Stochastic Calculus
Introduction
Let (\Omega, \mathcal{F}, \operatorname{P}) be a probability space. Remember that \Omega is the set of all the possible outcomes and \mathcal{F} contains all the events A \subset \Omega that we can assert if they happen or not.
In continuous time we define a stochastic process X_{t}(\omega) as a collection of random variables such that, given an outcome \omega \in \Omega, we can determine the path of the stochastic process over time. We can also think about the stochastic process in the opposite way. That is, for a given time t \leq T, how does the random variable X_{t}(\omega) behaves.
A filtration \{F_{t}\} determines how information is disseminated as we observe a stochastic process. At the very least, we want the filtration to remember what has happened before so that \mathcal{F}_{s} \subset \mathcal{F}_{t} when 0 \leq s < t.
Itô Processes
An Itô process \{X_{t}\} is a continuous-time stochastic process that can be written as the sum of an ordinary (pathwise) Lebesgue time integral and an Itô stochastic integral: X_{t}(\omega) = \int_{0}^{t} a(s, \omega) \mathop{}\!\mathrm{d}s + \int_{0}^{t} b(s, \omega) \mathop{}\!\mathrm{d}B_{s}(\omega), \tag{1} where the coefficient functions a(t,\omega) and b(t,\omega) are \mathcal{F}_{t}-adapted processes such that \int_{0}^{t} |a(s,\omega)| \mathop{}\!\mathrm{d}s < \infty, and \int_{0}^{t} b(s,\omega)^{2} \mathop{}\!\mathrm{d}s < \infty almost surely. For compact notation we commonly write \mathop{}\!\mathrm{d}X = a \mathop{}\!\mathrm{d}t + b \mathop{}\!\mathrm{d}B, with the understanding that this notation represents the integral representation in (1).
The stochastic integral is constructed by approximating the integrand with step processes on partitions. For a partition \Pi = \{t_{0}, t_{1}, \ldots, t_{n}\} of [0,T] with 0 = t_{0} < t_{1} < \dots < t_{n} = t, and \|\Pi\| \to 0 as n \to \infty, the Itô integral is defined as the mean-square (L^{2}) limit of Riemann-type sums: I_{t}(\omega) = \int_{0}^{t} b(s, \omega) \mathop{}\!\mathrm{d}B_{s}(\omega) = \lim_{n \to \infty} \sum_{j = 0}^{n - 1} b(t_{j}, \omega) \Delta B_{t_{j}}(\omega), \tag{2} where the limit is taken in L^{2} and, in particular, requires \operatorname{E}(I_{t}^{2}) < \infty.
The stochastic integral I_{t}(\omega) is a random process: its value at each time t depends on the sample point \omega. Under the usual measurability and square-integrability conditions on the integrand b(t, \omega), the integral admits a modification that is continuous in t for almost every \omega (i.e., there exists an indistinguishable version with continuous sample paths). Hence, without loss of generality, we may take I_{t} to have continuous sample paths.
Itô Integrals are Martingales
Consider first a simple, adapted integrand of the form b(t,\omega) = \sum_{j = 0}^{n - 1} b(t_{j}, \omega) \mathbf{1}_{(t_{j}, t_{j + 1}]}(t), so that the Itô integral on this partition is I_{t} = \sum_{j = 0}^{n - 1} b(t_{j}, \omega)\left(B_{t_{j + 1}} - B_{t_{j}}\right). Fix 0 \le s \le t and let k be the index with t_{k} \le s < t_{k+1}. Split the sum into the contributions up to time s and those after s so that I_{t} = \underbrace{\sum_{j = 0}^{k - 1} b(t_{j})\left(B_{t_{j + 1}} - B_{t_{j}}\right)}_{=I_{s}} \;+\; \sum_{j = k}^{n - 1} b(t_{j})\left(B_{t_{j + 1}} - B_{t_{j}}\right). The first term is \mathcal{F}_{s}-measurable. For each j \ge k, b(t_{j}) is measurable with respect to \mathcal{F}_{t_{j}} and the increment B_{t_{j + 1}} - B_{t_{j}} is independent of \mathcal{F}_{t_{j}} and has mean zero, so \operatorname{E}\left[b(t_{j})(B_{t_{j + 1}} - B_{t_{j}}) \mid \mathcal{F}_{s}\right] = 0. Taking conditional expectation yields \operatorname{E}(I_{t} \mid \mathcal{F}_{s}) = I_{s}, so I_{t} is a martingale.
