Stochastic Integration and Differential Equations. Second Edition.

By Anja Sturm, R. Malley. 2005

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Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model (2010, unknown): “Hence, by Proposition 8.22 of Cont and Tankov (2004) or Theorem 37 in Chapter 2 of Protter (2005), we solve (3.7) to get P (π)t = e R(π)t , t ≥ 0, where R(π)t = (1− π)r0t + πQt − 1 2 σ 2Qπ 2t + ∑ 0 s≤t (ln(1+ π∆Qs)− π∆Qs) , with∆Qs = Qs − Qs−.”
Long time asymptotics for constrained diffusions in polyhedral domains (2007, background): “(See page 211 of [18] for a definition.”
Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps (2008, background): “Introduction There exists an extensive literature dealing with stochastic differential equations with discontinuous paths incurred by Lévy processes (for instance, see monographs [2,6,19] and references therein).”
No Arbitrage Without Semimartingales (2009, methodology): “Proof: By standard results (eg, Theorem 10 of Chapter II of [16]) we know that simple predictable processes are dense in bL in ucp, where ucp denotes uniform convergence in probability on compact time sets.”