Why Return Derivatives Are Not Ordinary SDEs
Summary
The document asks whether the instantaneous return of a diffusion can itself be differentiated and modeled by another stochastic differential equation. It starts from a price process with constant drift and state-dependent volatility, then expresses the return informally using white noise. Differentiating that expression would require differentiating white noise, which is not an ordinary function. The author notes that generalized derivatives and Itô corrections may matter, and asks whether the resulting object can be represented as a Langevin equation or SDE.
An edit explores a proposed construction in which the log price is represented as an integral of a process that itself follows an SDE. It compares this with the Itô formula for the logarithm of the original price process. The material is a question and partial derivation rather than a resolved result: it supplies no proof that the proposed representation exists, and its formal white-noise manipulations require careful interpretation. It is useful for understanding the distinction between diffusion increments, instantaneous returns, and well-defined stochastic processes.
Key ideas
- Brownian-driven price paths do not have ordinary time derivatives.
- Writing white noise in a Langevin expression does not make its derivative an ordinary random variable.
- Itô's formula adds a volatility-dependent drift term when converting a price diffusion to log prices.
- The document proposes an SDE representation for instantaneous returns but does not establish its validity.
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Full text
# Is the time derivative of asset returns expressible as an SDE?
# Is the time derivative of asset returns expressible as an SDE?
Consider the following SDE for $(S_t)_{t\geq 0}$ under $\mathbb{Q}$, \begin{equation} \mathop{dS_t}=S_t\left(r\mathop{dt}+\sigma(t,S_t)\mathop{dW_t}\right), \end{equation} which (in Langevin form) may be expressed as \begin{equation} \dot{S}=S_t\left(r+\sigma(t,S_t)\xi(t)\right), \end{equation} such that $W_t$ is a wiener process and $\xi(t)$ is white noise with $r$ constant. Is it possible to take the time derivative of the returns of $S_t$, so that \begin{equation}\tag{1} \frac{d}{dt}\left(\frac{\dot{S}}{S}\right)=\frac{\partial \sigma(S_t,t)}{\partial t}\xi(t)+\sigma(S_t,t)\xi'(t), \end{equation} where $\xi'(t):=\left(\xi(t+h)-\xi(t)\right)/h$ as $h\rightarrow 0$? The expectation and covariance of $\xi'(t)=\eta(t)$ are given here. If so, can $\xi'(t)$ be expressed as an SDE, in terms of $\xi(t)$, since its distribution is also Gaussian? More succinctly, does there exist a Langevin equation ($a$ would intuitively be zero in this case) \begin{equation} \frac{dX_t}{dt}=a(X_t,t)+b(X_t,t)\xi(t), \end{equation} or its equivalent SDE where $X_t=\dot{S}/S=d(\ln S_t)/dt$?
Note that I am interpreting derivatives in the generalised sense here, since $\xi(t)$ is clearly nowhere differentiable. Also, if this has been discussed before, I would be grateful for a reference. Any help would be much appreciated.
Referring to the comments, there may be an additional $-\sigma(t,S_t)\frac{\partial \sigma(t,S_t)}{\partial t}$ term to add onto (1) when using Itô's lemma. From a historical context, Langevin equations were developed in the context of physics to study particles undergoing Brownian motion and so provide a less rigorous representation of SDEs when compared with Itô's formalism (see here).
Edit: Suppose that $\int_0^{t} X_s\mathop{dW_s}=\ln S_t-\ln S_0$ and $\mathop{dX_t}=a_t\mathop{dt}+b_t\mathop{dW_t}$. Then \begin{align} X_s&=X_0+\int_0^{s} a_u\mathop{du}+\int_0^{s} b_u\mathop{dW_u},\\ \int_0^{t} X_s\mathop{dW_s}&=\int_0^{t} X_0 \mathop{dW_s} + \int_0^{t}\int_0^{s} a_u\mathop{du}\mathop{dW_s}+\int_0^{t}\int_0^{s} b_u\mathop{dW_u}\mathop{dW_s}. \end{align} By Itô's lemma applied to GBM, \begin{equation} \ln S_t=\ln S_0+\int_0^{t}\left(r-\frac{\sigma_s^2}{2}\right)\mathop{ds}+\int_0^{t}\sigma_s\mathop{dW_s}. \end{equation} Hence, \begin{equation} rt-\frac{1}{2}\int_0^{t}\sigma_s^2\mathop{ds}+\int_0^{t}\sigma_s\mathop{dW_s}=X_0 W_t + \int_0^{t}\int_0^{s} a_u\mathop{du}\mathop{dW_s}+\int_0^{t}\int_0^{s} b_u\mathop{dW_u}\mathop{dW_s}. \end{equation}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.