Consider a general mortality-linked security (MLS) with a bounded payoff contingent on the evolution of the underlying mortality rate and the performance of associated risky assets. The mortality rate and asset prices are assumed to jointly follow a multivariate Itô process, driven by both a multivariate Brownian motion and a Poisson point process. We follow the utility indifference approach to pricing this MLS under the physical measure. To this end, we employ backward stochastic differential equations (BSDEs) to characterize the optimal investment strategy and the value function for the involved optimization problems. We then solve the resulting nonlinear BSDEs with a non-Lipschitz generator. This methodology, which combines the utility indifference approach with BSDE techniques, provides numerical tractability through Monte Carlo simulations. Finally, we conduct comprehensive numerical studies on the valuation of several concrete MLSs, with a focus on the sensitivity analysis of the indifference prices against various key model parameters, including, in particular, the correlation between the underlying mortality rate and asset price.

The large deviation principle in the small noise limit is derived for solutions of possibly degenerate Itô stochastic differential equations with predictable coefficients, which may also depend on the large deviation parameter. The result is established under mild assumptions using the Dupuis-Ellis weak convergence approach. Applications to certain systems with memory and to positive diffusions with square-root-like dispersion coefficient are included.

We pose an optimal control problem arising in a perhaps new model for retirement investing. Given a control function f and our current net worth X(t) for any t, we invest an amount f(X(t)) in the market. We need a fortune of M ‘superdollars’ to retire and want to retire as early as possible. We model our change in net worth over each infinitesimal time interval by the Itô process dX(t) = (1 + f(X(t)))dt + f(X(t))dW(t). We show how to choose the optimal f = f 0 and show that the choice of f 0 is optimal among all nonanticipative investment strategies, not just among Markovian ones.

We consider a classic competing-species model with the rates changed to include Gaussian white noise. We show that if the noise is not too large, then the stochastic version is ergodic. An explicit relation between the noise and the original competing-species parameters gives a sufficient condition for ergodicity.