Brownian Motion and Stochastic Calculus

By Xiongzhi Chen. 2008

5,611 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/99e37b4b689339da03493d01a853c00afc67f63d

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Citations in the corpus

Robust control and model misspecification (2006, background): “See Karatzas and Shreve [30], pages 4 and 5 for a discussion.”
Portfolios and risk premia for the long run. (2012, neutral): “Under Assumption 1 for f ∈ C20 (Rn × E), f(Xt)− f(X0)− ∫ t 0 (Lf)(Xu)du is a martingale also with respect to Ft; cf. Karatzas and Shreve (1991), Section 5.4.”
Existence and Uniqueness of Solutions to Fokker–Planck Type Equations with Irregular Coefficients (2008, neutral): “We refer to the excellent textbooks [11, 12, 20, 21, 23] for all the very important mathematical details of all the aspects adressed here.”
First and Second Order Necessary Conditions for Stochastic Optimal Control Problems (2012, background): “g [17]) we have the existence of a constant K > 0 such that for all i = 1, .”
Linear−quadratic optimal control and nonzero‐sum differential game of forward−backward stochastic system (2012, methodology): “By taking the expectation, and using Gronwall’s inequality, Burkholder– Davis–Gundy’s inequality (see Karatzas and Shreve [23]) to (7), we obtain E sup t∈[0,T ] |X̂t |2 ≤C { |â|2+E ∫ T 0 (|Bt P̂t +Ct K̂t +Dt Q̂t |2+|̂0t |2+|̂ 0 t |2)dt } .”
NONAUTONOMOUS KOLMOGOROV PARABOLIC EQUATIONS WITH UNBOUNDED COEFFICIENTS (2008, background): “In the case of Lipschitz continuous coefficients, there are many results stating conditions on µ and σ such that (1.1) is well posed. See, e.g., [ 12 , 13, 14].”
Properties of Realized Variance Under Alternative Sampling Schemes (2006, background): “Second, the equivalence between the diffusion process and the limiting case of the pure jump process does not rely on the assumed normality of ε, but holds for any iid random variable with finite variance (due to Donsker’s theorem; see, e.g., Karatzas and Shreve 1991, chap.”
Optimal Asset Allocation and Risk Shifting in Money Management (2007, methodology): “, Karatzas and Shreve (1988)] to a non-concave problem, has not been previously used in the literature.”
Quasi-stationary distributions and population processes (2011, background): “By the representation theorem proved in [38] Theorem III-4.”
Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries (2006, background): “Furthermore, for every t ≥ 0, Yt = Y0 + W̃s(t) and by (21) we have Xt = e Yt + α = e (”
Hölder regularity for operator scaling stable random fields (2007, neutral): “As in the proof of the Kolmogorov–Centsov Theorem (see [28]), we exhibit (xk)k a countable dense sequence of elements of K and a finite positive constant C such that for τE (xk − xk′) small enough, |Yα(xk)− Yα(xk′)| ≤ CτE (xk − xk′) |log τE (xk − xk′)| 1/α+1/2+ε almost surely.”