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Ornstein-Uhlenbeck
The Ornstein-Uhlenbeck SDE:
 * $$ dy_t = - \, a \, y_t \, dt + \sigma \, dW_t $$

New variable, new SDE, straightforward integration:
 * $$ x_t = y_t \, e^{at}, \; \; \; dx_t = \sigma \, e^{at} \, dW_t, \; \; \; x_t = x_0 + \sigma \int_0^t e^{as} dW_s $$

Back to the original variable:
 * $$ y_t = x_t \, e^{-at} = y_0 \, e^{-at} + \sigma \, e^{-at} \int_0^t e^{as} dW_s $$

Distribution of the integral:
 * $$ \int_0^t e^{as} dW_s \in N(0,\nu), \; \; \; \nu = \int_0^t e^{2as} ds = \frac{1}{2a} \left( e^{2at} - 1 \right) $$

Distribution of the original variable:
 * $$ y_t \in N(m,v), \; \; \; m = y_0 \, e^{-at}, \; \; \;

v = \sigma^2 e^{-2at} \nu = \frac{\sigma^2}{2a} \left( 1 - e^{-2at} \right) $$ Asymptotics:
 * $$ t = \infty: \; \; \; y_t \in N \left( 0,\frac{\sigma^2}{2a} \right) $$

Ornstein-Uhlenbeck integral

 * $$ z(0,T) = \int_0^T y_t \, dt $$
 * $$ z(0,T) = y_0 \, B(0,T) + \sigma \int_0^T B(t,T) \, dW_t, \; \; \;

B(t,T) = \frac{1}{a} \left[ 1 - e^{-a(T-t)} \right] $$
 * $$ z(0,T) \in N(m,v), \; \; \; m = y_0 \, B(0,T), \; \; \;

v = \sigma^2 \int_0^T B^2(t,T) \, dt = \frac{\sigma^2}{a^2} [T - B(0,T)] \, - \, \frac{\sigma^2}{2a} B^2(0,T) $$

Vasicek

 * $$ dr_t = (\theta - a\, r_t) \, dt + \sigma \, dW_t $$
 * $$ r_t = \alpha(t) + y_t, \; \; \; dy_t = - \, a \, y_t \, dt + \sigma \, dW_t, \; \; \; y_0 = r_0 $$
 * $$ \alpha(t) = \frac{\theta}{a} \left( 1 - e^{-at} \right), \; \; \;

y_t = r_0 \, e^{-at} + \sigma \, e^{-at} \int_0^t e^{as} dW_s$$
 * $$ z(0,T) = \int_0^T r_t \, dt = \int_0^T \alpha(t) \, dt + \int_0^T y_t \, dt $$

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