In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some results from optimal stopping theory, some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales, as well as an appropriate generalization of Itô's formula due to Gal'chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position ξ is assessed by an f -conditional expectation E f (·) (where f is a Lipschitz driver). We characterize the "value function" of the problem in terms of the solution to our RB-SDE. Under an additional assumption of left upper-semicontinuity along stopping times on ξ, we show the existence of an optimal stopping time. We also provide a generalization of Mertens decomposition to the case of strong E f -supermartingales.
In this paper, we compare equilibrium equity premium under discrete distributions of jump amplitudes. In particular, we consider the binomial and gamma distributions because of their applicability in finance. For the binomial, we assume that the price movement is allowed to either increase or decrease with probability p or 1 − p respectively. n is the trading period thereby forming a vector x of jump sizes (shifts) whose distribution is a binomial over time. For the gamma, the jumps are taken to be rare events following a Poisson distribution whose waiting times between them follows a gamma. In both distributions, the optimal consumption of the investor is affected by the deterministic time preference function ( ) y t but it has no effect on the diffusive and rare-events premia thereby not affecting the equilibrium equity premium. Also, for n k , 0 = , the volatility effect on the equity premium is the same in both the power and square root utility functions although the equity premium is not affected by the wealth process ( ) V t . However, the wealth process affects the equity premium of the quadratic utility fuction. We observe no significant differences in equity premium for the two discrete distributions.
In this paper, we study the risk averse investor's equilibrium equity premium in a semi martingale market with arbitrary jumps. We realize that, if we normalize the market, the equilibrium equity premium is consistent to taking the risk free rate ρ = 0 in martingale markets. We also observe that the value process affects both the diffusive and rare-event premia except for the CARA negative exponential utility function. The bond price always affect the diffusive risk premium for this risk averse investor.
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