BMO Martingales Research Articles

Dynamic risk measures: Time consistency and risk measures from BMO martingales

Time consistency is a crucial property for dynamic risk measures. Making use of the dual representation for conditional risk measures, we characterize the time consistency by a cocycle condition for the minimal penalty function.

BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

This paper is devoted to real valued backward stochastic differential equations (BSDEs for short) with generators which satisfy a stochastic Lipschitz condition involving BMO martingales. This framework arises naturally when looking at the BSDE satisfied by the gradient of the solution to a BSDE with quadratic growth in Z . We first prove an existence and uniqueness result from which we deduce the differentiability with respect to parameters of solutions to quadratic BSDEs. Finally, we apply these results to prove the existence and uniqueness of a mild solution to a parabolic partial differential equation in Hilbert space with nonlinearity having quadratic growth in the gradient of the solution.

Boundary behavior of SLE

We show that the normalized (pre-)Schwarzian derivative of SLE, after we subtract a negligible term, is a complex BMO martingale. Its BMO norm gives strong evidence for Duplantier’s duality conjecture. We also show that it has correlations that decay exponentially in the hyperbolic distance. We reexamine S. Rohde and O. Schramm’s derivative expectation to derive the conjectured sharp estimate for the Hölder exponent unless the parameter of SLE is 4.

An Exponential Martingale Equation

We prove an existence of a unique solution of an exponential martingale equation in the class of BMO martingales. The solution is used to characterize optimal martingale measures.

A REMARK ON A BMO MARTINGALE

In this paper, a negative answer to a question raised by Durrett(1984)[l] about a BMO martingale is given.

Two-Dimensional Conjugate Martingale Transforms

Characterizations of the two-dimensional H1, BMO and VMO martingale spaces generated by bounded Vilenkin systems via conjugate martingale transforms are studied.

Kähler diffusions, carleson measures and BMOA functions of several complex variables

In this paper we give some characterizations of BMOA functions in terms of Kahler diffusions and Carleson measures on the unit ball in C n n1 (see Theorem 1). This result implies Carleson measure characterizations of BMOA obtained recently by Choa and Choe [2] and Stanton [19]. Furthermore, as a corollary we prove that the spaces BMOA and BMO are isomorphic to closed complemented subspaces of a space of BMO martingales. From this it follows a non-isotropic version of Garnett-Jones theorem.

An Extension of Kazamaki's Results on BMO Differentials

Kazamaki has shown that if (M~)nz1 M are BMO martingales with continuous paths and lim M' = M in BMO, then e(Mf) converges in Al to &(M), where 6(M) denotes the stochastic exponential of M. While Kazamaki's result does not extend to the right continuous case, it does extend locally. It is shown here that if M', M are semimartingales and M' converges locally in A (a semimartingale BMO-type norm) to M then X' converges locally in AT (1 _ p < 0o) to X, where X', X are resepctively solutions of stochastic integral equations with Lipschitz-type coefficients and differentials dM', dM. (The coefficients are also allowed to vary.) This is a stronger stability than usually holds for solutions of stochastic integral equations, reflecting the strength of the Y norm.