Random oscillations and Brownian bridges with variable input (Q1205994)

Let \((B_ k)_{k\geq 1}\) be a sequence of independent Brownian bridges and \((\theta_ k)_{k\geq 1}\) be a sequence of positive i.i.d. random variables, which are also independent of \((B_ k)_{k\geq 1}\). Put \(T_ k=\sum^ k_{i=1}\theta_ i\), \(k\geq 1\), \(T_ 0=0\). Motivated by actuarial applications, the author is concerned with the stochastic process \[ C(s)=\sigma B_ k\left({s-T_{k-1}\over\theta_ k}\right)\quad\text{for } T_{k-1}\leq s<T_ k,\;k\geq 1, \] where \(\sigma>0\) is a parameter. He considers the special case where \(\theta_ 1\) is exponential of parameter \(\lambda\) and discusses the problem of estimating \(\lambda\) and \(\sigma\).