11.2 Stochastic differential equations

Stochastic differential equations (SDEs) blend deterministic dynamics with random fluctuations, extending ordinary differential equations to model systems with uncertainty. They consist of a deterministic term for average behavior and a stochastic term driven by a Wiener process, capturing random perturbations.

SDEs are crucial in various fields, from finance to physics. Solutions may be explicit or numerical, with existence and uniqueness theorems providing theoretical foundations. Key concepts include the Itô integral, Itô's lemma, and numerical methods like Euler-Maruyama for practical applications.

Definition of stochastic differential equations

• Stochastic differential equations (SDEs) model dynamical systems subject to random perturbations, extending ordinary differential equations to incorporate stochastic processes
• SDEs consist of a deterministic term, describing the system's average behavior, and a stochastic term, representing the random fluctuations affecting the system
• The stochastic term is typically driven by a Wiener process (Brownian motion), which is a continuous-time Gaussian process with independent increments

Solutions of stochastic differential equations

Existence of solutions

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• Existence theorems for SDEs establish conditions under which a solution to the equation exists
• Lipschitz continuity and linear growth conditions on the coefficients of the SDE are often sufficient to guarantee the existence of a solution
• Existence results can be global (valid for all times) or local (valid up to a certain time)

Uniqueness of solutions

• Uniqueness theorems for SDEs ensure that, under certain conditions, the solution to the equation is unique
• Lipschitz continuity of the coefficients is a common condition for uniqueness
• Uniqueness can be pathwise (holding for almost all sample paths) or in distribution (solutions have the same probability distribution)

Explicit solutions vs numerical methods

• Explicit solutions to SDEs can be obtained in some special cases, such as linear SDEs with constant coefficients
• In most cases, explicit solutions are not available, and numerical methods are employed to approximate the solution
• Numerical methods discretize the time interval and simulate the SDE step by step, using techniques such as the Euler-Maruyama method or the Milstein method

Itô integral

Definition of Itô integral

• The Itô integral extends the Riemann-Stieltjes integral to stochastic processes, allowing integration with respect to a Wiener process
• It is defined as a limit of Riemann-Stieltjes sums, where the integrand is evaluated at the left endpoint of each subinterval
• The Itô integral is a martingale and has zero mean

Properties of Itô integral

• Linearity: The Itô integral is linear in the integrand
• Adaptedness: The integrand must be adapted to the filtration generated by the Wiener process
• Continuity: The Itô integral is a continuous function of the upper limit of integration
• Zero quadratic variation: The quadratic variation of the Itô integral is zero

Itô isometry

• The Itô isometry relates the second moment of the Itô integral to the expected value of the squared integrand
• It states that $E[\int_0^T f(t) dW(t)]^2 = E[\int_0^T f(t)^2 dt]$, where $f$ is the integrand and $W$ is the Wiener process
• The Itô isometry is a key tool in the analysis of SDEs and the derivation of Itô's lemma

Itô's lemma

Statement of Itô's lemma

• Itô's lemma is a stochastic calculus analogue of the chain rule in ordinary calculus
• It provides a formula for the differential of a function of a stochastic process satisfying an SDE
• The formula includes an additional term involving the second derivative of the function, which accounts for the quadratic variation of the process

Applications of Itô's lemma

• Itô's lemma is used to derive the dynamics of functions of stochastic processes, such as the price of a derivative security in financial mathematics
• It allows the computation of moments, probability distributions, and other characteristics of transformed processes
• Itô's lemma is a fundamental tool in the study of stochastic optimal control problems and the Hamilton-Jacobi-Bellman equation

Generalized Itô's lemma

• The generalized Itô's lemma extends the original formula to functions of several stochastic processes and time
• It accounts for the cross-variations between the processes and the partial derivatives of the function with respect to each process and time
• The generalized Itô's lemma is particularly useful in the analysis of multi-dimensional SDEs and systems with multiple sources of uncertainty

Stochastic exponential

Definition of stochastic exponential

• The stochastic exponential (also known as Doléans-Dade exponential) is a stochastic process that generalizes the ordinary exponential function to the context of SDEs
• It is defined as the solution to a linear SDE with a specific form, involving the stochastic integral of the process itself
• The stochastic exponential is a local martingale and has the property of being strictly positive

Properties of stochastic exponential

• Multiplicative property: The product of two stochastic exponentials is another stochastic exponential
• Inverse property: The inverse of a stochastic exponential is also a stochastic exponential
• Relationship to ordinary exponential: In the absence of the stochastic term, the stochastic exponential reduces to the ordinary exponential function
• Martingale property: Under certain conditions on the integrand, the stochastic exponential is a martingale

