Stochastic Calculus

Overview

Stochastic calculus is the branch of mathematics that extends classical calculus to stochastic (random) processes. Where classical calculus deals with deterministic functions and their derivatives/integrals, stochastic calculus handles processes whose evolution involves randomness. The key insight is that stochastic processes like Brownian motion have infinite variation, requiring new integration and differentiation rules—particularly the Itô integral and Itô formula.

Fundamental Areas

• Foundations

  • Measure-Theoretic Probability - Probability spaces, filtrations, adapted processes, information flow
  • Brownian Motion - The canonical stochastic process (continuous-time random walk)
  • Martingale Theory - Fair-game processes fundamental to stochastic modeling
  • Stopping Times - Random times adapted to the filtration

• Quadratic Variation — How stochastic processes differ from classical ones. Brownian motion has infinite total variation but finite quadratic variation, requiring new calculus rules.

• Itô Integral — The stochastic version of the Riemann integral. Integration with respect to Brownian motion and other martingales. Properties (linearity, martingale property, Itô isometry).

• Itô Calculus (Change of Variables) — The stochastic generalization of the chain rule

  • Itô Formula (1D and multivariate) — how functions of stochastic processes evolve
  • Stochastic Integration By Parts — product rules for stochastic processes
  • Quadratic covariation — the extra terms that appear in stochastic calculus

• Stochastic Differential Equations — Definition and theory

  • Existence and uniqueness of solutions (Picard-Lindelöf conditions: growth and Lipschitz)
  • Strong vs weak solutions
  • Markov property and time-homogeneity of solutions

• SDE Solution Methods & Examples — Specific important processes

  • Geometric Brownian Motion — Exponential growth with random noise (stock prices)
  • Ornstein-Uhlenbeck Process — Mean-reverting processes (interest rates, commodities)
  • Brownian Bridge — Brownian motion conditioned to return to a fixed point
  • Linear SDEs and their explicit solutions

• Diffusion Processes — Time-homogeneous SDEs and their properties

  • Itô diffusions: definition and key properties (flow property, Markov property)
  • Continuous sample paths and regularity

• Infinitesimal Generator — Operator associated with a diffusion

  • Generator and semi-group theory
  • Dynkin’s Formula — relating expectations to the generator (martingale characterization)
  • Kolmogorov backward and forward equations

• Advanced Results — Tools for solving and transforming SDEs

  • Girsanov’s Theorem — Change of measure and drift transformation
  • Feynman-Kac Formula — Connection between SDEs and PDEs
  • Fokker-Planck Equation / Kolmogorov Equations — Evolution of probability densities
  • Martingale Representation Theorem — Every martingale is an Itô integral

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The Itô integral has several helpful properties that follow from its derivation as an approximation of sums of elementary functions. Letting $f,g\in \mathcal{V}(0,T)$ and let $0\le S<U<T$ we have that:

1. ${\int }_S^{T}fd{B}_t={\int }_S^{U}fd{B}_t+{\int }_U^{T}fd{B}_t$ almost surely (partition of integration area);
2. ${\int }_S^T(cf+g)d{B}_t=c\cdot {\int }_S^{T}fd{B}_t+{\int }_S^{T}gd{B}_t$ (constant $c$) almost surely (linearity);
3. $\mathbb{E}\left[{\int }_S^{T}fd{B}_t\right]=0$ (mean zero);
4. ${\int }_S^{T}fd{B}_t$ is ${\mathcal{F}}_T$-measurable (adapted); and
5. ${\int }_S^{T}fd{B}_t$ is a martingale.

The Itô integral can be defined for a larger class of integrands $f$ than $\mathcal{V}$. Specifically, the measurability condition (2) in our definition of $\mathcal{V}$ can be relaxed to the following:
2. There exists an increasing family of $\sigma$-algebras ${\mathcal{H}}_t;t\ge 0$ such that:
1. $B_t$ is a martingale with respect to ${\mathcal{H}}_t$; and
2. $f_t$ is ${\mathcal{H}}_t$-adapted.

