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We derive a characterization of equilibrium controls in continuous-time, time-inconsistent control (TIC) problems using the Malliavin calculus. For this, the classical duality analysis of adjoint BSDEs is replaced with the Malliavin integration by parts. This results into a necessary and sufficient maximum principle which is applied to a linear-quadratic TIC problem, recovering previous results obtained by duality analysis in the mean-variance case, and extending them to the linear-quadratic setting. We also show that our results apply beyond the linear-quadratic case by treating the generalized Merton problem. [ABSTRACT FROM AUTHOR]

This paper deals with optimal prediction in a regime-switching model driven by a continuous-time Markov chain. We extend existing results for geometric Brownian motion by deriving optimal stopping strategies that depend on the current regime state and prove a number of continuity properties relating to optimal value and boundary functions. Our approach replaces the use of closed form expressions, which are not available in our setting, with PDE arguments that also simplify the approach of [du Toit & Peskir (2009) Selling a stock at the ultimate maximum, Annals of Applied Probability 19 (3), 983-1014.] in the classical Brownian case. [ABSTRACT FROM AUTHOR]

We construct differential forms of all orders and a covariant derivative together with its adjoint on the probability space of a standard Poisson process, using derivation operators. In this framewok we derive a de Rham-Hodge-Kodaira decomposition as well as Weitzenböck and Clark-Ocone formulas for random differential forms. As in the Wiener space setting, this construction provides two distinct approaches to the vanishing of harmonic differential forms. [ABSTRACT FROM AUTHOR]

In this paper, we derive a formula for the expectation of random Hermite polynomials in Skorohod integrals, extending classical results in the adapted case. As an application we recover, under simple conditions and with short proofs, the anticipative Girsanov identity and quasi-invariance results obtained in Ref. 6 for quasi-nilpotent shifts on the Wiener space. [ABSTRACT FROM AUTHOR]

The solution of the Fokker–Planck equation for exponential Brownian functionals usually involves spectral expansions that are difficult to compute explicitly. In this paper, we propose a direct solution based on heat kernels and a new integral representation for the square modulus of the Gamma function. A financial application to bond pricing in the Dothan model is also presented. [ABSTRACT FROM AUTHOR]

We obtain lower and upper bounds on option prices in one-dimensional jump-diffusion markets with point process components. Our proofs rely in general on the classical Kolmogorov equation argument and on the propagation of convexity property for Markov semigroups, but the bounds on intensities and jump sizes formulated in our hypotheses are different from the ones already found in the literature (Finance and Stochastics4(2) (2000) 209–222; 10(2) (2006) 229–249). [ABSTRACT FROM AUTHOR]

A general method for deriving Girsanov or quasi-invariance formulas for classical stochastic processes with independent increments obtained as components of Lévy processes on real Lie algebras is presented. Letting a unitary operator arising from the associated factorizable current representation act on an appropriate commutative subalgebra, a second commutative subalgebra is obtained. Under certain conditions the two commutative subalgebras lead to two classical processes such that the law of the second process is absolutely continuous w.r.t. to the first. Examples include the Girsanov formula for Brownian motion as well as quasi-invariance formulas for the Poisson process, the Gamma process,[sup 15,16] and the Meixner process. [ABSTRACT FROM AUTHOR]

The compensated Poisson noise is expressed as a composite sum (splitting) of creation and annihilation operators, whose probabilistic interpretation relies on time changes. We construct an Itô table for this decomposition and obtain continuous and discrete time realizations of Lévy processes on the finite difference algebra fd and on sl[sub 2], e.g. the space-time dual of the Poisson process (compensated gamma process), and the continuous binomial process. [ABSTRACT FROM AUTHOR]