📂Stochastic Differential Equations

Bridge of Brown

Definition ^{1 2}

$$d Y_{t} = {{ b - Y_{t} } \over { 1 - t }} dt + d W_{t} \qquad, t \in [0,1), Y_{0} = a$$ Let’s denote it as $a, b \in \mathbb{R}$. The stochastic process $Y_{t}$, which is a solution of the $1$-dimensional stochastic differential equation, from ($a$ to $b$) is called a Brownian Bridge. $$Y_{t} = a (1-t) + bt + (1-t) \int_{0}^{t} {{ 1 } \over { 1 - s }} d W_{s}$$

Description

Brownian Bridge is a very special stochastic process that starts at $a$ and, no matter how much it wanders in the middle, eventually stops at $b$. When $t \to 1$, $Y_{t}$ almost surely converges to $b$.

The farther $Y_{t}$ gets from $b$, the more $b-Y_{t}$ significantly influences the numerator of the drift term, especially as the denominator approaches $0$ indefinitely close at $t \approx 1$, compensating for the wandering during the period.