Stochastic differential equations and noise: driven, drifting,...?

Question

This is a question on terminology but connected to basic intuitions. I would like to have a practitioner's point of view on the use of the term "driven by" some noise for stochastic (partial) differential equations. When i consider geometric brownian motion $$dS_t = \mu S_tdt + \sigma S_tdW_t$$ to me what drives $S_t$ is the drift term $\mu S_tdt$ and not the diffusion term $\sigma S_tdW_t=S_t\xi_tdt$ with $\xi$ a white noise, when interpreted appropriately (which can be done with the Hida-Malliavin calculus, if i understand correctly but im only learning about it). The diffusion term is rather a damping (or just a "passive" diffusion) of the drift term, the "drive", the main motion, of the stochastic process. So i wonder why many (most) papers talk of a S(P)DE "driven by" a noise $\xi$, instead of say "damp(en)ed", "blurred", "regulated", "moderated", or "smothered" by $\xi$. All the more so that noise is often touted for its regularizing property on differential equations, thus does not correspond to a drive (which would rather create singularities). For example, geometric brownian motion has the formula $S(0)\exp\left(\left(\mu-\frac{\sigma^2}{2}\right)t+\sigma W_t\right)$, so its growth is somehow slowed to the tune of $\exp\left(\frac{1}{2}\sigma^2t\right)$ by the white noise "driving" it, and when $\mu<\sigma^2/2$ it's solutions tend to $0$ almost surely. [EDIT: I made an error in writing the expectation of GBM, i was overestimating the effect of noise -this is a major use of stackexchange: writing wrong stuff and feeling silly afterwards. It is actually quite counterintuitive that GBM trajectories may tend to $0$ almost surely while its mean grows exponentially as for the "classical part" of the GBM equation. A quantitative formulation is the law of iterated logarithm which asymptotically bounds the supremum of BM, below $t$.]

What am i missing ? Is terminology good as is, or is it just well accepted but perhaps not ideal ? Thank you very much.

Answer 1

In stochastic (partial) differential equations (S(P)DEs), the term "driven by" noise is often used to describe the role of the stochastic term in the equation. However, as you have pointed out, this term can be misleading in some cases.

In the case of the geometric Brownian motion, the drift term $\mu S_t \ dt$ is what drives the process, while the diffusion term $\sigma S_t \ dW_t$ acts as a damping or "smoothing" effect on the motion. It is true that the noise term can regularize or stabilize the solution to the equation, but it does not necessarily "drive" the motion.

The use of the term "driven by" noise may be a convention in the field of stochastic calculus and S(P)DEs, and it may not always accurately reflect the underlying dynamics of the system. Alternative terms such as "dampened" or "moderated" may be more appropriate in some cases.

Answer 2

I am far from being an expert, but let me try to offer a possible perspective from which it makes sense to say that SDEs are driven by noise rather than drift :

When dealing with continuous time stochastic process $(X_t)$, which is really just a family of random variables indexed by (a subinterval of) $\mathbb R^+$, the first thing we need to do is to define a common probability space $(\Omega,\mathcal F,\Pr) $ on which these variables are defined. We quickly find out however that to be able to say interesting/precise things about these stochastic processes, we need to introduce extra concepts which characterize the fact that the process evolves through time. We want to be able to talk about events like "is $(X_t)$ left-continuous ?" or "Is the supremum of $(X_t)$ over the interval $[0,T]$ greater than $a$ ?"

To do so, a key concept that we need is that of a filtration : a family of nested $\sigma$-algebras $(\mathcal F_t) $ which encode "the information available up to time $t$" for the process $(X_t)$, provided $(X_t)$ is adapted to said filtration.
Now if we consider solution to SDEs, i.e. processes which are solutions of $$dX_t = \mu(t,X_t)\ dt + \sigma(t,X_t)\ dW_t $$ where $W_t$ is a standard Brownian motion, the question we want to ask, per above discussion, is "to which filtration is a solution of this SDE adapted ?" The very natural answer is that the solution $X_t$ is adapted to the natural filtration of the driving Browian motion, i.e. $(X_t)$ is adapted to $(\mathcal F_t):=(\sigma(W_t)) $. Therefore it is the process $(W_t)$ which gives us the information we need to answer meaningful question about the solution $(X_t)$, and in that sense, it is $(W_t)$ which drives the SDE.
(By the way, the above interpretation remains valid if we replace $W_t$ by any semimartingale.)

Another point I want to make is also that, in general, we are interested in the distribution of a solution $(X_t)$ at any given time $t$, or rather, in how that distribution changes at time evolves. When looking at it from a "purely distributional" viewpoint, the "main source of randomness" comes from the diffusion term $\sigma dW_t$ rather than the drift $\mu dt$.
I know I am being extremely handwavy here, so let me illustrate my point with this paper on diffusion based generative models : Maximum Likelihood Training of Score-Based Diffusion Models, Song et al. (2021). In this paper, the authors want to map a random variable with initial distribution $X_0$ to a standard normal random variable. They show that by gradually adding noise to $X_0$ they in fact create a stochastic process approximately solution of an SDE $$ dX_t = \mu(t,X_t)\ dt + \sigma(t,X_t)\ dW_t,\quad X_0\sim\rho_0$$ which converges in distribution to $\mathcal N(0,1)$ as $t\to\infty$, provided that the coefficients are well chosen. They call this the "forward SDE" which is driven by a Brownian motion given forward in time.
In the other direction, they explain how to take a $\mathcal N(0,1)$ r.v. and map it to a $\rho_0$ distributed r.v. by gradually adding noise as well, leading to an approximate solution of the "backward SDE" $$dX_t = \mu(t,X_t)\ dt + \sigma(t,X_t)\ d\bar W_t,\quad X_0\sim\mathcal N(0,1)$$ Where $\bar W_t := W_{T-t}$ is a Brownian motion going "backwards in time". In this case, it is particularly consistent to call a solution of such an SDE to be driven by a backwards Brownian motion.

I hope that at least some of the above makes sense to you.