11.1 Stochastic integrals

Stochastic integrals extend integration to random processes, enabling analysis of systems with unpredictable fluctuations. They're crucial for modeling complex phenomena in finance, physics, and engineering. Understanding stochastic integrals helps tackle problems involving uncertainty.

Itô and Stratonovich integrals are two main types, each with unique properties. Itô integrals are widely used in finance, while Stratonovich integrals preserve classical calculus rules. Both form the foundation for solving stochastic differential equations and modeling real-world random processes.

Definition of stochastic integrals

• Stochastic integrals extend the concept of integration to stochastic processes, which are mathematical models describing the evolution of random variables over time
• Stochastic integration is a fundamental tool in the study of stochastic processes, enabling the analysis of complex systems subject to random fluctuations
• Stochastic integrals are essential for understanding and solving problems in various fields, such as mathematical finance, physics, engineering, and biology

Intuition behind stochastic integration

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• Stochastic integration aims to define the integral of a stochastic process with respect to another stochastic process, typically a Brownian motion or a semimartingale
• The intuition behind stochastic integration is to approximate the integral by a sum of products of the integrand and the increments of the integrator process
• The key challenge in stochastic integration is that the integrator process, such as Brownian motion, has unbounded variation and is nowhere differentiable, requiring special techniques to define the integral properly

Formal definition

• Stochastic integrals are formally defined as limits of Riemann-Stieltjes sums, where the integrand is a predictable process and the integrator is a semimartingale
• The definition of stochastic integrals relies on the concept of quadratic variation, which measures the accumulated squared increments of a process
• The choice of the point at which the integrand is evaluated within each subinterval leads to different types of stochastic integrals, such as Itô integrals and Stratonovich integrals

Differences vs Riemann-Stieltjes integrals

• Stochastic integrals differ from Riemann-Stieltjes integrals in several key aspects:
  • The integrator in stochastic integrals is a stochastic process, while in Riemann-Stieltjes integrals, it is a deterministic function of bounded variation
  • Stochastic integrals require the integrand to be predictable, meaning it should be known just before the integration point, while Riemann-Stieltjes integrals have no such requirement
  • The quadratic variation of the integrator plays a crucial role in stochastic integrals, while it is not relevant in Riemann-Stieltjes integrals
• Stochastic integrals can be seen as a generalization of Riemann-Stieltjes integrals to the stochastic setting, enabling the integration of processes with respect to more general integrators

Itô integrals

• Itô integrals, named after the Japanese mathematician Kiyoshi Itô, are the most widely used type of stochastic integrals
• Itô integrals are defined using a non-anticipative approach, where the integrand is evaluated at the left endpoint of each subinterval in the Riemann-Stieltjes sum approximation
• The choice of the left endpoint in Itô integrals leads to a simpler calculus and more tractable mathematical properties compared to other stochastic integrals

Definition of Itô integrals

• Given a predictable process $X(t)$ and a Brownian motion $B(t)$, the Itô integral of $X$ with respect to $B$ is defined as: $\int_0^t X(s) dB(s) = \lim_{n \to \infty} \sum_{i=1}^n X(t_{i-1}) (B(t_i) - B(t_{i-1}))$ where $0 = t_0 < t_1 < \ldots < t_n = t$ is a partition of the interval $[0, t]$ with mesh size tending to zero
• The Itô integral is a martingale with respect to the natural filtration generated by the Brownian motion, which is a desirable property for many applications

Itô processes

• An Itô process is a stochastic process that can be represented as the sum of an Itô integral and a drift term
• Formally, a process $X(t)$ is an Itô process if it satisfies the stochastic differential equation (SDE): $dX(t) = \mu(t, X(t)) dt + \sigma(t, X(t)) dB(t)$ where $\mu$ and $\sigma$ are measurable functions, and $B(t)$ is a Brownian motion
• Itô processes are widely used to model various phenomena, such as stock prices in mathematical finance (geometric Brownian motion) and particle motion in physics (Langevin equation)

Itô's lemma

• Itô's lemma, also known as Itô's formula, is a fundamental result in stochastic calculus that provides a rule for computing the differential of a function of an Itô process
• Let $X(t)$ be an Itô process satisfying the SDE $dX(t) = \mu(t, X(t)) dt + \sigma(t, X(t)) dB(t)$, and let $f(t, x)$ be a twice continuously differentiable function. Then, Itô's lemma states that: $df(t, X(t)) = \left(\frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial x^2}\right) dt + \sigma \frac{\partial f}{\partial x} dB(t)$
• Itô's lemma is essential for deriving the dynamics of functions of Itô processes, such as option pricing formulas in mathematical finance (Black-Scholes equation)