Relationship to geometric Brownian motion

• Geometric Brownian motion (GBM) is a stochastic process that models the evolution of a stock price or other asset value
• GBM is obtained by taking the stochastic exponential of a Wiener process with drift
• The stochastic exponential ensures that the GBM remains positive, which is a desirable property for modeling asset prices
• Many financial models, such as the Black-Scholes option pricing model, rely on the properties of GBM and the stochastic exponential

Stratonovich integral

Definition of Stratonovich integral

• The Stratonovich integral is an alternative to the Itô integral for stochastic integration with respect to a Wiener process
• It is defined using a midpoint rule, evaluating the integrand at the average of the left and right endpoints of each subinterval
• The Stratonovich integral satisfies the ordinary chain rule, unlike the Itô integral

Comparison to Itô integral

• The Itô and Stratonovich integrals differ in the choice of the point at which the integrand is evaluated within each subinterval
• The Itô integral uses the left endpoint, while the Stratonovich integral uses the midpoint
• The two integrals lead to different versions of stochastic calculus, with the Stratonovich calculus being more intuitive for some applications

Stratonovich calculus

• Stratonovich calculus is the stochastic calculus based on the Stratonovich integral
• It satisfies the ordinary chain rule, making it more familiar to practitioners accustomed to ordinary calculus
• Stratonovich SDEs can be converted to Itô SDEs by adding a correction term involving the derivative of the diffusion coefficient
• The choice between Itô and Stratonovich calculus depends on the modeling assumptions and the specific application

Linear stochastic differential equations

Homogeneous linear equations

• Homogeneous linear SDEs are equations where the drift and diffusion coefficients are linear functions of the state variable, with no external forcing term
• The solution to a homogeneous linear SDE can be expressed using the stochastic exponential of the integral of the coefficients
• Homogeneous linear SDEs are used to model various phenomena, such as population dynamics with stochastic growth rates

Inhomogeneous linear equations

• Inhomogeneous linear SDEs include an external forcing term in addition to the linear drift and diffusion coefficients
• The solution to an inhomogeneous linear SDE consists of a homogeneous solution and a particular solution, which can be found using the variation of parameters formula
• Inhomogeneous linear SDEs are employed in the modeling of systems subject to random external influences, such as stochastic control problems

Variation of parameters formula

• The variation of parameters formula is a method for solving inhomogeneous linear SDEs
• It expresses the solution as the sum of a homogeneous solution and a stochastic integral involving the forcing term and the fundamental solution matrix
• The formula reduces the problem of solving an inhomogeneous SDE to the computation of stochastic integrals, which can be done using numerical methods or, in some cases, analytically

Numerical methods for SDEs

Euler-Maruyama method

• The Euler-Maruyama method is a simple and widely used numerical scheme for approximating the solution of SDEs
• It is an extension of the Euler method for ordinary differential equations, incorporating the stochastic term using a discretized Wiener process
• The method has a strong convergence order of 0.5 and a weak convergence order of 1, making it suitable for many applications

Milstein method

• The Milstein method is an improvement over the Euler-Maruyama method, achieving a strong convergence order of 1
• It includes an additional term in the discretization scheme, involving the derivative of the diffusion coefficient
• The Milstein method requires the computation of multiple stochastic integrals, which can be challenging in higher dimensions

Runge-Kutta methods for SDEs

• Runge-Kutta methods, originally developed for ordinary differential equations, can be adapted to solve SDEs
• Stochastic Runge-Kutta methods incorporate the stochastic term by using multiple evaluations of the drift and diffusion coefficients within each time step
• Higher-order Runge-Kutta methods for SDEs, such as the stochastic Runge-Kutta method of order 1.5, can provide more accurate approximations at the cost of increased computational complexity

Applications of stochastic differential equations

Financial mathematics

• SDEs are extensively used in financial mathematics to model the dynamics of asset prices, interest rates, and other financial variables
• The Black-Scholes model for option pricing is based on a geometric Brownian motion, which is a particular type of SDE
• Stochastic volatility models, such as the Heston model, employ SDEs to capture the time-varying and random nature of volatility in financial markets

Physics and engineering

• SDEs are applied in various fields of physics and engineering to model systems subject to random fluctuations
• In statistical mechanics, SDEs are used to describe the motion of particles in a fluid, such as the Langevin equation for Brownian motion
• In control engineering, SDEs are employed to model systems with stochastic disturbances and to design robust control strategies

Biology and ecology

• SDEs find applications in biology and ecology to model the dynamics of populations, epidemics, and other biological processes
• Stochastic population models, such as the Lotka-Volterra equations with random perturbations, use SDEs to account for environmental fluctuations and demographic noise
• In neuroscience, SDEs are employed to model the stochastic firing of neurons and the propagation of signals in neural networks