The result for general square-integrable adapted integrands follows by approximating arbitrary predictable integrands by simple ones and using the Itô isometry to pass to the limit.
Conversely, for the Brownian filtration there is the martingale representation theorem: any \mathcal{F}_{t}-adapted martingale \{M_{t}\} with \operatorname{E}(M_{t}^{2})<\infty for all t can be written as M_{t} = M_{0} + \int_{0}^{t}\varphi(s, \omega) \mathop{}\!\mathrm{d}B_{s}, for a predictable process \varphi satisfying \operatorname{E}\!\left(\int_{0}^{t}\varphi(s, \omega)^{2}\mathop{}\!\mathrm{d}s\right)<\infty. This gives the converse representation as a stochastic integral with respect to Brownian motion.
Itô Isometry
The Itô isometry expresses the second moment of the stochastic integral: \operatorname{E}(I_{t}^{2}) = \operatorname{E}\left(\int_{0}^{t} b(s, \omega)^{2} \, ds\right), i.e., the mean square of the integral equals the expectation of the time-integral of the squared integrand.
Consider a simple, adapted integrand b(s, \omega) = \sum_{j = 0}^{n - 1} b(t_{j}, \omega)\mathbf{1}_{(t_{j}, t_{j + 1}]}(s), for which I_{t} = \sum_{j = 0}^{n - 1} b(t_{j})\left(B_{t_{j + 1}} - B_{t_{j}}\right). Using independence and zero mean of non-overlapping Brownian increments and orthogonality of cross-terms, \begin{aligned} \operatorname{E}(I_{t}^{2}) &= \operatorname{E}\left(\sum_{j = 0}^{n - 1} b(t_j)^{2} (B_{t_{j + 1}} - B_{t_{j}})^{2}\right) = \operatorname{E}\left(\sum_{j = 0}^{n - 1} b(t_j)^{2} (t_{j + 1} - t_{j})\right) \\ &= \operatorname{E}\left(\int_{0}^{t} b(s, \omega)^{2} \mathop{}\!\mathrm{d}s\right). \end{aligned} The general result follows by approximating a square-integrable predictable integrand by such simple processes and passing to the limit in L^{2}. The isometry therefore gives an isometric linear map from the space of square-integrable predictable integrands (with norm given by \operatorname{E}\int_{0}^{t} b(s)^2 \mathop{}\!\mathrm{d}s) into L^{2} of the resulting stochastic integrals.
In particular, this yields the square-integrability requirement \operatorname{E}(I_{t}^{2}) = \int_{0}^{t} b_{s}^{2} ds < \infty.
Quadratic Variation
The quadratic variation of a continuous semimartingale X is the (pathwise) limit of squared increments along a refining sequence of partitions \Pi = \{0 = t_{0} < t_{1} < \dots < t_{n} = t\}: [X, X]_{t} = \lim_{n \to \infty} \sum_{j = 0}^{n - 1} \left(\Delta X_{t_{j}}\right)^{2}, whenever the limit exists in probability (or almost surely for continuous local martingales).
Quadratic Variation of the Stochastic Integral
For the Itô integral I_{t} = \int_{0}^{t} b(s, \omega) \mathop{}\!\mathrm{d}B_{s}, one has the explicit expression for its quadratic variation: [I, I]_{t} = \int_{0}^{t} b^{2}(s, \omega) \mathop{}\!\mathrm{d}s. We usually summarize this infinitesimally as d[I, I]_{t} = (dI)^{2} = b^{2} \mathop{}\!\mathrm{d}t.
