+\)), the price is expressible via a mixture of log-normal distributions, convolving the classical pricing formula with the law of the inverse subordinator. Explicitly,
\[
C(S, t) = e^{-r(T-t)} \int_0^\infty \E\big[ ( S e^{ r(T-t) - \frac{1}{2} y + \sqrt{y}\,Z } - K )^+ \big]\, f_{E(T-t)}(y)\, dy,
\]
where \(Z \sim N(0,1)\) and \(f_{E(\tau)}\) is the density of \(E(\tau)\). When \(E(\tau)\) is inverse-stable, its density admits explicit representations via Wright or Mittag-Leffler functions, permitting analytic series solutions.

## 7. Market Completeness and Arbitrage Implications

The existence of an EMM under the Girsanov transform guarantees the absence of arbitrage. For \(\alpha < 1\), the random structure of \(E(t)\) implies market incompleteness, as the non-deterministic time changes preclude hedging every contingent claim with traded assets. The unique martingale measure that preserves the sub-diffusive stochastic integral structure provides a canonical construction, closely paralleling the classical Black-Scholes argument but within the framework of time-changed processes and fractional calculus [2511.10371].

Source: https://www.emergentmind.com/topics/girsanov-transform-for-sub-diffusions