PRH | Essay | 7.10 • Blur Versus Itô Calculus

Blur Versus Itô Calculus Stochastic Uncertainty and Epistemological Blur Aleksandar Perišić November 2025 Abstract Both the blur method and Itô stochastic calculus are ways of reasoning under incomplete information. At first sight they even use some of the same analytic tools (Gaussian kernels, heat semigroups, quadratic variation). But they live on different levels. Itô calculus starts from a probability space and a genuine stochastic process; its task is to separate and control a deterministic part and a random part inside one and the same object. Blur, in contrast, is a meta–method: it never splits any particular process into “known” and “unknown” pieces, but instead budgets what our theory and resources can reliably resolve, and what must remain epistemically blurred for the purposes of the current argument. The goal of this note is to compare these two views systematically. We emphasize: (i) what kind of “unknown” each formalism talks about; (ii) how Gaussian objects and heat kernels play different roles in the two settings; and (iii) in what sense Itô calculus would appear, in retrospect, as a perfectly natural theorem inside a preexisting blur framework. 1 Introduction Itô calculus was invented to describe time evolution when uncertainty is built into the model from the start. A typical process Xt=X0+Zt 0 b(Xs, s)ds +Zt 0 σ(Xs, s)dWs is a mixture of a predictable drift and a Brownian noise term. The calculus tells us how such processes move, how functionals f ( Xt )behave, and how to track statistics such as expectations, variances, and distributions. The blur method was developed for a different purpose. It starts from the observation that in serious mathematics we almost never work with perfectly sharp objects. We implicitly tolerate hidden processes, approximations, and coarse summaries as long as certain invariants survive unchanged. Blur makes this tolerance explicit: one chooses a small, positive averaging (an “approximate identity” or kernel), and a budget that states how much unresolved structure one agrees to carry. Blur insists that the meaning we care about—the invariant we are reading from the system—is robust under such admissible regularizations. On the surface there is a temptation to say: blur is “just like” a probabilistic method, and many blur examples even use Gaussian kernels or other familiar distributions. The purpose of this article is to clarify why this identification is misleading and to draw a clean contrast between Itô calculus and blur. Very roughly: • Itô calculus is a calculus on stochastic processes: it assumes randomness as part of the ontology and learns to live with it. • Blur is a calculus on our knowledge: it assumes that some coordinates of reality are inaccessible at the chosen budget and designs positive, normalized blurs that hide those coordinates while preserving the invariants we can test. 1 The two worlds do meet: an Itô process is a perfectly valid input to a blur argument. But the logical direction is important: Itô calculus stays inside probability theory; blur sits one level above and could just as well be applied to PDEs, prime numbers, Collatz dynamics, or classical geometry. 2 A very short reminder of Itô calculus We recall just enough classical material to have something concrete to compare with blur. Standard references include Itô’s original paper and the textbooks by Øksendal and by Karatzas and Shreve [1,2,3]. 2.1 Itô processes and semimartingales Fix a filtered probability space (Ω ,F, ( Ft ) t≥0,P )satisfying the usual conditions, and let ( Wt ) t≥0 be a standard Brownian motion. A (scalar) Itô process is a process of the form Xt=X0+Zt 0 bsds +Zt 0 σsdWs, where X0 is F0 -measurable, ( bt )and ( σt )are adapted processes satisfying mild integrability conditions, and the second integral is an Itô integral. In the general language of stochastic calculus, Xis a semimartingale, i.e. a sum of a local martingale and a finite-variation process. Already here we see a sharp split: •At := Rt 0bsds is of finite variation and encodes the drift: the part of the motion that is predictable from the past at the given level of modeling. •Mt := Rt 0σsdWs is a local martingale and encodes the noise: its conditional expectation given the past is zero. The randomness is not a tool; it is an intrinsic part of the object. 2.2 Itô integrals and quadratic variation The Itô integral is first defined for simple adapted processes, Ht= n X k=1 Hk1(tk−1,tk](t), by ZT 0 HtdWt:= n X k=1 HkWtk−Wtk−1, and then extended in the L2-sense. A characteristic property is isometry: EhZT 0 HtdWt2i=EhZT 0 H2 tdti. The square-integrable martingales Mt = Rt 0HsdWs admit a quadratic variation ⟨M⟩t = Rt 0H2 sds . For Brownian motion itself we have ⟨W⟩t = t , reflecting the scaling of Gaussian increments. The presence of quadratic variation is the most visible way in which the Itô world differs from classical smooth calculus: one can still write down an informal “ dW2 t = dt ”, but this is a statement about random fluctuations that survive in the limit and about which the calculus is very precise. 