Chapter 3 Stochastic Differential Equations
3.1 Introduction to SDEs
A stochastic differential equation (SDE) describes how a random process evolves over time, combining deterministic trends with random fluctuations.
The general form is:
\[dX(t) = \mu(t, X) \, dt + \sigma(t, X) \, dB(t)\]
where:
- \(\mu(t, X)\) is the drift coefficient - the deterministic trend
- \(\sigma(t, X)\) is the diffusion coefficient (or volatility) - controls the random fluctuations
- \(B(t)\) is standard Brownian motion
This differential equation is shorthand for the integral equation:
\[X(t) = X(0) + \int_0^t \mu(s, X(s)) \, ds + \int_0^t \sigma(s, X(s)) \, dB(s)\]
The first integral is ordinary (deterministic), the second is an Itô integral (random).
3.2 Geometric Brownian Motion
The most famous SDE in finance models stock prices:
\[dS = \mu S \, dt + \sigma S \, dB\]
where \(\mu\) and \(\sigma\) are constants.
3.2.1 Interpretation
Rewrite as:
\[\frac{dS}{S} = \mu \, dt + \sigma \, dB\]
The return \(dS/S\) has two components:
- Deterministic drift: \(\mu \, dt\) per unit time
- Random shock: \(\sigma \, dB\) with volatility \(\sigma\)
This says returns (not prices!) have a constant drift and volatility.
3.2.2 Solving GBM
We can’t just “integrate” the SDE directly. Instead, we use Itô’s lemma to find the right transformation!
Strategy: Try \(f(S) = \log S\).
Then: - \(f'(S) = 1/S\) - \(f''(S) = -1/S^2\)
From the SDE: \(dS = \mu S \, dt + \sigma S \, dB\)
Therefore: \((dS)^2 = (\mu S \, dt + \sigma S \, dB)^2 = \sigma^2 S^2 (dB)^2 = \sigma^2 S^2 \, dt\)
(The \(dt \cdot dB\) and \((dt)^2\) terms vanish.)
By Itô’s lemma:
\[d(\log S) = \frac{1}{S} dS + \frac{1}{2}\left(-\frac{1}{S^2}\right)(dS)^2\]
\[= \frac{1}{S}(\mu S \, dt + \sigma S \, dB) - \frac{1}{2} \cdot \frac{1}{S^2} \cdot \sigma^2 S^2 \, dt\]
\[= \mu \, dt + \sigma \, dB - \frac{\sigma^2}{2} \, dt\]
\[= \left(\mu - \frac{\sigma^2}{2}\right) dt + \sigma \, dB\]
This is linear in \(\log S\)! Integrating from 0 to \(t\):
\[\log S(t) - \log S(0) = \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma B(t)\]
Exponentiating:
\[\boxed{S(t) = S(0) \exp\left[\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma B(t)\right]}\]
3.2.3 Key Observations
- Always positive: \(S(t) > 0\) always (good for stock prices!)
- Log-normal: \(\log S(t)\) is normally distributed
- Itô correction: The drift in the exponent is \(\mu - \sigma^2/2\), not \(\mu\)
- Expected value: \(E[S(t)] = S(0)e^{\mu t}\) (using the moment generating function)
- Median: \(\text{median}[S(t)] = S(0)e^{(\mu - \sigma^2/2)t}\)
The difference between \(\mu\) and \(\mu - \sigma^2/2\) is sometimes called Jensen’s inequality correction or the convexity adjustment.
3.2.4 The Itô Correction Explained
Why the \(-\sigma^2/2\) term?
The exponential function is convex. Because of volatility, \(S(t)\) spends more time in regions where the exponential curves upward. To maintain the correct expected drift rate, we need to adjust downward by \(\sigma^2/2\).
This is a general feature: non-linear transformations of stochastic processes pick up correction terms proportional to the second derivative (curvature).
3.3 Ornstein-Uhlenbeck Process
This SDE models mean reversion - a tendency to return to a long-term average:
\[dX = -\theta X \, dt + \sigma \, dB\]
where \(\theta > 0\) is the mean reversion speed.
3.3.1 Interpretation
- When \(X > 0\): drift is negative (pulled toward zero)
- When \(X < 0\): drift is positive (pushed toward zero)
- Strength of pull proportional to distance from zero
This models: - Interest rates (tend to revert to long-term average) - Commodity prices (mean reversion due to supply/demand) - Temperature (seasonal mean reversion)
3.3.2 Solving the OU Process
Try the transformation \(f(t, X) = e^{\theta t} X\).