4 Itô Formula

Similar to Riemann integrals, we wish to establish an Itô integral version of the chain rule without differentiation theory, known as the Itô formula. The Itô formula provides a method informing us about the behavior of a function of BM $g(B_t)$ or indeed other processes by expanding it into stochastic integrals, also known as Itô processes.

1-Dimensional Itô Process

A (1-dimensional) Itô process (or stochastic integral) is a stochastic process $X_t$ on $(\Omega ,\mathcal{F},\mathbb{P})$ of the form $X_t=X_0+{\int }_0^{t}u(s,\omega )ds+{\int }_0^{t}v(s,\omega )d{B}_s,$where $v\in {\mathcal{W}}_{\mathcal{H}}$, so that $\mathbb{P}\left({\int }_0^{t}v(s,\omega {)}^{2}ds<\infty \text{ for all }t\ge 0\right)=1.$We further assume that $u$ is ${\mathcal{H}}_t$-adapted and $\mathbb{P}\left({\int }_0^t|u(s,\omega )|ds<\infty \text{ for all }t\ge 0\right)=1$. The differential form of the Itô process $(X_t)$ is $d{X}_t=udt+vd{B}_t.$

1-Dimensional Itô Formula

Let $X_t$ be an Itô process given by $d{X}_t=udt+vd{B}_t.$Further, let $g(t,x)\in C^2([0,\infty )\times \mathbb{R})$ (i.e. twice differentiable on $[0,\infty )\times \mathbb{R}$). Then $Y_t=g(t,X_t)$ is again an Itô process and $d{Y}_t=g_t(t,X_t)dt+g_x(t,X_t)d{X}_t+\frac{1}{2}{g}_{xx}(t,X_t)d{⟨X,X⟩}_t.$

Here $d{⟨X,X⟩}_t=(d{X}_t)\cdot (d{X}_t)$ is computed according to the following:

$$dt\cdot dt=dt\cdot d{B}_t=d{B}_t\cdot dt=0,d{B}_t\cdot d{B}_t=0.$$

Proof:

Example (Itô Formula Application I)
Consider the integral

$$Y_t={\int }_0^t{B}_{s}d{B}_s.$$

Assuming we can write $Y_t=g(t,B_t)$ from Itô’s formula we have

$$d{Y}_t=\left[g_t(t,B_t)+\frac{1}{2}{g}_{BB}(t,B_t)\right]dt+g_B(t,B_t)d{B}_t,$$

which written in integral form gives

$$Y_t-Y_0={\int }_0^t\left[g_t(s,B_s)+\frac{1}{2}{g}_{BB}(s,B_s)\right]ds+{\int }_0^t{g}_B(s,B_s)d{B}_s.$$

Hence we need some $g(t,B_t)$ such that

$$(1)g_t(t,B_t)+\frac{1}{2}{g}_{BB}(t,B_t)=0\&(2)g_B(t,B_t)=B_t.$$

Since $g_B=B_t⟹g_{BB}=1$ and so $g_t=-\frac{1}{2}$. An obvious candidate therefore is

$$g(t,B_t)=\frac{1}{2}{B}_t^2-\frac{1}{2}t,$$

which in fact is the result, as can be verified by plugging into Itô’s formula.

Example (Itô Formula Application II)
Consider the Itô process

$$g(t,B_t)=\exp (bt+\sigma B_t),$$

where $b,\sigma$ are constants for drift and diffusion. To compute $dg$ we apply the Itô formula

$$\begin{array}{rl}dg(t,B_t)& =b\exp (bt+\sigma B_t)dt+\sigma \exp (bt+\sigma B_t)d{B}_t+\frac{1}{2}{\sigma }^2\exp (bt+\sigma B_t)d{⟨B,B⟩}_t\\ & =g(t,B_t)\left\{\left(b+\frac{1}{2}{\sigma }^2\right)dt+\sigma d{B}_t\right\}.\end{array}$$

The process $X_t=g(t,B_t)$ defines geometric BM and is used to simulate the change of the stock price in the Black-Scholes-Merton Model ($\mu$ as mean return and $\sigma$ as volatility).