2 2.3 Itô’s formula For a twice continuously differentiable function f : R→R and an Itô process X as above, Itô’s formula reads f(Xt)=f(X0) + Zt 0 f′(Xs)bsds +1 2Zt 0 f′′(Xs)σ2 sds +Zt 0 f′(Xs)σsdWs. There is a clean separation of roles: the two dt -integrals form a finite-variation process (drift plus Itô correction), while the last term is a martingale. The formula does not remove the randomness; it describes how the random and nonrandom parts of f(Xt)evolve. The same structure appears in vector-valued settings and for more general semimartingales; in each case the central object is still a process whose randomness is part of the ontology and persists to the end of the computation. 2.4 A toy example: linear SDE and Gaussian blurring To have a concrete model in mind, consider the scalar SDE dXt=µ dt +σ dWt, X0=x, with constants µ, σ ∈R,σ= 0. Its unique strong solution is Xt=x+µt +σWt. For a test function f∈C2 b(R)define u(t, x) := E[f(Xt)|X0=x]. By Itô’s formula applied to f ( Xt )and then taking expectations, one finds that u solves the linear parabolic PDE ∂tu(t, x) = µ ∂xu(t, x) + σ2 2∂xxu(t, x), u(0, x) = f(x). Equivalently, u(t, ·) = Ptf, where (Pt)t≥0is the Brownian semigroup with drift, and one can write explicitly (Ptf)(x) = ZR f(y)κt(x−y)dy, κt(z) = 1 √2πσ2texp −(z−µt)2 2σ2t. From the Itô point of view, this is a statement about the distribution of Xt : Xt is Gaussian with mean x+µt and variance σ2t, and Ptcomputes expectations with respect to that law. From the blur point of view, a different emphasis is possible. At each fixed t > 0the operator Pt is convolution with a positive, normalized kernel κt at spatial scale √t : it is a Gaussian blur of the initial profile f . The quantity u ( t, x )is then a blur-stable observable of the system: it depends only on f as seen through a Gaussian window of variance σ2t , and the precise microscopic path structure of (Xs)0≤s≤tis invisible at this resolution. In other words, the same formula plays two roles: Itô calculus reads it as a description of a probability law, while blur can read it as a prescription for which invariants of f survive after Gaussian blurring at a given scale. 3 Blur in a nutshell We now summarize the blur framework just enough for a comparison. In what follows, “blur” always means a positive, normalized averaging with an explicit resolution parameter and a corresponding epistemic budget. 3 3.1 Blur-invariant meaning and information budgets The starting point is the notion of blur-invariant meaning of a constant [ 4 ]. Very informally: when we say “ π ” or “ e ”, we do not mean a specific infinite string of digits; we mean an entire isomorphism class of admissible procedures that all land on the same semantic invariant. Different limits, algorithms, and integrals are all tolerated as long as they preserve the same underlying meaning (circle/circumference ratio, exponential flow, Fourier kernel normalization, and so on). This principle is made concrete by splitting information into two components relative to a theory Th and a budget b(time/space/proof–length bound): bits(π↾N)=deducibleTh,b(N)+randomTh,b(N). Here “bits” is conceptual bookkeeping: one can make this precise using various notions of algorithmic or proof complexity, but for the present discussion we only need the idea that some bits admit certificates inside (Th, b)while the rest do not. The first term counts bits whose values admit certificates in Th that fit into budget b ; the second term counts bits that are epistemically random at that budget [ 4 ]. There is nothing probabilistic here: these bits may be fully computable in principle. They are “random” only in the sense that our current resources do not suffice to resolve them. Blur, in this sense, is always relative: it separates what we can control from what we currently cannot, and it demands that our methods respect this separation explicitly. 3.2 Blur as a method: kernels, safety margins, and invariants On the analytic side, blur is implemented by approximate identities: positive kernels that converge to a Dirac mass and whose convolution does not change the invariants we care about. In the additive channel one uses kernels of the form (B+ τf)(x) = ZR κτ(x−u)f(u)du, where ( κτ ) τ>0 is a family of even, nonnegative functions with Rκτ = 1 and κτ→δ0 as τ↓ 0. In the multiplicative channel one uses kernels on the log–line with the Haar measure du/u, (B× εf)(x) = Z∞ 0 Kεx uf(u)du u, again with Kεforming an approximate identity in log x[5]. In elementary and undergraduate settings the same philosophy appears as simple averaging plus an explicit safety margin: inside/outside polygon means for the circle, moving averages for noisy counts, symmetric differences for numerical derivatives, finite truncations for prime products, and so on. The blur is the averaging; the safety margin is the budget for the error that remains after blur [6]. The key point is methodological: • You declare, up front, how you will blur (which kernel, at what scale) and how much error you are willing to carry. • You insist that the quantity you are reading (an invariant, a sign, a bound) is stable under such blur—if it is not, the method itself tells you that your current description of the system is insufficient. 