Partial derivatives: - \(\frac{\partial f}{\partial t} = \theta e^{\theta t} X\) - \(\frac{\partial f}{\partial X} = e^{\theta t}\) - \(\frac{\partial^2 f}{\partial X^2} = 0\)
By Itô’s lemma:
\[d(e^{\theta t} X) = \theta e^{\theta t} X \, dt + e^{\theta t} dX + 0\]
Substitute \(dX = -\theta X \, dt + \sigma \, dB\):
\[d(e^{\theta t} X) = \theta e^{\theta t} X \, dt + e^{\theta t}(-\theta X \, dt + \sigma \, dB)\]
\[= \sigma e^{\theta t} \, dB\]
The drift terms cancel! Integrating:
\[e^{\theta t} X(t) - X(0) = \sigma \int_0^t e^{\theta s} \, dB(s)\]
Therefore:
\[\boxed{X(t) = X(0)e^{-\theta t} + \sigma \int_0^t e^{-\theta(t-s)} \, dB(s)}\]
3.3.3 Properties of the Solution
- Exponential decay: The initial condition \(X(0)\) decays as \(e^{-\theta t}\)
- Half-life: Time to halve: \(t_{1/2} = \frac{\log 2}{\theta}\)
- Gaussian: \(X(t)\) is normally distributed (it’s a linear combination of Brownian increments)
- Mean: \(E[X(t)] = X(0)e^{-\theta t} \to 0\) as \(t \to \infty\)
- Variance: \(\text{Var}(X(t)) = \frac{\sigma^2}{2\theta}(1 - e^{-2\theta t}) \to \frac{\sigma^2}{2\theta}\) as \(t \to \infty\)
The process reaches a stationary distribution \(N(0, \sigma^2/(2\theta))\) as \(t \to \infty\).
3.4 General Strategy for Solving SDEs
Not all SDEs have closed-form solutions, but when they do, the strategy is:
- Guess a transformation \(f(X)\) or \(f(t, X)\) that might linearize or simplify things
- Apply Itô’s lemma to find \(df\)
- Hope that \(df\) is simpler (ideally linear or separable)
- Integrate and solve for \(f\)
- Transform back to find \(X\)
Common tricks:
- For multiplicative noise: try \(\log X\)
- For mean reversion: try \(e^{\theta t} X\)
- For time-dependent drift: try integrating factors
- For Bessel processes: try \(X^2\)
This is an art! Intuition comes with practice.
3.5 Linear SDEs
SDEs of the form:
\[dX = [a(t) + b(t)X] \, dt + [c(t) + d(t)X] \, dB\]
where \(a, b, c, d\) are deterministic functions, can always be solved using integrating factors.
The solution involves: 1. Solving the homogeneous equation 2. Variation of parameters for the particular solution 3. Combining with the stochastic integral
This parallels the theory of linear ODEs, but with stochastic integrals replacing ordinary integrals.
3.6 Existence and Uniqueness
When does an SDE have a solution? When is it unique?
Theorem (Existence and Uniqueness): If \(\mu(t, x)\) and \(\sigma(t, x)\) satisfy:
Lipschitz condition: There exists \(K > 0\) such that \[|\mu(t,x) - \mu(t,y)| + |\sigma(t,x) - \sigma(t,y)| \leq K|x - y|\] for all \(t, x, y\)
Linear growth: There exists \(C > 0\) such that \[|\mu(t,x)| + |\sigma(t,x)| \leq C(1 + |x|)\] for all \(t, x\)
Then the SDE has a unique strong solution for any initial condition \(X(0)\).
These conditions ensure the drift and diffusion are “nice enough” - not too wild or growing too fast.
3.7 Strong vs. Weak Solutions
- Strong solution: Constructed pathwise on a given probability space with a given Brownian motion
- Weak solution: The probability distribution of the solution is specified, but the construction may vary
Most applications in finance use weak solutions - we care about distributions of prices, not individual paths.
3.8 Multidimensional SDEs
SDEs can be vectors:
\[dX_i = \mu_i(t, \mathbf{X}) \, dt + \sum_{j=1}^d \sigma_{ij}(t, \mathbf{X}) \, dB_j\]
where \(\mathbf{X} = (X_1, \ldots, X_n)\) and \(B_1, \ldots, B_d\) are independent Brownian motions.
The matrix \(\sigma = (\sigma_{ij})\) is the diffusion matrix. The covariance structure is:
\[d\langle X_i, X_j \rangle = \sum_{k=1}^d \sigma_{ik} \sigma_{jk} \, dt = (\sigma \sigma^T)_{ij} \, dt\]
Multidimensional Itô’s lemma becomes:
\[df(\mathbf{X}) = \sum_i \frac{\partial f}{\partial x_i} dX_i + \frac{1}{2}\sum_{i,j} \frac{\partial^2 f}{\partial x_i \partial x_j} d\langle X_i, X_j \rangle\]
This is essential for modeling portfolios, interest rate curves, and other multivariate phenomena.