Preface

Lp solutions of backward stochastic differential equations with jumps

“…For any scenario ω ∈ Ω, let D p(ω) collect all jump times of the path p(ω), which is a countable subset of (0, T ] (see e.g. Section 1.9 of [39]). We assume that for some finite measure ν on X , F X , the counting measure…”

Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation

“…See [5] for background on this topic. In this case the driving path is a d-dimensional Brownian motion W (t) = (W 1 (t), · · · , W d (t)) where W i (t) are independent standard Brownian motions defined on [0, ∞).…”

“…In this context we investigate the smoothness requirements on f for existence and uniqueness of solutions. In section 5 we give examples to show that the results of sections 2 and 3 are sharp, to the extent that that uniqueness can fail with f ∈ C γ whenever 1 < γ < p < 3, and existence can fail for f ∈ C p−1 whenever 1 < p < 2 or 2 < p < 3. Section 6 treats global existence questions.…”

Stochastic hamiltonian dynamical systems

“…The map z ∈ M −→ F t (z, ω) ∈ M is a local diffeomorphism of M , for each t ≥ 0 and almost all ω ∈ Ω in which this map is defined (see [IW89]). In the following result, we show that, in the symplectic context, Hamiltonian flows preserve the symplectic form and hence the associated volume form θ = ω ∧ n ... ∧ ω.…”

“…The mathematical formulation of Brownian motions (or Wiener processes) on manifolds has been the subject of much research and it is a central topic in the study of stochastic processes on manifolds (see [IW89, Chapter 5], [E89, Chapter V], and references therein for a good general review of this subject).…”