No abstract

We study time-consistency questions for processes of monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a process of monetary risk measures time-consistent if it assigns to a process of financial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time, and we show how this property manifests itself in the corresponding process of acceptance sets. For processes of coherent and convex monetary risk measures admitting a robust representation with sigma-additive linear functionals, we give necessary and sufficient conditions for time-consistency in terms of the representing functionals.

We show that the sum of a Brownian motion and a non-trivial multiple of an independent fractional Brownian motion with Hurst parameter H P (0, 1] is not a semimartingale if H P (0, 1 2 ) ( 1 2 , 3 4 ], that it is equivalent to a multiple of Brownian motion if H 1 2 and equivalent to Brownian motion ifAs an application we discuss the price of a European call option on an asset driven by a linear combination of a Brownian motion and an independent fractional Brownian motion.

No abstract

For a d-dimensional diffusion of the form d X t = µ(X t )dt + σ (X t )dW t and continuous functions f and g, we study the existence and uniqueness of adapted processes Y , Z , , and A solving the second-order backward stochastic differential equation (2BSDE)If the associated PDEhas a sufficiently regular solution, then it follows directly from Itô's formula that the processessolve the 2BSDE, where L is the Dynkin operator of X without the drift term. The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z , , A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Y t = v(t, X t ), t ∈ [0, T ]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods.