5. (SDEs with Analytical Solutions)

In some cases (we might get lucky) it is possible to find analytical solutions to SDEs

(Kloeden and Platen, 1995, pp. 118–126). Some examples of SDEs and their solutions are:

dX = 1 2

a2X dt + aX dW X(t) = X0 exp(aW(t)).

dX = aX dt + bX dW X(t) = X0 exp((a − 1 2 b2)t + bW(t)).
dX = 1 2 X dt + X dW X(t) = X0 exp(W(t)).
Here W(t) is a Wiener process.
Answer the following questions:

a) Integrate these equations into the framework and decide how to determine the difference between the explicit solution and the solution defined by a given finite difference scheme.

b) Experiment with the schemes by varying the number of time steps. Is there some way to determine what the order of convergence of a given scheme is?

c) We focus on the explicit Euler method. Produce a multi-curve graph in the style of Figure 14.5 in which each curve is a path produced by the Euler method with a given number of time steps, for example step sizes NT=10, 20, 50, 100, 200, 300. Do you see monotonic convergence to the ‘exact path’ as NT increases?