4 3.3 Blur between addition and multiplication A guiding example is the seam between addition and multiplication. From the Fourier/Mellin viewpoint, addition lives on ( R, dx )with its Fourier transform, while multiplication lives on ((0 ,∞ ) , dx/x )with its Mellin transform. After the change of variables x = eu the Mellin transform becomes a Fourier transform on the log–line; the standard uncertainty principle then forbids us from keeping both channels perfectly sharp at the same time. In concrete terms, if we insist on performing part of a computation in the multiplicative channel and reading it in the additive channel, we are obliged to blur on the log–line first. This is encoded in the construction of soft addition and soft multiplication, where one works with blurred identities B+ τand B× εand tracks a small channel–switch error that inevitably appears when both uncertainty knobs are pushed toward zero simultaneously. The central moral is that blur is not cosmetic: it is the toll one must pay to mix +and × in a mathematically honest way, with Euler’s constant γ appearing as the finite part that survives when two different blurs are taken to zero along any legal schedule [5]. 4 What kind of “unknown” do we talk about? We can now compare the two formalisms along several axes. 4.1 Ontology: random world vs. blurred knowledge In Itô calculus the randomness is part of the world: •There is a fixed probability space, a filtration, and one or more Brownian motions. • The processes we study are random variables indexed by time. Their paths are almost surely nowhere differentiable and have nontrivial quadratic variation. • The calculus is about describing and manipulating these genuinely random objects and their laws. Even if one uses Itô calculus only to compute expectations or probabilities of events, the underlying ontology remains probabilistic. In blur, the “unknown” is not an assumed random noise; it is the portion of the system that we have chosen not to resolve at the current budget. We work with: •A current reservoir of knowledge Kcur (theorems, algorithms, bounds); •A target Kdes (some statement or numerical invariant we want); •A deep, often unformulated Kinacc (the true but unmanageable structure of the object). Blur is a controlled relaxation By [ Kinacc ]which hides the unmanageable coordinates but preserves the invariants we can test. The method never commits to any specific probabilistic model of the unknown part; it only requires that the blurred objects behave consistently with the invariants being tracked. Thus the unknown in Itô calculus is ontological (the world is assumed random, at least in the model); the unknown in blur is epistemological (we choose not to resolve certain coordinates and are honest about that choice). 5 4.2 Splitting the object versus splitting our knowledge Itô calculus explicitly splits an object into deterministic and random parts. In the semimartingale decomposition Xt=X0+Mt+At the finite-variation part At and the martingale part Mt play different roles, and a large part of stochastic calculus is dedicated to preserving and exploiting this split. Blur never splits the process or quantity in this way. Instead it splits our knowledge: bits =deducible at (Th, b) + epistemically random at (Th, b). The same numerical object may migrate from the “epistemically random” side to the “deducible” side as our budget b grows; blur is explicitly designed to model this migration. In Itô calculus, by contrast, a Brownian motion does not become “less random” if we increase computing power: the randomness is not a temporary lack of knowledge but part of the definition. 