The Diffusion(X) calling sequence computes the diffusion term of an Ito process X. That is, given a process $X\left(t\right)$ governed by the stochastic differential equation (SDE)

The parameter X can be either a stochastic process or an expression involving stochastic variables. In the first case a Maple procedure is applied for computing the diffusion term. This procedure will accept two parameters: the value of the state variable and the time, and return the corresponding value of the diffusion. In the second case, Ito's lemma will be applied to calculate the diffusion term of X. Note that the Diffusion command will perform formal computations; the validity of these computations for a given function f will not be verified.

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

$X≔\mathrm{OrnsteinUhlenbeckProcess}\left(0\,\mathrm{\theta }\,\mathrm{\kappa }\,\mathrm{\sigma }\right)$

$X≔\mathrm{\_X0}$

$\mathrm{Diffusion}\left(X\right)$

$\left(x\,t\right)\mapsto \mathrm{\sigma }$

$\mathrm{Diffusion}\left(X\left(t\right)\right)$

$\mathrm{\sigma }$

$W≔\mathrm{WienerProcess}\left(\right)$

$W≔\mathrm{\_W}$

$\mathrm{Diffusion}\left(at+bW\left(t\right)\right)$

$b$

$X≔t\mapsto \exp \left(a\cdot t+b\cdot W\left(t\right)\right)$

$X≔t\mapsto {\ⅇ}^{a\cdot t+b\cdot W\left(t\right)}$

$\mathrm{simplify}\left(\frac{\mathrm{Diffusion}\left(X\left(t\right)\right)}{X\left(t\right)}\right)$

$b$

$U≔\mathrm{WienerProcess}\left(\right)$

$U≔\mathrm{\_W0}$

$Y≔t\mapsto \exp \left(a\cdot t+b\cdot W\left(t\right)+c\cdot U\left(t\right)\right)$

$Y≔t\mapsto {\ⅇ}^{a\cdot t+b\cdot W\left(t\right)+c\cdot U\left(t\right)}$

$\mathrm{simplify}\left(\frac{\mathrm{Diffusion}\left(Y\left(t\right)\right)}{Y\left(t\right)}\right)$

$\left[\begin{array}{cc}b& c\end{array}\right]$

$\mathrm{Diffusion}\left(⟨X\left(t\right)\,Y\left(t\right)⟩\right)$

$\left[\begin{array}{cc}b{\ⅇ}^{at+b\mathrm{\_W}\left(t\right)}& 0\\ b{\ⅇ}^{at+b\mathrm{\_W}\left(t\right)+c\mathrm{\_W0}\left(t\right)}& c{\ⅇ}^{at+b\mathrm{\_W}\left(t\right)+c\mathrm{\_W0}\left(t\right)}\end{array}\right]$

$\mathrm{\Sigma }≔⟨⟨1.0|0.5⟩\,⟨0.5|1.0⟩⟩$

$\mathrm{\Sigma }≔\left[\begin{array}{cc}1.0& 0.5\\ 0.5& 1.0\end{array}\right]$

$V≔\mathrm{WienerProcess}\left(\mathrm{\Sigma }\right)$

$V≔\mathrm{\_W1}$

$Z≔t\mapsto \exp \left(a\cdot t+b\cdot V\left(t\right)\left[1\right]+c\cdot V\left(t\right)\left[2\right]\right)$

$Z≔t\mapsto {\ⅇ}^{a\cdot t+b\cdot {V\left(t\right)}_1+c\cdot {V\left(t\right)}_2}$

$\mathrm{simplify}\left(\frac{\mathrm{Diffusion}\left(Z\left(t\right)\right)}{Z\left(t\right)}\right)$

$\left[\begin{array}{cc}b+0.500000000000000c& 0.500000000000000b+c\end{array}\right]$

Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.

Kloeden, P., and Platen, E., Numerical Solution of Stochastic Differential Equations, New York: Springer-Verlag, 1999.

The Finance[Diffusion] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.