Stochastic differential equation

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A stoachastic differential equation (SDE) is a paritcular type of differential equation which includes at least one stochastic term. SDEs are particularly useful in financial mathematics for modelling the random evolution of underlying market prices, volatility term structures, interest rate term structures and other random financial variables. A particularly common example of an SDE is one which models some underlying price as geometric Brownian motion (GBM):

\[dS = \mu S\,dt + \sigma S\, dW\]

This SDE is the basis of the Black-Scholes model. Here \(\mu S\, dt\) is the deterministic part and \(\sigma S\,dW\) is the stochastic part, with \(dW\) being an instantaneous random increment from the normal distribution. The letter X is also used sometimes i.e. \(dX\) but the meaning is essentially the same in most cases. The letter W denotes Weiner, after Norbert Weiner, who formalised the mathematics behind Brownian motion.

The study and solution of SDE's requires the application of Stochastic Calculus with Ito's Lemma lying at the centre of that field.

For more information see:

  • Bernt K Øksendal, Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer.
  • Paul Wilmott, Paul Wilmott on Quantitative Finance, Volume I, Ch. 4.
  • Steven E Shreve, Stochastic Calculus for Finance, Vols I & II.