Applications of Itô calculus

• Itô calculus, which encompasses Itô integrals and Itô's lemma, has numerous applications in various fields:
  • In mathematical finance, Itô calculus is used to model asset prices, derive option pricing formulas, and develop risk management strategies
  • In physics, Itô calculus is employed to study stochastic dynamics, such as Brownian motion, diffusion processes, and stochastic resonance
  • In engineering, Itô calculus is applied to model and control systems subject to random perturbations, such as in signal processing and robotics
• The widespread use of Itô calculus highlights the importance of stochastic integrals in solving real-world problems involving uncertainty and randomness

Stratonovich integrals

• Stratonovich integrals, named after the Russian physicist Ruslan Stratonovich, are an alternative type of stochastic integrals
• Stratonovich integrals are defined using a midpoint approach, where the integrand is evaluated at the midpoint of each subinterval in the Riemann-Stieltjes sum approximation
• The choice of the midpoint in Stratonovich integrals leads to a more complex calculus but preserves the standard chain rule and ordinary calculus rules

Definition of Stratonovich integrals

• Given a predictable process $X(t)$ and a Brownian motion $B(t)$, the Stratonovich integral of $X$ with respect to $B$ is defined as: $\int_0^t X(s) \circ dB(s) = \lim_{n \to \infty} \sum_{i=1}^n X\left(\frac{t_{i-1} + t_i}{2}\right) (B(t_i) - B(t_{i-1}))$ where $0 = t_0 < t_1 < \ldots < t_n = t$ is a partition of the interval $[0, t]$ with mesh size tending to zero
• The Stratonovich integral is not a martingale, unlike the Itô integral, which can be a disadvantage in some applications

Comparison vs Itô integrals

• Stratonovich integrals and Itô integrals are related by a correction term involving the quadratic covariation of the integrand and the integrator
• The relationship between Stratonovich and Itô integrals is given by: $\int_0^t X(s) \circ dB(s) = \int_0^t X(s) dB(s) + \frac{1}{2} \int_0^t \frac{\partial X}{\partial x}(s) d\langle B \rangle_s$ where $\langle B \rangle_s$ denotes the quadratic variation of the Brownian motion
• The choice between Stratonovich and Itô integrals depends on the specific problem and the desired mathematical properties

Stratonovich calculus

• Stratonovich calculus is the calculus based on Stratonovich integrals, analogous to Itô calculus for Itô integrals
• Stratonovich calculus has the advantage of preserving the standard chain rule and ordinary calculus rules, making it more intuitive for practitioners familiar with classical calculus
• However, Stratonovich calculus often leads to more complex calculations and less tractable mathematical properties compared to Itô calculus

Chain rule for Stratonovich integrals

• The chain rule for Stratonovich integrals is similar to the standard chain rule in ordinary calculus
• Let $X(t)$ be a process satisfying the Stratonovich SDE $dX(t) = \mu(t, X(t)) dt + \sigma(t, X(t)) \circ dB(t)$, and let $f(t, x)$ be a twice continuously differentiable function. Then, the chain rule for Stratonovich integrals states that: $df(t, X(t)) = \left(\frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x}\right) dt + \sigma \frac{\partial f}{\partial x} \circ dB(t)$
• The chain rule for Stratonovich integrals is simpler than Itô's lemma, as it does not involve the second-order term with the second partial derivative of $f$

Properties of stochastic integrals

• Stochastic integrals, both Itô and Stratonovich, possess several important properties that are crucial for their application and analysis
• These properties highlight the similarities and differences between stochastic integrals and classical Riemann-Stieltjes integrals, and they provide the foundation for developing stochastic calculus

Linearity of integration

• Stochastic integrals are linear operators, meaning that for any predictable processes $X(t)$ and $Y(t)$, and any constants $a$ and $b$, the following holds: $\int_0^t (aX(s) + bY(s)) dB(s) = a \int_0^t X(s) dB(s) + b \int_0^t Y(s) dB(s)$
• The linearity property allows for the decomposition and superposition of stochastic integrals, simplifying calculations and analysis

Isometry property

• The isometry property, also known as Itô isometry, relates the expected value of the square of a stochastic integral to the expected value of the quadratic variation of the integrand
• For any predictable process $X(t)$, the Itô isometry states that: $\mathbb{E}\left[\left(\int_0^t X(s) dB(s)\right)^2\right] = \mathbb{E}\left[\int_0^t X(s)^2 ds\right]$
• The isometry property is essential for computing moments of stochastic integrals and establishing convergence results

