Solution of Stochastic Differential Equations in Finance

Fig. 1. Solution to the Black Scholes stochastic differential equation (4). The exact solution (5) is plotted as a black curve. The Euler-Maruyama approxima- tion with time step ∆t = 0.2 is plotted as circles. The drift and diffusion parameters are set to µ = 0.75 and σ = 0.30, respectively. Shown in grey is the actual stock price series, from which µ and σ were inferred.
As another example, consider the Langevin equation


dX(t) = −µX(t) dt + σ dW_t (11)
where µ and σ are positive constants. In this case, it is not possible to ana- lytically derive the solution to this equation in terms of simple processes. The solution of the Langevin equation is a stochastic process called the Ornstein- Uhlenbeck process. Fig. 2 shows one realization of the approximate solution.
It was generated from an Euler-Maruyama approximation, using the steps


w₀ = X₀ (12)
w_i+1 = w_i − µw_i∆t_i + σ∆W_i
for i = 1, . . . , n. This stochastic differential equation is used to model systems that tend to revert to a particular state, in this case the state X = 0, in the presence of a noisy background. Interest-rate models, in particular, often contain mean-reversion assumptions.

3. Strong convergence of SDE solvers.

The definition of convergence is similar to the concept for ordinary differential equation solvers, aside from the differences caused by the fact that a solution

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