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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.

plm
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    I’m voting to close this question because pondering about what is a correct nomenclature is pointless in mathematics. – Kurt G. May 15 '23 at 18:26
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    @KurtG. I strongly disagree with your statement. Also, this is definitely not a reason to close a question. We have a "terminology" tag for a reason after all. – Stratos supports the strike May 15 '23 at 20:45
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    @StratosFair, Thank you. Kurt G, perhaps this is more a question for Mathoverflow. If you close it i will probably ask there. To me there is an interesting mathematical issue here, but it may be more an open-ended questions, where there is no "yes/no" answer, formula, or proof to offer. But it is certainly not pointless, like say understand why "white noise" is called that, or a nilpotent group of matrices is not made up of nilpotent matrices. Terminology is very important to understand a mathematical topic and work with it. But it's ok if you close it -i don't know about politics here. – plm May 15 '23 at 23:59
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    I do not close a question but can (and did) vote for that. Secondly, to answer your question: that terminology is well accepted. If it is ideal or not is a matter of opinion. What counts is that we know what the terms in an SDE stand for mathematically - not how we baptize them. – Kurt G. May 16 '23 at 10:17
  • I always understood the driving terminology as a shorthand for the fact that the source of randomness comes from a particular stochastic process, so the randomness is driven by such. For GBM, which has a unique solution $X_t$ expressible as a Borel function of $W_t$, this is even more intuitive. I agree with @KurtG. that this question might create pointless fuss. This also surely does not belong on MO. Maybe a good reformatting would be asking about the story of such term using the tag [math-history]. – Snoop May 16 '23 at 18:09
  • Thank you @Snoop. Fortunately StratosFair was kind enough to answer. I have seen several such questions on Mathoverflow, and they used to be quite welcome in the beginning, but perhaps the people frequenting MO have become less welcoming so i don't know how it would be received now. About your statement "which has a unique solution $X_t$ expressible as a Borel function of $W_t$": you mean (strong) uniqueness almost surely wrt the measure on $(\mathbb R^{[0,\infty)},borels)$ or $(C([0,\infty)),borels)$ induced by $W$ [Cf Karatzas-Shreve] -Wiener measure in the 2nd case ? Or a stronger result ? – plm May 18 '23 at 12:55
  • I am referring to the simple fact that $X_t=x e^{(\mu-\sigma^2/2)t+\sigma W_t}$ when $X_0=x$ so we have $X_t=f(x,W_t)$. – Snoop May 18 '23 at 14:12
  • Thank you. But this is not the only solution, at least not among square integrable martingales on $(\mathbb R^{[0,\infty)},\mathcal B(\mathbb R^{[0,\infty)}))$ -one can for instance add a point discontinuity to the "canonical" $X_t$ above without changing the Itô integral, in various ways, for instance in a sample path. I am not sure if it is unique among continuous martingales, it must be easy to answer. Perhaps with some trick which preserves the Itô integral. – plm May 18 '23 at 21:38

2 Answers2

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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.

Stratos supports the strike
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  • [1/n] Thank you very much StratosFait, you provide alot of interesting comments. I had not looked at the "information content" perspective. I am not sure though it is the origin of the term: people would see diffusion of a brownian particle in a fluid as driven by white noise, one of the rare cases where there is basically no "deterministic force" it becomes a driving force applied to the "particle". The idea is that noise is always a "weak" force and when there is a "strong" force it will always oppose it, and be seen as a friction force. But the information content view is really useful too, – plm May 16 '23 at 10:05
  • [2/n] perhaps more mathematical, less physical. So im happy we could discuss all that here. Regarding your paper a few remarks: 1 They don't use the term "drive(n)". 2 It seems to me that the terminology is that it the (Hida or formal) time derivative of the diffusion process that is said to be driving the SDE; in the case of brownian motion, that is white noise. If you look at uses of the word "driven" they usually write the equation not with "differentials", implicitly as an Itô integral equation wrt to brownian motion or some martingales, but as honest differential equaltions with respect – plm May 16 '23 at 10:07
  • [3/n] to time (and more derivatives for SPDEs) and thus the noise comes in naturally from brownian motion as the derivative wrt time of $B_t$, whence the formulation i chose in my question. So i think in your article if they actually used "driven" they would rewrite the equation dividing by the time derivative, so $dt$ would not appear anywhere as factor (but only as denominator though they use $partial_t$ instead so we don't see the denominator); then they would say "driven by white noise". If we think of brownian motion as the motion of a particle its derivative wrt time – plm May 16 '23 at 10:08
  • [4/n] would be speed or momentum if we assume unit mass, and we usually think of dynamics (in particular of drive) in terms of force, but because the force exerted by our brownian particle on the solution is by conservative hit, we can think in terms of momenta and see the momentum, that is the noise, as driving the solution. But if we can forget one time derivative (compared to the usual newtonian dynamics) we can forget a second one, so we can as well say that the brownian motion does drive the solution process and indeed i've encountered also this terminology. – plm May 16 '23 at 10:08
  • [5/n, n=5] So "drive(n)" is here an overloaded word and im glad i thought through those things, however trivial. Here are examples first SDE then SPDE where "driven" used, with a derivative formulation (as opposed to the differential-Itô integral formulation of your paper): https://www.researchgate.net/publication/312781681_Stochastic_Processes_Driven_by_Colored_Noise_A_Path_Integral_Point_of_View SPDE examples: https://arxiv.org/pdf/1501.03360.pdf , https://perso.univ-rennes1.fr/ismael.bailleul/files/GuideRS.pdf Please correct me or comment if i messed or missed anything. Thanks again. :) – plm May 16 '23 at 10:09
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    You're welcome. Unfortunately, the comment section is not for extended discussions, so let me quickly address some of your points. After which, if there is more you want to ask, you should just ask a separate, specific, question. (or maybe you could create a chatroom and ping me specifically, but I'm almost surely not knowledgeable enough about the subject and I have not much interest in continuing this discussion further anyway). – Stratos supports the strike May 17 '23 at 11:48
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    I first wanted to say that by no means my explanation was based on any historical reference, but simply a possible way to think about it. The idea being that the Brownian motion (or source of randomness, as Snoop said in a comment), is what carries most of the "distributional information" on $X_t$ when we think of it as a random variable. Perhaps when we think of it from a purely physical perspective, this doesn't make as much sense, but it seems like ultimately, people have stuck with that terminology. – Stratos supports the strike May 17 '23 at 11:53
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    Secondly (and lastly), the paper I referenced may have not mentioned that explicitly (my bad), but if you read more literature on diffusion models, people do refer to the random noise as what "drives" the model. The idea is that the random noise "drives" the distribution $\rho_0$ to $\mathcal N(0,1)$ (and vice versa). Ultimately, yes, this is a slightly overloaded term which can be confusing, so if you want a definitive answer as to why it stuck around based on a mathematical history perspective, I suggest you ask a separate question specifically about that. – Stratos supports the strike May 17 '23 at 11:58
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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.

Sean Roberson
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Kobrakixx
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