Stochastic calculus extends classical analysis to random processes, providing the mathematical framework for modeling continuous-time phenomena in finance, physics, and biology. Unlike ordinary calculus, which deals with deterministic functions, stochastic calculus handles processes whose evolution is driven by randomness. This course develops the theory and techniques needed to integrate with respect to random paths, differentiate stochastic processes, and solve equations whose solutions are inherently random. The fundamental challenge — and the core innovation — is that random paths lack the regularity of classical curves, requiring entirely new definitions of integral and derivative.

The course begins by establishing the measure-theoretic foundations through the Lebesgue–Stieltjes integral, which generalizes classical integration to functions of bounded variation and serves as the stepping stone to stochastic integration. Semimartingales are then introduced as the natural class of processes that can be decomposed into a predictable component and a purely random component, making them ideal candidates for integration theory. The stochastic integral is then constructed, first for simple integrands and extended to adapted processes, with the central result being Itô's formula — the chain rule for stochastic processes. Finally, stochastic differential equations are treated as integral equations, providing both existence–uniqueness theorems and techniques for solving important classes of equations that appear throughout applications.

Each chapter builds on the previous: measure theory enables precise definitions of variation and integrability; semimartingales organize processes by their decomposition properties; the stochastic integral operates on this structure to produce a usable calculus; and SDEs are then solved by appealing to all prior machinery.

# Introduction

This course begins where ordinary differential equations leave off: the world is noisy, and the deterministic framework of classical analysis is insufficient to model systems subject to random perturbations. Stochastic calculus provides the rigorous machinery to handle such systems, and this introductory chapter motivates the entire enterprise — explaining why white noise is not a function, how Brownian motion enters naturally as the integral of white noise, and how the Wiener integral gives a first taste of what the Itô integral will achieve in full generality.

[motivation] **From ODEs to SDEs.** Classical analysis revolves around ordinary differential equations of the form \begin{align*} \dot{x}(t) = F(x(t)). \end{align*} Many physical systems, however, are subject to random perturbations: a particle in a fluid, stock prices, neural firing rates. A natural way to model such a system is to add a random forcing term, \begin{align*} \dot{x}(t) = F(x(t)) + \eta(t), \end{align*} where $\eta$ is a random function representing noise. The question is: what properties should $\eta$ have? If we are modeling physical noise — thermal fluctuations, quantum randomness, measurement error — we expect the noise at widely separated times to be essentially independent. Noise at time $t$ carries no memory of what happened at time $s$ when $|t - s|$ is large. The idealization of this observation is to demand that $\eta(t)$ and $\eta(s)$ are independent for every $t \neq s$. Such a process is called **white noise**. **Why White Noise Is Not a Function.** The independence requirement places a severe constraint on $\eta$. If $\eta(t)$ and $\eta(s)$ were independent for every $t \neq s$ and $\eta$ were a measurable function, then $\eta$ would have to be almost everywhere constant — a contradiction. More precisely, white noise turns out to exist only as a Schwartz distribution, not as a genuine function. This is the first fundamental obstacle the course must overcome. **The Integral Formulation and Brownian Motion.** To understand the simplest case, set $F = 0$. The equation reduces to $\dot{x} = \eta$, or in integral form, \begin{align*} x(t) = x(0) + \int_0^t \eta(s)\, ds. \end{align*} For this integral to make sense, $\eta$ should at least be a signed measure. But white noise is not even that, so the integral cannot be interpreted classically. We proceed nonetheless by examining what properties such an $x$ would have to satisfy. For any partition $0 = t_0 < t_1 < \cdots < t_n$, the increments \begin{align*} x(t_i) - x(t_{i-1}) = \int_{t_{i-1}}^{t_i} \eta(s)\, ds \end{align*} should be independent (since the noise values on disjoint time intervals are independent), and their variance should scale as $|t_i - t_{i-1}|$. These are precisely the defining properties of Brownian motion increments. Thus, the "integral of white noise" should be Brownian motion. **Why Work in Continuous Time?** One might ask: why not simply discretize time and work with finite sums? The answer parallels the relationship between Riemann sums and the Lebesgue integral. The Lebesgue integral requires significant foundational work to construct, but once built, it is a vastly more powerful tool: integrating $1/x^3$ requires no special tricks, whereas summing $\sum_{n=1}^\infty 1/n^3$ in closed form is a much harder problem. Similarly, stochastic calculus in continuous time, once constructed, yields explicit computations and structural results — Itô's formula, the Lévy characterization, Girsanov's theorem — that have no easy discrete analogues. Furthermore, many important continuous-time processes are naturally described as solutions to stochastic differential equations, just as trigonometric functions and Bessel functions are characterized as solutions to ordinary differential equations. The SDE viewpoint gives both a computational handle and structural insight. **The Two Integrals: Itô and Stratonovich.** There are two principal approaches to defining stochastic integrals: the **Itô integral** and the **Stratonovich integral**. This course focuses on the Itô integral. One key advantage is that the Itô integral of an adapted process with respect to a martingale is again a martingale — a property that makes it the natural tool for probabilistic analysis. The Stratonovich integral, while more natural from the perspective of differential geometry and chain rules, is less convenient for the probabilistic techniques that dominate this course. [/motivation]