Itô Integration by Parts

For a continuous function $f(s,\omega )$ that is of bounded variation with respect to $s\in [0,t]$ almost surely we have that ${\int }_0^{t}f(s)d{B}_s=f(t)B_t-{\int }_0^t{B}_{s}d{f}_s.$

4.1 Multi-dimensional Itô Formula

Considering the situation in higher dimensions we define a $d$-dimensional Itô process as the integral with respect to the $m$-dimensional BM as

$$X_t^i=x_0^i+{\int }_0^t{\psi }_s^{i}ds+\sum _{k=1}^m{\int }_0^t{\phi }_s^{i,k}d{B}_s^k,$$

for $k=1,\dots ,m$ and $i=1,\dots ,d$ where we note that such an Itô process exists in ${\mathbb{R}}^d$ and is constructed with respect to BM $B_s=(B_s^1,\dots ,B_s^m)$. To write this in a more compact form

$$X_t=x_0+{\int }_0^t{\psi }_{s}ds+{\int }_0^t{\phi }_s\cdot d{B}_s,$$

where $X_t,x_0,{\psi }_s\in {\mathbb{R}}^d$ and ${\phi }_s\in {\mathbb{R}}^{d\times m}$ and here ${\phi }_s^{i,k}$ stands for the process $\phi$ used to construct the $i$-th coordinate of the Itô process as a stochastic integral with respect to the $k$-th coordinate of the $m$-dimensional BM.

Multi-Dimensional Itô Formula

For some vector-valued function $g:{\mathbb{R}}_{+}\times {\mathbb{R}}^d\to {\mathbb{R}}^p\in C^{1,2}$, then the multi-dimensional Itô formula holds \begin{align}dg^i(t,X_{t}) & =\partial_{t}g^i(t,X_{t})dt+\nabla_{x} g^i(t,X_{t})\cdot dX_{t}+ \frac{1}{2}\mathrm{Tr}(\varphi_{t}^T H \varphi_{t})dt \\ & = \partial_{t}g^i(t,X_{t})dt+\sum_{k=1}^{d}\partial_{x_{k}}g^i(t,X_{t})dX_{t}^k+ \frac{1}{2}\sum_{j,k = 1}^{d}\partial_{x_{j}, x_{k}}g^i(t,X_{t})d\left< X^j, X^k \right> _{t},\end{align}where ${\nabla }_{x}f$ denotes the gradient of $f$ with respect to vector $x$, ${\phi }_t\in {\mathbb{R}}^{d\times m}$ is a matrix and $H_{d\times d}$ is the Hessian of $g$ restricted on its action on $x\in {\mathbb{R}}^d$ at $(t,X_t)$. The bracket $d{⟨X^j,X^k⟩}_t={\sum }_{l=1}^m{\phi }_t^{j,l}{\phi }_t^{k,l}dt$.

Proof:
The structure of the proof is still exactly the same as the 1-dimensional case. We need only verify that $d{B}_t^{p}d{B}_t^q=0$ for $p\ne q$ and $d{⟨X^j,X^k⟩}_t={\sum }_{l=1}^m{\phi }_t^{j,l}{\phi }_t^{k,l}dt$.

4.2 The Martingale Representation Theorem

The martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.

Martingale Representation Theorem

Let $B(t)$ be $n$-dimensional BM. Suppose $M_t$ is an ${\mathcal{F}}_t^{(n)}$-martingale (w.r.t.) $\mathbb{P}$ and that $M_t\in {\mathcal{L}}_2(\mathbb{P})$ for all $t\ge 0$. Then there exists a unique stochastic process $g(s,\omega )$ such that $g\in {\mathcal{V}}^{(n)}(0,t)$ for all $t\ge 0$ and $M_t(\omega )=\mathbb{E}[M_0]+{\int }_0^{t}g(s,\omega )dB(s),$almost surely for all $g\ge 0$.

The proof of this result is involved and is omitted from this note. Details can be found in our note on this theorem here.

5 Stochastic Differential Equations

References: Stochastic Differential Equations 1.

