Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion
Let \(B\) be a Brownian motion starting from 0 and \(-\infty <\alpha ,\beta <1\); two stochastic functional equations are studied in this paper. They are \[ (1) \quad W_t=B_t+\alpha M_t^W+\frac 12L_t^W \quad \text{and}\quad (2)\;X_t=B_t+\alpha M_ t^X+\beta I_t^X, \] where \(M_t^Y\), \(I_t^Y\) and \(L_t^Y\) are the maximal process, minimal process of the process \(Y\) before time \(t\) and the local time of \(Y\) at 0, respectively. It is proven that for these two equations there exist pathwise unique solutions \(W\) and \(X\), respectively, and are adapted to the filtration of \(B\). Furthermore, \(W\) behaves like reflected Brownian motion (RBM) except when it attains a new maximum and it is then called \(\alpha\)-perturbed RBM. Similarly, \(X\) behaves like Brownian motion (BM) except when it attains a new maximum or minimum and it is then called \(\alpha,\beta\)-doubly perturbed BM. The time change of the positive (respectively, negative) part of an \(\alpha,\beta\)-doubly perturbed BM is an \(\alpha\)-perturbed RBM (respectively, \(\beta\)-perturbed RBM). And these two processes are independent.
\(\alpha \)-perturbed reflected Brownian motion
\(\alpha, \beta \)-doubly perturbed Brownian motion
scientific article; zbMATH DE number 1300444