Sketch of proof. Take a simple adapted integrand b(s, \omega) = \sum_{j} b(t_{j}, \omega) \mathbf{1}_{(t_{j}, t_{j + 1}]}(s), so that I_{t} = \sum_{j = 0}^{n - 1} b(t_{j}) \Delta B_{t_{j}}. In this case the quadratic variation along the partition is [I, I]_{t} = \lim_{n \to \infty} \sum_{j = 0}^{n - 1} b^{2}(t_{j}) (\Delta B_{t_{j}})^{2}.
To show this limit equals \int_{0}^{t} b^{2}(s,\omega)\,ds in mean square, consider the mean-square difference \begin{aligned} \operatorname{E}\left[\left(\sum_{j = 0}^{n - 1} b(t_{j})^{2}(\Delta B_{t_{j}})^{2} - \sum_{j = 0}^{n - 1} b(t_{j})^{2} \Delta t_{j}\right)^{2}\right] & = \operatorname{E}\left[\left(\sum_{j = 0}^{n - 1} b(t_{j})^{2}((\Delta B_{t_{j}})^{2} - \Delta t_{j})\right)^{2}\right] \\ & = \operatorname{E}\left[\sum_{j = 0}^{n - 1} \sum_{k = 0}^{n - 1} b(t_{j})^{2} b(t_{k})^{2} ((\Delta B_{t_{j}})^{2} - \Delta t_{j}) ((\Delta B_{t_{k}})^{2} - \Delta t_{k})\right] \\ & = \sum_{j = 0}^{n - 1} \sum_{k = 0}^{n - 1} \operatorname{E}[b(t_{j})^{2} b(t_{k})^{2} ((\Delta B_{t_{j}})^{2} - \Delta t_{j}) ((\Delta B_{t_{k}})^{2} - \Delta t_{k})]. \end{aligned} If j<k the factors involving disjoint increments are independent, so the cross-terms vanish: \operatorname{E}\left[b(t_{j})^{2} b(t_{k})^{2} ((\Delta B_{t_{j}})^{2} - \Delta t_{j}) ((\Delta B_{t_{k}})^{2} - \Delta t_{k})\right] = 0. Using \Delta B_{t_{j}} \sim \mathcal{N}(0, \Delta t_{j}) we have \operatorname{E}[(\Delta B_{t_{j}})^{2}] = \Delta t_{j} and \operatorname{V}[(\Delta B_{t_{j}})^{2}] = 2 (\Delta t_{j})^{2}. Hence \begin{aligned} \operatorname{E}\left[\left(\sum_{j = 0}^{n - 1} b(t_{j})^{2}(\Delta B_{t_{j}})^{2} - \sum_{j = 0}^{n - 1} b(t_{j})^{2} \Delta t_{j}\right)^{2}\right] & = \sum_{j = 0}^{n - 1} \operatorname{E}[b(t_{j})^{4} ((\Delta B_{t_{j}})^{2} - \Delta t_{j})^{2}] \\ & = \sum_{j = 0}^{n - 1} \operatorname{E}[b(t_{j})^{4}] \operatorname{E}[((\Delta B_{t_{j}})^{2} - \Delta t_{j})^{2}] \\ & = \sum_{j = 0}^{n - 1} \operatorname{E}[b(t_{j})^{4}] 2 (\Delta t_{j})^{2} \\ & \le \lVert{\Pi}\rVert^2 \sum_{j = 0}^{n - 1} \operatorname{E}[b^{4}(t_{j})] (\Delta t_{j}). \end{aligned}
Under the integrability assumption \int_{0}^{t} \operatorname{E}\left[b^{4}(s)\right] \mathop{}\!\mathrm{d}s = \operatorname{E}\left(\int_{0}^{t} b^{4}(s) \mathop{}\!\mathrm{d}s\right) < \infty, the right-hand side tends to zero as \lVert{\Pi}\rVert \to 0. Therefore \sum_{j = 0}^{n - 1} b(t_{j})^{2}(\Delta B_{t_{j}})^{2} \xrightarrow[\,L^{2}\,]{} \int_{0}^{t} b^{2}(s, \omega) \mathop{}\!\mathrm{d}s, which yields the desired quadratic variation identity. \square
Quadratic Variation of the Itô Process
For the general Itô process defined in (1), the quadratic variation is equivalent to the stochastic integral, expressed as: [X, X]_{t} \;=\; \int_{0}^{t} b(s,\omega)^{2}\mathop{}\!\mathrm{d}s. The drift term \int_{0}^{t} a(s, \omega) \mathop{}\!\mathrm{d}s contributes zero to the quadratic variation because it is of bounded variation. Specifically, its increments are of order \Delta t, leading to their squares being of order (\Delta t)^{2}, which vanish in the limit. Thus, only the stochastic integral contributes to the quadratic variation.
