1. Introduction
Dynamical systems are commonly modeled using Ordinary Differential Equations (ODE) or Partial Differential Equations (PDE). While ODE and PDE serve as effective tools for expressing and analyzing dynamical systems, setting up coefficients and components can be challenging.
2. Population Growth Model
Consider the simple population growth model $\frac{dX}{dt} = r(t)X(t)$, where $X(0) = x$. Finding an appropriate $r(t)$ to reflect growth dynamics, especially in the presence of random factors, can be challenging. Introducing a term $\text{Noise}$ allows for a more flexible representation: $r(t) = a(t) + \text{Noise}$. The resulting growth model is $\frac{dX}{dt} = a(t)X(t) + \text{Noise} \times X(t)$.
3. Stochastic Representation
The general form of the equation becomes $\frac{dX}{dt} = b(t, X_t) + \sigma(t, X_t)\ \text{Noise}$. Multiplying both sides by $dt$, we get $dX = b(t, X_t)dt + \sigma(t, X_t)\ \text{Noise}\ dt$.
4. Challenges in Defining the Equation
Defining the equation $dX = b(t, X_t)dt + \sigma(t, X_t)\ \text{Noise}\ dt$ poses two challenges:
- Definition of Noise
- Definition of $\int_{t}^{t+\Delta}\sigma(s, X_s)\ \text{Noise}\ ds$.
5. Modeling Noise
To model Noise, we use white noise $W_t$ with the same distribution at $t$. Typically, white Gaussian noise $W_t$ is adopted for ease of analysis, where $E[W_t] = 0$ and $\text{Cov}(W_t, W_s) = I \delta(t-s)$.
6. Integration using Brownian Motion
Defining $\int_{t}^{t+\Delta}\sigma(s, X_s)\ W_s\ ds$ is ambiguous due to the properties of $W_t$. Therefore, an alternative integration method uses Brownian motion $B_t$. The discretized version of $dX = b(t, X_t)dt + \sigma(t, X_t)\ W_t\ dt$ on $[0, t]$ is $X_{k+1} - X_k = b(t_k, X_k)\Delta t_k + \sigma(t_k, X_k)\ W_k\ \Delta t_k$.
7. Stochastic Differential Equation (SDE)
The SDE $X_t - X_0 = \int_{0}^{t} b(s, X_s)\ ds + \int_{0}^{t} \sigma(s, X_s)\ dB_s$ models a dynamical system with noise influencing state changes. Alternatively, it can be expressed as $dX = b(t, X_t)dt + \sigma(t, X_t)\ dB_t$.
8. Applications of SDE
SDEs are exploited to model various dynamics containing random phenomena, such as random growth models, physical dynamics, or stock prices that cannot be explained without incorporating randomness.