Solving the Ornstein–Uhlenbeck SDE
We are asked to solve the stochastic differential equation (SDE):
\[ dr_t = \lambda(\theta - r_t)\,dt + \sigma\,dW_t, \]
where \( r_0 \) is the initial value, \( \lambda, \theta, \sigma \in \mathbb{R} \), and \( W_t \) is a standard Brownian motion.
Step 1: Linearization and Integrating Factor
Rewriting the SDE:
\[ dr_t + \lambda r_t\,dt = \lambda \theta\,dt + \sigma\,dW_t. \]
Multiply both sides by the integrating factor \( e^{\lambda t} \):
\[ e^{\lambda t}dr_t + \lambda e^{\lambda t} r_t\,dt = \lambda \theta e^{\lambda t}\,dt + \sigma e^{\lambda t}\,dW_t. \]
Recognize the left-hand side as a total derivative:
\[ d(e^{\lambda t} r_t) = \lambda \theta e^{\lambda t}\,dt + \sigma e^{\lambda t}\,dW_t. \]
Step 2: Integration
Integrate both sides from 0 to \( t \):
\[ e^{\lambda t} r_t - r_0 = \lambda \theta \int_0^t e^{\lambda s}\,ds + \sigma \int_0^t e^{\lambda s}\,dW_s. \]
Evaluate the deterministic integral:
\[ \int_0^t e^{\lambda s}\,ds = \frac{1}{\lambda}(e^{\lambda t} - 1), \quad \text{so} \quad \lambda \theta \int_0^t e^{\lambda s}\,ds = \theta(e^{\lambda t} - 1). \]
Thus:
\[ e^{\lambda t} r_t = r_0 + \theta(e^{\lambda t} - 1) + \sigma \int_0^t e^{\lambda s}\,dW_s. \]
Step 3: Solve for \( r_t \)
Divide through by \( e^{\lambda t} \):
\[ r_t = e^{-\lambda t} r_0 + \theta(1 - e^{-\lambda t}) + \sigma e^{-\lambda t} \int_0^t e^{\lambda s}\,dW_s. \]
Equivalently, this can be written using the convolution form:
\[ r_t = e^{-\lambda t} r_0 + \theta(1 - e^{-\lambda t}) + \sigma \int_0^t e^{-\lambda(t - s)}\,dW_s. \]
Mean Reversion Property
Taking expectations:
\[ \mathbb{E}[r_t] = e^{-\lambda t} r_0 + \theta(1 - e^{-\lambda t}), \]
since the stochastic integral has mean zero.
As \( t \to \infty \), we have:
\[ \lim_{t \to \infty} \mathbb{E}[r_t] = \theta. \]
This confirms that the Ornstein–Uhlenbeck process is mean-reverting to level \( \theta \), regardless of the initial value \( r_0 \).
Final Result
The explicit solution to the Ornstein–Uhlenbeck SDE is:
\[ r_t = e^{-\lambda t} r_0 + \theta(1 - e^{-\lambda t}) + \sigma \int_0^t e^{-\lambda(t - s)}\,dW_s. \]