A general stochastic differential equation in ${\mathbb{R}}^n$ with initial condition has the form

$$\{\begin{array}{l}d{X}_t=b(t,X_t)dt+\sigma (t,X_t)dBt\\ X_0=x,\end{array}$$

where $x,X_t\in {\mathbb{R}}^n$, $b:{\mathbb{R}}_{+}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, $\sigma :{\mathbb{R}}_{+}\times {\mathbb{R}}^n\to {\mathbb{R}}^{n\times m}$ and $B_t$ is an $m$-dimensional BM. Here $b$ is the drift coefficient and $\sigma$ is the volatility coefficient. We note the difference between SDEs and a general Itô process lies in the fact that $b,\sigma$ are functions of the unknown process $X_t$.

Existence and Uniqueness of SDE Solutions

Fixing time $T>0$ and assuming $b:[0,T]\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, $\sigma :[0,T]\times {\mathbb{R}}^n\to {\mathbb{R}}^{n\times m}$ are deterministic functions with bounded time variable $T$ which satisfy the following conditions:

1. $\exists c>0,\forall t\in [0,T],\forall x\in {\mathbb{R}}^n,|b(t,x)|+|\sigma (t,x)|\le c(1+|x|)$ (growth condition);
2. $\exists D>0,\forall t\in [0,T],\forall x,y\in {\mathbb{R}}^n,|b(t,x)-b(t,y)|+|\sigma (t,x)-\sigma (t,y)|\le D|x-y|$ (Lipschitz condition),

then the SDE has unique solution in $L^2([0,T]\times \Omega )$ that has continuous sample path.

The proof of this theorem can be found detailed in our specific note on Stochastic Differential Equations 1.

5.1 Strong and Weak SDE Solutions

A strong solution to a stochastic differential equation is defined as the solution $X_t\in {\mathcal{F}}_t$ adapted to the filtration generated by Brownian Motion 1 and the initial condition (if it is random)

$${\mathcal{F}}_t=\sigma (B_s,0\le s\le t)\vee \sigma (X_0).$$

The uniqueness and global existence theorem stated for Stochastic Differential Equations 1 is just proving that for strong solutions.

5.2 SDE Example: Linear SDE

References: Linear Stochastic Differential Equations.

Linear SDE

Let $W_t$ be $m$-dimensional BM, $A$ be an $n\times n$ matrix and $B$ an $n\times m$ matrix. Then a linear stochastic differential equation $X_t$ has dynamics given by $\{\begin{array}{l}d{X}_t=A{X}_{t}dt+Bd{W}_t,\\ X_0=x.\end{array}$

To find the general solution of a linear SDE we first note that by setting the noise term $B$ to the $0$-matrix we have the ODE $d{X}_t=A{X}_{t}dt$ which has the solution $X_t=X_0{e}^{At}$. Setting this as our

which are given explicitly by

$$X_t=e^{At}x+{\int }_0^t{e}^A(t-s)Bd{W}_s.$$

5.3 SDE Example: Black Scholes Model

References: Black-Scholes-Merton Model

The Black-Scholes model provides the structure on which we are able to price contingency claims (specifically options). Specifically, the Black-Scholes model is a mathematical model for the dynamics of a financial market containing derivative investment instruments such as stock shares or futures contracts assumed to have a lognormal distribution of prices following a random walk with constant drift and volatility.

Black-Scholes Model

The Black-Scholes model consists of two assets, a risk free asset $B$ and a stock $S$ with price dynamics given by \begin{align}dB_{t}&=rB_{t}dt \\dX_{t}& =\mu X_{t}dt+\sigma X_{t}dB_{t}\end{align}
where $r,\mu ,\sigma \in \mathbb{R}$ are deterministic constants.

So in our notation we have $b(t,x)=\mu x$ and $\sigma (t,x)=\sigma x$ which are both Lipschitz in in $x$ and satisfy the growth condition.

