Girsanov Transform for Sub-Diffusions

• Girsanov transform for sub-diffusions is a stochastic change-of-measure method applied to time-changed Brownian motion models driven by an inverse stable subordinator.
• It introduces a geometric sub-diffusive asset pricing model that generalizes the classical Black-Scholes framework by incorporating anomalous diffusion and extended waiting times.
• The approach leads to a time-fractional PDE for option pricing and novel explicit solutions, ensuring absence of arbitrage despite market incompleteness.

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 (Zhang et al., 13 Nov 2025).

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 (Zhang et al., 13 Nov 2025).