524 research outputs found
Solvability and numerical simulation of BSDEs related to BSPDEs with applications to utility maximization.
In this paper we study BSDEs arising from a special class of backward stochastic partial differential equations (BSPDEs) that is intimately related to utility maximization problems with respect to arbitrary utility functions. After providing existence and uniqueness we discuss the numerical realizability. Then we study utility maximization problems on incomplete financial markets whose dynamics are governed by continuous semimartingales. Adapting standard methods that solve the utility maximization problem using BSDEs, we give solutions for the portfolio optimization problem which involve the delivery of a liability at maturity. We illustrate our study by numerical simulations for selected examples. As a byproduct we prove existence of a solution to a very particular quadratic growth BSDE with unbounded terminal condition. This complements results on this topic obtained in [6, 7, 8].numerical scheme; stochastic optimal control; utility optimization; quadratic growth; distortion transformation; logarithmic transformation; BSPDE; BSDE;
A Functional Approach to FBSDEs and Its Application in Optimal Portfolios
In Liang et al (2009), the current authors demonstrated that BSDEs can be
reformulated as functional differential equations, and as an application, they
solved BSDEs on general filtered probability spaces. In this paper the authors
continue the study of functional differential equations and demonstrate how
such approach can be used to solve FBSDEs. By this approach the equations can
be solved in one direction altogether rather than in a forward and backward
way. The solutions of FBSDEs are then employed to construct the weak solutions
to a class of BSDE systems (not necessarily scalar) with quadratic growth, by a
nonlinear version of Girsanov's transformation. As the solving procedure is
constructive, the authors not only obtain the existence and uniqueness theorem,
but also really work out the solutions to such class of BSDE systems with
quadratic growth. Finally an optimal portfolio problem in incomplete markets is
solved based on the functional differential equation approach and the nonlinear
Girsanov's transformation.Comment: 26 page
Second Order Backward Stochastic Differential Equations with Quadratic Growth
We extend the wellposedness results for second order backward stochastic
differential equations introduced by Soner, Touzi and Zhang \cite{stz} to the
case of a bounded terminal condition and a generator with quadratic growth in
the variable. More precisely, we obtain uniqueness through a representation
of the solution inspired by stochastic control theory, and we obtain two
existence results using two different methods. In particular, we obtain the
existence of the simplest purely quadratic 2BSDEs through the classical
exponential change, which allows us to introduce a quasi-sure version of the
entropic risk measure. As an application, we also study robust risk-sensitive
control problems. Finally, we prove a Feynman-Kac formula and a probabilistic
representation for fully nonlinear PDEs in this setting.Comment: 31 page
Existence and uniqueness results for BSDEs with jumps: the whole nine yards
This paper is devoted to obtaining a wellposedness result for
multidimensional BSDEs with possibly unbounded random time horizon and driven
by a general martingale in a filtration only assumed to satisfy the usual
hypotheses, i.e. the filtration may be stochastically discontinuous. We show
that for stochastic Lipschitz generators and unbounded, possibly infinite, time
horizon, these equations admit a unique solution in appropriately weighted
spaces. Our result allows in particular to obtain a wellposedness result for
BSDEs driven by discrete--time approximations of general martingales.Comment: 48 pages, final version, forthcoming in the Electronic Journal of
Probabilit
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