Solution of Stochastic Differential Equations in Finance

Fig. 2. Solution to Langevin equation (11). The upper path is the solution approximation for parameters µ = 10, σ = 1, computed by the Euler-Maruyama method.
to an SDE is a stochastic process, and each computed trajectory is only one realization of that process. Each computed solution path w(t), using Euler- Maruyama for example, gives a random value at T , so that w(T ) is a random variable as well. The difference between the values at time T , e(T ) = X(T ) − w(T ), is therefore a random variable.
A discrete-time approximation is said to converge strongly to the solution
X(t) at time T if

lim
∆t→0
E{|X(T ) − w_∆t(T )|} = 0

where w_∆t is the approximate solution computed with constant stepsize ∆t, and E denotes expected value. For strongly convergent approximations, we further quantify the rate of convergence by the concept of order. An SDE solver converges strongly with order m if the expected value of the error is of mth order in the stepsize, i.e. if for any time T ,
E{|X(T ) − w_∆t(T )|} = O((∆t)^m)
for sufficiently small stepsize ∆t. This definition generalizes the standard con- vergence criterion for ordinary differential equations, reducing to the usual definition when the stochastic part of the equation goes to zero.
Although the Euler method for ordinary differential equations has order 1, the strong order for the Euler-Maruyama method for stochastic differential equations is 1/2. This fact was proved in Gikhman and Skorokhod (1972), under appropriate conditions on the functions a and b in (6).
In order to build a strong order 1 method for SDEs, another term in the “stochastic Taylor series” must be added to the method. Consider the

stochastic differential equation
⁽ dX(t) = a(X, t)dt + b(X, t)dW_t X(0) = X₀.

(13)