Martingale property

• Itô integrals with respect to a Brownian motion are martingales, meaning that their expected value at any future time, conditioned on the current information, is equal to their current value
• Formally, for any predictable process $X(t)$, the Itô integral $\int_0^t X(s) dB(s)$ is a martingale with respect to the natural filtration generated by the Brownian motion
• The martingale property is crucial for applications in mathematical finance, as it allows for the use of martingale techniques in pricing and hedging financial derivatives

Quadratic variation

• The quadratic variation of a stochastic process measures the accumulated squared increments of the process over time
• For a Brownian motion $B(t)$, the quadratic variation is given by: $\langle B \rangle_t = t$
• The quadratic variation plays a fundamental role in stochastic calculus, as it appears in the definition of stochastic integrals and in key results such as Itô's lemma
• The quadratic variation of a stochastic integral can be computed using the Itô isometry: $\left\langle \int_0^\cdot X(s) dB(s) \right\rangle_t = \int_0^t X(s)^2 ds$

Stochastic differential equations (SDEs)

• Stochastic differential equations (SDEs) are differential equations driven by stochastic processes, typically Brownian motion or more general semimartingales
• SDEs are used to model the evolution of systems subject to random fluctuations, where the randomness is introduced through the stochastic integral term
• The study of SDEs is a central topic in stochastic calculus, as it combines the concepts of stochastic integrals, Itô's lemma, and differential equations

Definition of SDEs

• An SDE is a differential equation of the form: $dX(t) = \mu(t, X(t)) dt + \sigma(t, X(t)) dB(t)$ where $X(t)$ is the stochastic process of interest, $\mu$ and $\sigma$ are measurable functions called the drift and diffusion coefficients, and $B(t)$ is a Brownian motion
• The SDE can be interpreted as a shorthand notation for the integral equation: $X(t) = X(0) + \int_0^t \mu(s, X(s)) ds + \int_0^t \sigma(s, X(s)) dB(s)$
• SDEs can be defined using either Itô or Stratonovich integrals, leading to different interpretations and properties

Strong vs weak solutions

• There are two main types of solutions to SDEs: strong solutions and weak solutions
• A strong solution to an SDE is a stochastic process that satisfies the integral equation almost surely and is adapted to the filtration generated by the Brownian motion
• A weak solution to an SDE is a stochastic process that satisfies the integral equation in distribution and may be defined on a different probability space than the original Brownian motion
• Strong solutions are pathwise unique, while weak solutions are unique in distribution

Existence and uniqueness of solutions

• The existence and uniqueness of solutions to SDEs depend on the regularity conditions imposed on the drift and diffusion coefficients
• The most common conditions for ensuring the existence and uniqueness of strong solutions are the Lipschitz continuity and linear growth conditions:
  • Lipschitz continuity: $|\mu(t, x) - \mu(t, y)| + |\sigma(t, x) - \sigma(t, y)| \leq K|x - y|$ for some constant $K$
  • Linear growth: $|\mu(t, x)|^2 + |\sigma(t, x)|^2 \leq K(1 + |x|^2)$ for some constant $K$
• Under these conditions, the SDE has a unique strong solution, which can be approximated using numerical methods

Numerical methods for SDEs

• Numerical methods for SDEs aim to approximate the solution of an SDE by discretizing time and computing the stochastic integral incrementally
• The most common numerical methods for SDEs are:
  • Euler-Maruyama method: a simple explicit scheme that approximates the stochastic integral using the left endpoint rule
  • Milstein method: an improved scheme that includes a correction term based on the Itô-Taylor expansion
  • Higher-order methods: schemes that achieve better convergence rates by including more terms in the Itô-Taylor expansion
• The choice of the numerical method depends on the desired accuracy, computational efficiency, and the specific properties of the SDE

Applications of stochastic integrals

• Stochastic integrals and the associated stochastic calculus have found numerous applications in various fields, where randomness and uncertainty play a crucial role
• The ability to model and analyze systems subject to random fluctuations has led to significant advances in science, engineering, and finance

Mathematical finance

• Stochastic integrals are extensively used in mathematical finance to model asset prices, derive pricing formulas for financial derivatives, and develop risk management strategies
• The Black-Scholes model, which is based on an SDE with geometric Brownian motion, has become a standard tool for pricing and hedging options
• Stochastic volatility models, such as the Heston model, employ SDEs to capture the time-varying nature