## Gaussian Spaces and Isometric Embeddings

The first obstacle in building a stochastic integral is that we need to land in the right space. An integral against Brownian motion should produce a random variable, and that random variable should carry Gaussian statistics — but how can we guarantee this algebraically, without ad hoc calculations? The answer is to build the integral as an isometric embedding into a Gaussian subspace of $L^2(\Omega)$. We begin by making this target space precise.

[definition:GaussianSpace] Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. A subspace $S \subseteq L^2(\Omega, \mathcal{F}, \mathbb{P})$ is called a **Gaussian space** if $S$ is a closed linear subspace and every $X \in S$ is a centered Gaussian random variable (i.e., $X \sim N(0, \sigma^2)$ for some $\sigma^2 \geq 0$). [/definition]

The point of the definition is that Gaussian spaces are stable under $L^2$ limits: if $(X_n)$ is a sequence of centered Gaussians in $S$ converging in $L^2$ to $X$, then $X$ is also centered Gaussian (since $L^2$ convergence implies convergence of characteristic functions, and the characteristic function of $N(0, \sigma_n^2)$ converges to that of a Gaussian). Closed linear subspaces of $L^2$ that consist entirely of centered Gaussians are therefore the natural arena in which to build stochastic integrals via Hilbert space methods.

The crucial existence result shows that every separable Hilbert space can be "realized" as a Gaussian space, with the Hilbert space inner product corresponding exactly to the covariance structure of the associated random variables.

[quotetheorem:2067]

[citeproof:2067]

This theorem is the foundation on which everything else rests, so it is worth pausing to understand exactly what it says — and what it does not say.

First, why separability? The proof constructs the isometry by mapping each element of a countable orthonormal basis to an independent standard Gaussian, then extending by linearity and $L^2$ closure. If $H$ were not separable, no countable basis would exist and this construction would break down. Non-separable Hilbert spaces do admit Gaussian isometries (by a transfinite argument), but the proof is substantially harder and we never encounter non-separable spaces in this course.

Second, uniqueness: the isometry $I$ is not unique. Different choices of orthonormal basis yield different assignments $e_i \mapsto X_i$, and different probability spaces are possible as well. What is unique — up to isomorphism of Gaussian spaces — is the covariance structure: any two Gaussian isometries $I, I': H \to L^2(\Omega)$ that agree on inner products are related by a unitary transformation on the target Gaussian space. For our purposes, we fix one such isometry once and for all.

Third, why an isometry rather than merely a bounded linear map? The isometry condition $\mathbb{E}[I(f)I(g)] = (f,g)_H$ is exactly the Itô isometry in embryonic form. It ensures that $L^2$ convergence of integrands implies $L^2$ convergence of integrals — the property that allows us to extend from step functions to all of $L^2$ by density. A bounded map that is not an isometry would distort inner products and destroy this extension.

Taking $H = L^2(\mathbb{R}_+)$ — the $L^2$ space of square-integrable functions on $[0, \infty)$ — we obtain an isometry that sends each $L^2$ function to a centered Gaussian random variable while preserving the inner product exactly. This isometry is precisely what we will call Gaussian white noise.

## Gaussian White Noise

Given that Brownian motion is supposed to be the "integral of white noise", one naturally asks: what is white noise itself? The difficulty is that white noise cannot be a function — as the motivation block explained, any measurable function with independent values at every pair of distinct times must be essentially constant. What white noise can be is a linear functional on $L^2$, and the Gaussian space framework tells us exactly what kind.