To solve this system we note that if we ignore the stochastic term $d{X}_t=\mu X_{t}dt$ is an ODE with solution $X_t=X_0\cdot e^{\mu t}$. We therefore consider changing variables with $Y_t=\log (X_t)$ and applying Itô’s formula we obtain

$$d{Y}_t=\frac{1}{X_t}d{X}_t-\frac{1}{2X_t^2}d{⟨X,X⟩}_t=\frac{1}{X_t}d{X}_t-\frac{1}{2}{\sigma }^{2}dt,$$

where we have used that $d{⟨X,X⟩}_t={\sigma }^2{X}_t^{2}d{⟨B,B⟩}_t={\sigma }^2{X}_t^{2}dt$. Substituting in our initial expression we have

$$\begin{array}{rl}d{Y}_t& =\mu dt+\sigma d{B}_t-\frac{1}{2}{\sigma }^{2}dt=\left(\mu -\frac{1}{2}{\sigma }^2\right)dt+\sigma d{B}_t\\ ⟹Y_t-Y_0& ={\int }_0^t\left(\mu -\frac{1}{2}{\sigma }^2\right)dt+{\int }_0^t\sigma d{B}_t\\ & =\left(\mu -\frac{1}{2}{\sigma }^2\right)t+\sigma B_t\\ ⟹X_t& =X_0\cdot e^{\left(\mu -\frac{1}{2}{\sigma }^2\right)t+\sigma B_t}.\end{array}$$

5.4 SDE Example: OU Process

References: Ornstein-Uhlenbeck Process

An Ornstein-Uhlenbeck process is a mean reverting stochastic process that is a Gaussian process, is Markov and is temporally stationary. This process can be used to describe the fluctuation of interest rate around the mean interest rate $M$ with the speed of the regression described by $\theta$.

Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck process $(X_t)$ with drift has dynamics described by the stochastic differential equation $d{X}_t=\theta (\mu -X_t)dt+\sigma d{B}_t$where $\theta ,\sigma >0$, $\mu \in \mathbb{R}$ are all constant and $B_t$ is a BM.

Consider such a process with a deterministic starting value $X_0$. To find a solution to this SDE we first change the variable to set the regression level to 0 with $Y_t=X_t-\mu$. From Itô’s formula we have

$$d{Y}_t=d{X}_t=\theta (\mu -X_t)dt+\sigma d{B}_t=-\theta Y_{t}dt+\sigma d{B}_t.$$

Considering when $\sigma =0$ we have the ODE $d{Y}_t=-\theta Y_{t}dt$ with the solution $Y_t=Y_0\cdot e^{-\theta t}$ and so replacing this constant $Y_0$ with some yet to define process $C_t$ we have $Y_t=C_t{e}^{-\theta t}$ and from Itô’s formula we have

$$d{Y}_t=-\theta e^{-\theta t}{C}_{t}dt+e^{-\theta t}d{C}_t=-\theta Y_{t}dt+e^{-\theta t}d{C}_t.$$

We therefore can define $d{C}_t=\sigma e^{\theta t}d{B}_t$ and so $C_0=Y_0=X_0-\mu$ and

$$\begin{array}{rl}{C}_t-C_0& ={\int }_0^t\sigma e^{\theta s}d{B}_s\\ ⟹C_t& =X_0-\mu +\sigma {\int }_0^t{e}^{\theta s}d{B}_s\\ ⟹Y_t& =(X_0-\mu )e^{-\theta t}+\sigma e^{-\theta t}{\int }_0^t{e}^{\theta s}d{B}_s\\ ⟹X_t& =(X_0-\mu )e^{-\theta t}+\mu +\sigma e^{-\theta t}{\int }_0^t{e}^{\theta s}d{B}_s.\end{array}$$

5.5 SDE Example: Brownian Bridge

References: Brownian Bridge.

A Brownian bridge is a continuous-time gaussian process $(X_t)$ whose probability distribution is the conditional probability distribution of a standard Brownian Motion $B_t$ subject to the condition (when standardized) that $W_T=0$, so that the process is pinned to the same value at both t = 0 and t = T.

Brownian Bridge

The Brownian Bridge is an SDE $(X_t)$ characterized by the dynamics $\{\begin{array}{l}d{X}_t=\frac{b-X_t}{1-t}dt+d{B}_t\\ X_0=a,\end{array}$ for $a,b\in \mathbb{R}$ and $0\le t<1$.

