Stochastic Calculus and Financial Applications

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Introduction to the Economics and Mathematics of Financial Markets (2004, background): “We follow here a proof given in Steele (2000). We consider only the simplest case of Itô’s rule: ∫ t 1 ∫ t f ′′ f (W (t)) − f (0) = f ′(W (s)) dW (s) + 2 (W (s))ds (3.”
Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series (2013, background): “Steele [37]); however the major differences here are that no filtration is involved, Zω is a random element of L 2 ([0, 1],C), and the functions Aω are operator-valued.”
Pricing Black–Scholes options with correlated interest rate risk and credit risk: an extension (2005, background): “Many stochastic interest rate models, such as Vasicek (1977) and Heath et al.”
Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets (2007, unknown): “By the definition of the standard stochastic integral (see, for example, page 96 of Steele [88]) and Theorem B.”
Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations (2009, background): “Many basic models are introduced in [16] with an abundance of interpretations and insights.”
Portfolio Choice Problems (2010, background): “Mathematically more rigorous treatments of the material can be found in Karatzas and Shreve (1991) and Steele (2001). subject to the terminal condition V (0,Wt+τ , zt+τ ) = u(Wt+τ ).”