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A Radically Elementary Theory of Itô Diffusions and Associated Partial Differential Equations

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2067)

Abstract

Let \({t}_{0} \in \mathbf{T} \setminus \{ 1\}\) and \(a,b : \mathbf{R} \times [0,1] \rightarrow \mathbf{R}\).

Keywords

  • Classical Ordinary Differential Equation
  • Infinitesimal Approximation
  • Infinitesimal Space
  • Diffusion Coefficient Function
  • Second-order Taylor Expansion

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 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. 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

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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

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