This is the Brownian bridge starting from $a$ and ending at $b$ that has the same finite-dimensional distribution as the BM conditioning on starting from $a$ at time $0$ and ending at $b$ at time $1$. The Brownian bridge starting from $0$ at time $0$ and ending at $0$ at time $1$ has the simple form as $B_t-t{B}_1$.

We can verify that the solution is given by

$$Y_t=a(1-t)+bt+(1-t){\int }_0^t\frac{1}{1-s}d{B}_s.$$

First note that since

$$G:={\int }_0^t\frac{1}{1-s}s{B}_s\sim \mathcal{N}\left(0,\frac{1}{1-t}\right),$$

its characteristic function is

$$\varphi (p)={\varphi }_G(p(1-t))=e^{\frac{1}{2(1-t)}{p}^2(1-t{)}^2}=e^{\frac{1}{2}{p}^2(1-t)}\to 1\text{as}t\to 1.$$

Hence $(1-t)G\stackrel{\mathcal{L}}{\to }0$ as $t\to 1$ and since the limit is a constant $(1-t)G\stackrel{\mathbb{P}}{\to }0$ as $t\to 1$. Finally, we notice that this stochastic integral is the sum of countably many independent random variables, hence from Durrett 2.5.10 we have $(1-t)G\stackrel{\text{a.s.}}{\to }0$ as $t\to 1$ and $Y_t\stackrel{\text{a.s.}}{\to }b$.

5.6 SDE Example: Geometric Mean-Reverting Process

Geometric Mean-Reverting Process

The geometric mean-reverting process $X_t$ is defined as the solution to the SDE $\{\begin{array}{l}d{X}_t=\kappa (\alpha -\log X_t)X_{t}dt+\sigma X_{t}d{B}_t\\ X_0=x>0.\end{array}$

By using the substitution $Y_t=\log X_t$ we can verify that the solution to the SDE is

$$X_t=\exp \left\{e^{-\kappa t}\log x+\left(\alpha -\frac{{\sigma }^2}{2\kappa }\right)(1-e^{-\kappa t})+\frac{{\sigma }^2}{2\kappa }(1-e^{-2\kappa t})\right\}.$$

6 Diffusion Processes

Itô Diffusion

A (time-homogeneous) Itô diffusion in ${\mathbb{R}}^n$ is a stochastic process $(X_t)$ defined on space $(\Omega ,\mathcal{F},\mathbb{P})$ that satisfies the stochastic differential equation $\{\begin{array}{l}d{X}_t=b(X_t)dt+\sigma (X_t)d{B}_t\\ X_s=x\end{array}$ where $X_t\in {\mathbb{R}}^n$, $B_t\in {\mathbb{R}}^m$, $B:{\mathbb{R}}^n\to {\mathbb{R}}^n$, $\sigma :{\mathbb{R}}^n\to {\mathbb{R}}^{n\times m}$ where it is assumed that $b,\sigma$ are both Lipschitzs on ${\mathbb{R}}^n$ (to ensure the existence and uniqueness of strong solutions).

The distinguishing feature of diffusions is that the drift and diffusion coefficients $b,\sigma$ are time independent and the Lipschitz condition is here to ensure the existence and uniqueness of the strong solution. Next we detail the results outlining the key properties of Itô diffusions.

Time Homogeneity of Itô DIffusions

Denoting the solution to the SDE above with initial condition $X_s=x$ as an Itô diffusion by $X_{s+h}^{s,x}$ then $\forall s\ge 0,x\in {\mathbb{R}}^n,\{X_{s+t}^{s,x}{\}}_{t\ge 0}\stackrel{\mathcal{L}}{=}\{X_t^{0,x}{\}}_{t\ge 0}.$

Proof:

Flow Property of Itô Diffusion

For an Itô diffusion $X$ we have $\forall t,h\ge 0,X_h^{0,X_t^{0,x}}\stackrel{a.s.}{X_{t+h}^{0,x}}.$

Proof: By construction, both are strong solutions to the same SDE

$$X_{t+h}^{0,x}=X_t^{0,x}+{\int }_t^{t+h}b(X_s^{0,x})ds+{\int }_t^{t+h}\sigma (X_s^{0,x})d{B}_s.$$

By the uniqueness in modification we conclude