To illustrate this, consider simple adapted integrands: a(s, \omega) = \sum_{j} a(t_{j}, \omega) \mathbf{1}_{(t_{j}, t_{j + 1}]}(s), \qquad b(s, \omega) = \sum_{j} b(t_j, \omega) \mathbf{1}_{(t_{j}, t_{j + 1}]}(s). The increment of X over the interval [t_{j}, t_{j+1}] is given by: \Delta X_{t_{j}}(\omega) = a(t_{j}, \omega) \Delta t_{j} + b(t_{j}, \omega) \Delta B_{t_{j}}(\omega). Consequently, we have: \left(\Delta X_{t_{j}}\right)^{2} = a(t_{j})^{2}(\Delta t_{j})^{2} + 2 a(t_{j}) b(t_{j})(\Delta t_{j})(\Delta B_{t_{j}}) + b(t_{j})^{2}(\Delta B_{t_{j}})^{2}.
In the limit as n \to \infty, we analyze the expression: \lim_{n \to \infty} \sum_{j = 0}^{n - 1} \left(\Delta X_{t_{j}}\right)^{2}. The first term vanishes because: \sum_{j = 0}^{n - 1} a(t_{j}, \omega)^{2}(\Delta t_{j})^{2} \leq \lVert{\Pi}\rVert \int_{0}^{t} a(s, \omega)^{2} \mathop{}\!\mathrm{d}s \to 0 \quad \text{as } n \to \infty, assuming that \int_{0}^{t} a(s, \omega)^{2} \mathop{}\!\mathrm{d}s < \infty almost surely.
Next, we analyze the second term by considering the sum: S_{n} = \sum_{j = 0}^{n - 1} 2 a(t_{j}, \omega) b(t_{j}, \omega) \Delta t_{j} \Delta B_{t_{j}}. We can express the expected value as: \operatorname{E}(S_{n}^{2}) = \operatorname{E}\left[\left( \sum_{j = 0}^{n - 1} 2 a(t_{j}) b(t_{j}) \Delta t_{j} \Delta B_{t_{j}} \right)^{2}\right] = \operatorname{E}\left[\sum_{j = 0}^{n - 1} \sum_{k = 0}^{n - 1} 4 a(t_{j}) b(t_{j}) a(t_{k}) b(t_{k}) \Delta t_{j} \Delta t_{k} \Delta B_{t_{j}} \Delta B_{t_{k}}\right]. Since for j < k, the terms a(t_{j}) b(t_{j}) a(t_{k}) b(t_{k}) \Delta t_{j} \Delta t_{k} \Delta B_{t_{j}} are independent of \Delta B_{t_{k}}, all cross-terms vanish, leading to: \begin{aligned} \operatorname{E}(S_{n}^{2}) & = \operatorname{E}\left[\sum_{j = 0}^{n - 1} 4 a^{2}(t_{j}) b^{2}(t_{j}) (\Delta t_{j})^{2} (\Delta B_{t_{j}})^{2} \right] \\ & = \sum_{j = 0}^{n - 1} \operatorname{E}\left[4 a^{2}(t_{j}) b^{2}(t_{j}) (\Delta t_{j})^{2}\right] \operatorname{E}\left[(\Delta B_{t_{j}})^{2}\right] \\ & = \sum_{j = 0}^{n - 1} \operatorname{E}\left[4 a^{2}(t_{j}) b^{2}(t_{j}) (\Delta t_{j})^{3}\right] \\ & \le \lVert{\Pi}\rVert^{2} \sum_{j = 0}^{n - 1} \operatorname{E}\left[4 a^{2}(t_{j}) b^{2}(t_{j}) \Delta t_{j}\right]. \end{aligned} As a result, we find: \lim_{n \to \infty} \operatorname{E}(S_{n}^{2}) \le \lim_{n \to \infty} \lVert{\Pi}\rVert^{2} \int_{0}^{t} \operatorname{E}\left[4 a^{2}(s) b^{2}(s)\right] ds = 0, which implies that \lim_{n \to \infty} S_{n} = 0 in L^{2} (and thus in probability).
