Survival probabilities in the Sisyphus random walk model with absorbing traps

We analyze the dynamics of the Sisyphus random walk model, a discrete Markov chain in which the walkers may randomly return to their initial position $x_0$. In particular, we present a remarkably compact derivation of the time-dependent survival probability function $S(t;x_0)$ which characterizes the random walkers in the presence of an absorbing trap at the origin. The survival probabilities are expressed in a compact mathematical form in terms of the $x_0$-generalized Fibonacci-like numbers $G^{(x_0)}_t$. Interestingly, it is proved that, as opposed to the standard random walk model in which the survival probabilities depend linearly on the initial distance $x_0$ of the walkers from the trap and decay asymptotically as an inverse power of the time, in the Sisyphus random walk model the asymptotic survival probabilities decay exponentially in time and are characterized by a non-trivial (non-linear) dependence on the initial gap $x_0$ from the absorbing trap. We use the analytically derived results in order to analyze the underlying dynamics of the `survival-game', a highly risky investment strategy in which non-absorbed agents receive a reward $P(t)$ which gradually increases in time.