4.3 Gaussian kernels: distribution vs. blur Both worlds use Gaussians and heat kernels, but in different roles. In the Itô world: • The Gaussian law describes the distribution of increments of Brownian motion. The heat equation appears as the backward (or forward) Kolmogorov equation governing transition densities. •The Brownian semigroup (Pt)t≥0is a Markov semigroup acting on functions via Ptf(x) = E[f(x+Wt)]. Here Ptis not a blur we freely choose; it is part of the specification of the model. In the blur world: • Gaussian kernels are used as approximate identities on the time or log–axis. They are tools we choose because they have good analytic properties: positivity, normalization, fast decay in frequency, and well-controlled limits as the variance tends to zero. • A Gaussian blur κτ on the log–line is not a claim that the underlying multiplicative structure is random with a Gaussian law. It is a regularizer that we know how to remove in the limit while preserving the blur-invariant meaning of the quantities we care about. Formally the integrals and convolutions may look very similar. But the direction of interpretation is opposite: Itô calculus starts from a probability law and derives consequences; blur starts from invariants and admissible kernels and is agnostic about whether the underlying data are random or not. 4.4 Does the result remain random? Applying Itô calculus to an Itô process never removes its randomness. If X is driven by Brownian motion, then f ( Xt )is again a random variable; Itô’s formula merely reorganizes its evolution into a drift and a martingale part. Even asymptotic limits (law of large numbers, central limit theorems, invariant measures) talk about convergence in distribution or almost sure convergence; the randomness is still there. Blur, by design, does not decide whether the underlying object is random. It only insists that whatever we claim at the end is stable under the chosen blurs and consistent with the information budget. The same blur argument may apply equally well to: 6 •primes and explicit formulas; •Collatz dynamics and residue-class averages; •approximate constants such as π,e,orγderived from multiple representations; • a stochastic process, if we are interested in blur-invariant observables rather than the full law. In all these cases blur refuses to classify the outcome as “random” or “nonrandom”; it simply enforces that the conclusion is supported by invariants that do not depend on the unresolved coordinates. 5 How Itô calculus looks from the blur perspective It is instructive to imagine a world in which the blur method was developed first, and Itô calculus appears later as a special instance. 5.1 SDEs as systems with inaccessible microstructure Consider an SDE dXt=b(Xt, t)dt +σ(Xt, t)dWt on Rd. Blur would treat this as a black-box evolution with two visible properties: •a deterministic part encoded by the generator (Ltf)(x) = X i bi(x, t)∂xif(x) + 1 2X i,j aij(x, t)∂xi∂xjf(x), where a=σσ⊤; • a blur scale in time and space induced by the quadratic variation and the heat kernel associated with Lt. From this viewpoint, the Brownian noise is simply the mechanism that forces us to read the process through a Gaussian blur. The Itô correction 1 2σ2f′′ in the scalar case is exactly the part of the generator that cannot be seen by a purely deterministic ( dt )-based calculus. The fact that the noise is Gaussian is one possible choice of blur kernel that makes the invariants of interest accessible. In other words, blur would say: the true microscopic motion of Xt is inaccessible; what we can resolve at our budget are the blur-invariant objects such as the semigroup Ps,t , the generator Lt , and expectations of functions of Xt . The Itô formula is then a precise statement about how these invariants behave under test functions f. 5.2 Itô’s formula as a blur-stable identity From the blur perspective, Itô’s formula is exactly the kind of identity one likes: f(Xt)=f(X0)+(blur-stable drift) +(zero-mean fluctuation). The fluctuation term (the martingale integral) has expectation zero and can often be bounded in L2 or almost surely. The drift term is what survives after heavy blurring in time, for instance when we integrate against smooth test functions or pass to expectations. In many applications of stochastic calculus one only keeps the drift term: one takes expectations and reads off a PDE satisfied by u ( t, x ) = E [ f ( Xt ) |X0 = x ]. This is precisely a blur move: we sacrifice pathwise information and keep only the invariant encoded in the evolution of u. 7 5.3 Random walks, scaling limits, and blur The classical way to construct Brownian motion is as a limit of scaled random walks. In such constructions the Gaussian law appears as a blur invariant: many different microscopic step distributions, under suitable normalization, produce the same macroscopic limit. From the blur vantage point what matters is not the exact distribution of increments but the convergence of their rescaled sums to a blur-invariant object (Brownian motion) whose law is characterized by a small number of parameters. Here the same epistemological pattern appears as in number theory: • Many different micro-level configurations (paths, primes, digit strings) are compatible with the same macro-level invariants. • Blur claims that these invariants are the right unit of knowledge; the rest is epistemically random at the current budget. • Itô calculus adds the strong statement that the macro-level invariants can be encoded by a probability law on path space and manipulated via a calculus. The calculus thus refines one particular blur-invariant object (Brownian motion) into a full probabilistic theory of continuous-path martingales. 