Optimal consumption and investment in incomplete markets with general constraints

We study an optimal consumption and investment problem in a possibly incomplete market with general, not necessarily convex, stochastic constraints. We give explicit solutions for investors with exponential, logarithmic and power utility. Our approach is based on martingale methods which rely on recent results on the existence and uniqueness of solutions to BSDEs with drivers of quadratic growth

Multidimensional quadratic and subquadratic BSDEs with special structure

We study multidimensional BSDEs of the form $$Y_t = \xi + \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s$$ with bounded terminal conditions $\xi$ and drivers $f$ that grow at most quadratically in $Z_s$. We consider three different cases. In the first one the BSDE is Markovian, and a solution can be obtained from a solution to a related FBSDE. In the second case, the BSDE becomes a one-dimensional quadratic BSDE when projected to a one-dimensional subspace, and a solution can be derived from a solution of the one-dimensional equation. In the third case, the growth of the driver $f$ in $Z_s$ is strictly subquadratic, and the existence and uniqueness of a solution can be shown by first solving the BSDE on a short time interval and then extending the solution recursively.Comment: 16 page

BS\Delta Es and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS$\Delta$Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS$\Delta$Es and BSDEs are governed by drivers $f^N(t,\omega,y,z)$ and $f(t,\omega,y,z),$ respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BS$\Delta$Es are based on d-dimensional random walks $W^N$ approximating the d-dimensional Brownian motion W underlying the BSDE and that $f^N$ converges to f. Conditions are given under which for any bounded terminal condition $\xi$ for the BSDE, there exist bounded terminal conditions $\xi^N$ for the sequence of BS$\Delta$Es converging to $\xi$, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when $f^N$ and f are convex in z. We show that in this situation, the solutions of the BS$\Delta$Es converge to the solution of the BSDE for every uniformly bounded sequence $\xi^N$ converging to $\xi$. As a consequence, one obtains that the BSDE is robust in the sense that if $(W^N,\xi^N)$ is close to $(W,\xi)$ in distribution, then the solution of the Nth BS$\Delta$E is close to the solution of the BSDE in distribution too.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ445 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

BSDEs with terminal conditions that have bounded Malliavin derivative

We show existence and uniqueness of solutions to BSDEs of the form $$Y_t = \xi + \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s$$ in the case where the terminal condition $\xi$ has bounded Malliavin derivative. The driver $f(s,y,z)$ is assumed to be Lipschitz continuous in $y$ but only locally Lipschitz continuous in $z$. In particular, it can grow arbitrarily fast in $z$. If in addition to having bounded Malliavin derivative, $\xi$ is bounded, the driver needs only be locally Lipschitz continuous in $y$. In the special case where the BSDE is Markovian, we obtain existence and uniqueness results for semilinear parabolic PDEs with non-Lipschitz nonlinearities. We discuss the case where there is no lateral boundary as well as lateral boundary conditions of Dirichlet and Neumann type

Dynamic monetary risk measures for bounded discrete-time processes

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.Comment: 41 page

Equivalent and absolutely continuous measure changes for jump-diffusion processes

We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jump-diffusion processes that can explode and be killed by a potential.Comment: Published at http://dx.doi.org/10.1214/105051605000000197 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org