Weak limits of perturbed random walks and the equation \(Y_ t = B_ t+\alpha\sup\{Y_ s:s \leq t\} + \beta\inf\{Y_ s:s\leq t\}\)
The analytical problem is the following: Let \(\alpha\) and \(\beta\) be two fixed real parameters. Given the continuous function \(f: [0,\infty)\to (-\infty,\infty)\) with \(f(0)=0\), solve the functional equation \[ g(t) = f(t) +\alpha\sup\{f(s):s\leq t\}+\beta\inf\{f(s):s\leq t\}. \leqno (*) \] Denote \(\rho:=\alpha\beta/(1-\alpha)(1-\beta)\). \textit{Ph. Carmona, F. Petit} and \textit{M. Yor} [``Beta variables as times spent in \([0,\infty[\) by certain perturbed Brownian motions'' (to appear in J. Lond. Math. Soc., II. Ser. (1997)] showed that (a) if \(\max\{\alpha,\beta\}\geq 1\), then the equation \((*)\) is not solvable for most functions \(f\); (b) if \(\max\{\alpha,\beta\}<1\) and \(|\rho|<1\), then the equation \((*)\) admits a unique solution for all functions \(f\). The paper under review improves on result (b) of Carmona, Petit and Yor in the following sense: Existence of solution of equation \((*)\) for all functions \(f\) is proved under condition \(\max\{\alpha,\beta\}<1\), without any restriction on \(\rho\). Unicity of this solution holds under the relaxed additional condition \(|\rho|\leq1\). It is also shown that if \(|\rho|>1\), then unicity of solution fails for particular choices of the function \(f\). If in equation \((*)\) we insert \(f(t)=B_t\) a one-dimensional Brownian path, and if \(|\rho|\leq 1\), the (unique) solution \(g(t)=Y_t\) will be the so-called `Brownian motion perturbed at its past extrema'. Various properties of this stochastic process are stated. It is also proved that \(Y_t\) arises as weak limit of certain `reinforced' random walks. For further recent results on the subject see also the paper by \textit{M. Perman} and the reviewer [``Perturbed Brownian motions'' (to appear in Probab. Theory Relat. Fields (1997)].
Brownian motion perturbed at its past extrema
scientific article; zbMATH DE number 986795