We consider a discrete-time Markov chain with state space {1, 1 Δx, . . . , 1 kΔx N}. We compute explicitly the probability p j that the chain, starting from 1 jΔx, will hit N before 1, as well as the expected number d j of transitions needed to end the game. In the limit when Δx and the time Δt between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that p j and d j Δt tend to the corresponding quantities for the geometric Brownian motion.

The static second hyperpolarizability is derived from the space-fractional Schrödinger equation in the particle-centric view. The Thomas-Reiche-Kuhn sum rule matrix elements and the three-level ansatz determines the maximum second hyperpolarizability for a space-fractional quantum system. The total oscillator strength is shown to decrease as the space-fractional parameter α decreases, which reduces the optical response of a quantum system in the presence of an external field. This damped response is caused by the wavefunction dependent position and momentum commutation relation. Although the maximum response is damped, we show that the one-dimensional quantum harmonic oscillator is no longer a linear system for α = 1, where the second hyperpolarizability becomes negative before ultimately damping to zero at the lower fractional limit of α → 1/2.

We consider the so-called mean-variance portfolio selection problem in continuous time under the constraint that the shortselling of stocks is prohibited where all the market coefficients are random processes. In this situation the Hamilton-Jacobi-Bellman (HJB) equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value function often does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical) sense; however, in such cases can be interpreted as a viscosity solution. Here we show the unicity of the viscosity solution and we see that the optimal and the value functions are piecewise linear functions based on some Riccati differential equations. In particular we solve the open problem posed

We consider the so-called gambler's ruin problem for a discrete-time Markov chain that converges to a Cox-Ingersoll -Ross (CIR) process. Both the probability that the chain will hit a given boundary before the other and the average number of transitions needed to end the game are computed explicitly. Furthermore, we show that the quantities that we obtained tend to the corresponding ones for the CIR process. A reallife application to a problem in hydrology is presented.

We investigate the probability of the first hitting time of some discrete Markov chain that converges weakly to the Bessel process. Both the probability that the chain will hit a given boundary before the other and the average number of transitions are computed explicitly. Furthermore, we show that the quantities that we obtained tend (with the Euclidian metric) to the corresponding ones for the Bessel process.