Consider the (operatorname{ODE} partial_{t} xi(t, x)=x+b(xi(t, x)), xi(0, x)=x), where (b in mathcal{C}_{b}^{2}left(mathbb{R}^{d}, mathbb{R}^{d}ight)) such that (left|partial_{x}^{alpha}
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Consider the \(\operatorname{ODE} \partial_{t} \xi(t, x)=x+b(\xi(t, x)), \xi(0, x)=x\), where \(b \in \mathcal{C}_{b}^{2}\left(\mathbb{R}^{d}, \mathbb{R}^{d}ight)\) such that \(\left|\partial_{x}^{\alpha} b(x)ight|,|\alpha|=2\), is Lipschitz continuous. Show that all \(\partial_{x}^{\alpha} \xi(t, x),|\alpha| \leqslant 2\), are polynomially bounded.
Assume that the derivatives \(\partial_{x}^{\alpha} \xi(t, x)\) exist and solve the formally differentiated ODE.
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Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook
ISBN: 9783110741254
3rd Edition
Authors: René L. Schilling, Björn Böttcher
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