Finally, we have already established that: \lim_{n \to \infty} \sum_{j = 0}^{n - 1} b^{2}(s, \omega) (\Delta B_{t_{j}})^{2} = \int_{0}^{t} b^{2}(s, \omega) \mathop{}\!\mathrm{d}s in L^{2}. Therefore, we conclude: \sum_{j} \left(\Delta X_{t_{j}}\right)^{2} \xrightarrow[\,L^{2}\,]{} \int_{0}^{t} b(s, \omega)^{2} ds. This result can be extended to general square-integrable adapted coefficients through approximation with simple processes.
Itô’s Formula
Itô’s formula generalizes the chain rule to stochastic processes. It provides the precise way to compute the differential of a smooth function of an Itô process.
Sketch of Proof. We begin with a second-order Taylor expansion of f(X_{t + \Delta t}, t + \Delta t) around (X_{t}, t): \begin{aligned} f(X_{t + \Delta t}, t +\Delta t) \approx f(X_{t}, t) & + \frac{\partial f}{\partial t} \Delta t + \frac{\partial f}{\partial x}\Delta X_{t} \\ & + \frac{1}{2}\frac{\partial^{2} f}{\partial x^{2}}(\Delta X_{t})^{2} + \frac{1}{2}\frac{\partial^{2} f}{\partial t^{2}}(\Delta t)^{2} + \frac{\partial^{2} f}{\partial x\partial t}\Delta X_{t}\Delta t + \ldots \end{aligned}
Therefore, the increment in f is \Delta f = f(X_{t+\Delta t}, t+\Delta t) - f(X_{t},t) \approx \frac{\partial f}{\partial t}\Delta t + \frac{\partial f}{\partial x}\Delta X_{t} + \frac{1}{2}\frac{\partial^{2} f}{\partial x^{2}}(\Delta X_{t})^{2} + \text{higher order terms}.
For an Itô process with \mathop{}\!\mathrm{d}X_{t} = a(t,\omega) \mathop{}\!\mathrm{d}t + b(t,\omega) \mathop{}\!\mathrm{d}B_{t}, the increment over [t, t+\Delta t] is \Delta X_{t} = a(t, \omega) \Delta t + b(t, \omega) \Delta B_{t}. We already saw that (\Delta X_{t})^{2} \approx b^{2}(t) \Delta t. Thus, \Delta f \approx \left(\frac{\partial f}{\partial t} + a(t, \omega) \frac{\partial f}{\partial x} + \frac{1}{2} b^{2}(t, \omega) \frac{\partial^{2} f}{\partial x^{2}}\right) \Delta t + b(t, \omega) \frac{\partial f}{\partial x} \Delta B_{t}.
Passing to the limit as \Delta t \to 0 yields Itô’s formula in differential form: \mathop{}\!\mathrm{d}f = \left(\frac{\partial f}{\partial t} + a(t, \omega) \frac{\partial f}{\partial x} + \frac{1}{2}b^{2}(t, \omega) \frac{\partial^{2} f}{\partial x^{2}}\right)\mathop{}\!\mathrm{d}t + b(t, \omega) \frac{\partial f}{\partial x} \mathop{}\!\mathrm{d}B_{t}.
The additional term \frac{1}{2}b^{2}\frac{\partial^{2} f}{\partial x^{2}}\mathop{}\!\mathrm{d}t arises because of the non-zero quadratic variation (\mathop{}\!\mathrm{d}X_{t})^{2} = b^{2}\mathop{}\!\mathrm{d}t. This term has no classical analogue—in ordinary calculus where paths are smooth, the chain rule contains no such correction. The presence of this drift term is the defining feature of stochastic calculus and reflects the fractal, rough nature of Brownian motion. \square