Brownian motion, pinned at both ends. This uses a Brownian bridge.
A Brownian bridge is a continuoustime stochastic process B(t) whose probability distribution is the conditional probability distribution of a Wiener process W(t) (a mathematical model of Brownian motion) given the condition that B(1) = 0. More precisely:

B_t := (W_tW_1=0),\;t \in [0,1]
The expected value of the bridge is zero, with variance t(1 − t), implying that the most uncertainty is in the middle of the bridge, with zero uncertainty at the nodes. The covariance of B(s) and B(t) is s(1 − t) if s < t. The increments in a Brownian bridge are not independent.
Contents

Relation to other stochastic processes 1

Intuitive remarks 2

General case 3

References 4
Relation to other stochastic processes
If W(t) is a standard Wiener process (i.e., for t ≥ 0, W(t) is normally distributed with expected value 0 and variance t, and the increments are stationary and independent), then

B(t) = W(t)  t W(1)\,
is a Brownian bridge for t ∈ [0, 1]. It is independent of W(1)^{[1]}
Conversely, if B(t) is a Brownian bridge and Z is a standard normal random variable independent of B, then the process

W(t) = B(t) + tZ\,
is a Wiener process for t ∈ [0, 1]. More generally, a Wiener process W(t) for t ∈ [0, T] can be decomposed into

W(t) = B\left(\frac{t}{T}\right) + \frac{t}{\sqrt{T}} Z.
Another representation of the Brownian bridge based on the Brownian motion is, for t ∈ [0, 1]

B(t) = (1t) W\left(\frac{t}{1t}\right).
Conversely, for t ∈ [0, ∞]

W(t) = (1+t) B\left(\frac{t}{1+t}\right).
The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as

B_t = \sum_{k=1}^\infty Z_k \frac{\sqrt{2} \sin(k \pi t)}{k \pi}
where Z_1, Z_2, \ldots are independent identically distributed standard normal random variables (see the Karhunen–Loève theorem).
A Brownian bridge is the result of Donsker's theorem in the area of empirical processes. It is also used in the Kolmogorov–Smirnov test in the area of statistical inference.
A standard Wiener process satisfies W(0) = 0 and is therefore "tied down" to the origin, but other points are not restricted. In a Brownian bridge process on the other hand, not only is B(0) = 0 but we also require that B(1) = 0, that is the process is "tied down" at t = 1 as well. Just as a literal bridge is supported by pylons at both ends, a Brownian Bridge is required to satisfy conditions at both ends of the interval [0,1]. (In a slight generalization, one sometimes requires B(t_{1}) = a and B(t_{2}) = b where t_{1}, t_{2}, a and b are known constants.)
Suppose we have generated a number of points W(0), W(1), W(2), W(3), etc. of a Wiener process path by computer simulation. It is now desired to fill in additional points in the interval [0,1], that is to interpolate between the already generated points W(0) and W(1). The solution is to use a Brownian bridge that is required to go through the values W(0) and W(1).
General case
For the general case when B(t_{1}) = a and B(t_{2}) = b, the distribution of B at time t ∈ (t_{1}, t_{2}) is normal, with mean

a + \frac{tt_1}{t_2t_1}(ba)
and the covariance between B(s) and B(t), with s is

\frac{(t_2t)(st_1)}{t_2t_1}.
References

^ Aspects of Brownian motion, Springer, 2008, R. Mansuy, M. Yor page 2

Glasserman, Paul (2004). Monte Carlo Methods in Financial Engineering. New York: SpringerVerlag.

Revuz, Daniel; Yor, Marc (1999). Continuous Martingales and Brownian Motion (2nd ed.). New York: SpringerVerlag.
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