Abstract
Let \({t}_{0} \in \mathbf{T} \setminus \{ 1\}\) and \(a,b : \mathbf{R} \times [0,1] \rightarrow \mathbf{R}\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Indeed, by adding both \(0\,= - f\left (y,t\right )g\left (y,t +\mathrm{ d}t\right ) + f\left (y,t\right )g\left (y,t +\mathrm{ d}t\right )\) and \(0 = -\mathrm{d}f\left (y,t\right )g\left (y,t\right ) +\mathrm{ d}f\left (y,t\right )g\left (y,t\right )\) on each side of the equation, one obtains
$$\begin{array}{rcl} \mathrm{d}(fg)(y,t)& =& f\left (y,t +\mathrm{ d}t\right )g\left (y,t +\mathrm{ d}t\right ) - f\left (y,t\right )g\left (y,t\right ) \\ & =& f\left (y,t +\mathrm{ d}t\right )g\left (y,t +\mathrm{ d}t\right ) - f\left (y,t\right )g\left (y,t +\mathrm{ d}t\right ) \\ & & +f\left (y,t\right )g\left (y,t +\mathrm{ d}t\right ) - f\left (y,t\right )g\left (y,t\right ) \\ & =& \mathrm{d}f\left (y,t\right )g\left (y,t +\mathrm{ d}t\right ) + f\left (y,t\right )\mathrm{d}g\left (y,t +\mathrm{ d}t\right ) \\ & =& \mathrm{d}f\left (y,t\right )\mathrm{d}g\left (y,t\right ) +\mathrm{ d}f\left (y,t\right )g\left (y,t\right ) + f\left (y,t\right )\mathrm{d}g\left (y,t +\mathrm{ d}t\right ).\end{array}$$ - 2.
Indeed,
$$\begin{array}{rcl} \frac{\mathrm{d}\mathrm{{e}}^{-tV (x)}} {\mathrm{d}t} & =& \left (\mathrm{{e}}^{-tV (x)}\mathrm{{e}}^{-\mathrm{d}tV (x)} -\mathrm{ {e}}^{-tV (x)}\right )/\mathrm{d}t \\ & =& \mathrm{{e}}^{-tV (x)}\left ({\underbrace{\mathrm{{e}}^{-\mathrm{d}tV (x)} - 1}}_{ =-\mathrm{d}tV (X)+\mathcal{O}\left ({(\mathrm{d}t)}^{2}\right )}\right )/\mathrm{d}t \\ & & \\ & =& -\mathrm{{e}}^{-tV (x)}V (x).\end{array}$$
References
Benoît, E.: Random walks and stochastic differential equations. In: Diener, F., Diener, M. (eds.) Nonstandard Analysis In Practice. Universitext, pp. 71–90. Springer, Berlin (1995)
Berg, I.v.d.: On the relation between elementary partial difference equations and partial differential equations. Ann. Pure Appl. Logic 92(3), 235–265 (1998)
Berg, I.v.d.: Asymptotic solutions of nonlinear difference equations. Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6 17(3), 635–660 (2008)
Berg, I.v.d.: Asymptotics of families of solutions of nonlinear difference equations. Logic Anal. 1(2), 153–185 (2008)
Sari, T.: Petite histoire de la stroboscopie. In: Colloque Trajectorien à la Mémoire de Georges Reeb et Jean-Louis Callot (Strasbourg-Obernai, 1995), no. 1995/13 in Prépublications de l’Institut de Recherche Mathématique Avancée, pp. 5–15. Université Louis Pasteur, Strasbourg (1995)
Sari, T.: Stroboscopy and averaging. In: Colloque Trajectorien à la Mémoire de Georges Reeb et Jean-Louis Callot (Strasbourg-Obernai, 1995), no. 1995/13 in Prépublications de l’Institut de Recherche Mathématique Avancée, pp. 95–124. Université Louis Pasteur, Strasbourg (1995)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Herzberg, F.S. (2013). A Radically Elementary Theory of Itô Diffusions and Associated Partial Differential Equations. In: Stochastic Calculus with Infinitesimals. Lecture Notes in Mathematics, vol 2067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33149-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-33149-7_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33148-0
Online ISBN: 978-3-642-33149-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)