Harmonic analysis of stochastic equations and backward stochastic differential equations

By Freddy Delbaen, Shanjian Tang. 2008

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The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in $${\mathcal{R}^p}$$ ($${p\in [1,\infty)}$$) and backward stochastic differential equations (BSDEs) in $${\mathcal{R}^p\times \mathcal{H}^p}$$ ($${p\in (1, \infty)}$$) and in $${\mathcal{R}^\infty\times\overline{L^\infty}^{\rm BMO}}$$ , with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman’s inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear cases for stochastic differential equations (SDEs) and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Hölder inequality for some suitable exponent p ≥ 1. Finally, some relations are established between Kazamaki’s quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the stochastic integral operator with respect to M, which lead to a characterization of Kazamaki’s quadratic critical exponent of BMO martingales being infinite.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/d6b4d2f4046631a385a42ee0a44189d1c5db5aab

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