Given (T>0), let (left(X_{t}ight)_{t in[0, T)}) denote the solution of the stochastic differential equation [begin{equation*}d X_{t}=sigma d
Question:
Given \(T>0\), let \(\left(X_{t}ight)_{t \in[0, T)}\) denote the solution of the stochastic differential equation
\[\begin{equation*}d X_{t}=\sigma d B_{t}-\frac{X_{t}}{T-t} d t, \quad t \in[0, T) \tag{4.42}\end{equation*}\]
under the initial condition \(X_{0}=0\) and \(\sigma>0\).
a) Show that
\[
X_{t}=(T-t) \int_{0}^{t} \frac{\sigma}{T-s} d B_{s}, \quad 0 \leqslant t
b) Show that \(\mathbb{E}\left[X_{t}ight]=0\) for all \(t \in[0, T)\).
c) Show that \(\operatorname{Var}\left[X_{t}ight]=\sigma^{2} t(T-t) / T\) for all \(t \in[0, T)\).
d) Show that \(\lim _{t ightarrow T} X_{t}=0\) in \(L^{2}(\Omega)\). The process \(\left(X_{t}ight)_{t \in[0, T]}\) is called a Brownian bridge.
Step by Step Answer:
Introduction To Stochastic Finance With Market Examples
ISBN: 9781032288277
2nd Edition
Authors: Nicolas Privault