6 Didactic and philosophical remarks 6.1 Blur as the generic learning pattern Long before probability theory, mathematicians and philosophers were already using blur. Greek geometry routinely sandwiched a circle between polygons, averaged over symmetric configurations, and took limits that ignored small fluctuations. The method was always the same: identify a quantity that lies between two simpler expressions, blur away the details, and read off the invariant. The same pattern appears in school mathematics. A seven-year old tolerates a lot of blur in their first contact with numbers and shapes. A PhD candidate tolerates much less blur in their research, but the schema is unchanged: start from what you know, leave a margin for what you do not, and refine only where the argument actually demands it. The blur framework makes this pattern explicit and turns it into a formal policy: state your budget, pick blur kernels that respect the invariants you care about, and stop refining when the marginal information per unit budget drops below your threshold. 6.2 Where Itô calculus fits in this picture If blur had been available as a standard meta-language before the rise of probability, Itô calculus would likely have been introduced as one of its canonical theorems: • Start from a class of blur-invariant processes characterized by their quadratic variation and drift. • Show that under mild conditions this class admits a canonical representation as solutions to SDEs driven by Brownian motion. • Develop a calculus for functionals of such processes (Itô’s formula, Girsanov’s theorem, martingale representation) that respects the blur-invariant objects (semigroups, generators, invariant measures). 8 In that sense Itô calculus and blur belong to the same “family” of ideas: both are about reading stable invariants from noisy or partially hidden systems. The difference is in the meta-level. Itô calculus takes a probabilistic description of the world as given and works within it. Blur is neutral about the underlying ontology and therefore remains applicable to whatever new theory arises, stochastic or not. 7 Conclusion We can summarize the contrast as follows. • Level. Itô calculus is object-level: it is a calculus of random processes. Blur is meta-level: it is a discipline for tracking what can and cannot be known at a given budget. • Unknown. In Itô calculus the unknown is modeled as stochastic noise with a specified law (typically Gaussian). In blur the unknown is whatever part of reality we elect not to resolve; it need not be probabilistic and may even be fully deterministic in principle. • Decomposition. Itô calculus decomposes processes into drift plus martingale and keeps that split throughout. Blur decomposes information into deducible and epistemically random bits and allows bits to migrate as resources grow. • Kernels. Both use Gaussian or related kernels, but for different reasons: Itô uses them to define laws of motion; blur uses them as flexible, removable regularizers. • Scope. Itô calculus is tailored to stochastic differential equations and martingales. Blur applies equally to constants, primes, combinatorial dynamics, and yes, also to stochastic processes, as long as one is interested in invariants that survive blurring. None of this is a criticism of the probabilistic ontology itself: Itô calculus is extraordinarily successful within that framework, both conceptually and in applications. The point is rather that blur can be formulated without committing to any such ontology and therefore remains available as a meta-language even when the “noise” is combinatorial, arithmetic, or purely epistemic. From a didactic and epistemological point of view, blur is, in this sense, the more primitive notion: it describes the way knowledge grows and stabilizes in the presence of inaccessibility, whether that inaccessibility is stochastic, combinatorial, or analytic. Itô calculus is a deep and beautiful specialization of this general schema to one particularly rich kind of system—Brownian motion and its descendants. References [1] K. Itô, Stochastic integral, Proc. Imperial Acad. Tokyo 20 (1944), 519–524. [2] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 6th ed., Springer, 2010. [3] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Springer, 1991. [4] A. Perišić, Epistemological Blur, Zenodo, 2025. [5] A. Perišić, Soft-Addition and Soft-Multiplication and the Channel–Switch Error, Zenodo, 2025. [6] A. Perišić, Blur-as-a-Method: Small, Hands-On Problems (Elementary → Undergraduate), Zenodo, 2025. 9