Introduction : In this Euler Mayurma Method is applied to solve stochastic differential equation dx=(5-x)/(7-t)dt+dWt
Question : Using Euler Mayurma Method to solve SDE
\[dx=\frac{5-x}{7-t}dt+dW_t\]
subject to initial condition
[ x\left(0\right)=1]Solution :
\[dx=\frac{5-x}{7-t}dt+dW_t\]
Subject to \[ x\left(0\right)=1\]
So
\[t_0=0 , X_0=1\]
Euler Mayurma Method
Step 1:
\[X_1=X_0+\left(\frac{5-X_0}{7-t_0}\right)∆t+W1-W0\]
Using time-interval ∆t=0.1 and random numbers for Weiner Process
\[W_0=0,\ W_1=\frac{1}{2}\]
\[X_1=1+\left(\frac{5-1}{7-0}\right)0.1+\left(1-0\right)=2.057\]
Step 2:
\[X_2=X_1+\left(\frac{5-X_1}{7-t_1}\right)∆t+W2-W1\]
Now
\[t_1=t_0+∆t=0.1\]
and random numbers
\[W_1=\frac{1}{3},\ W_2=\frac{1}{2}\]
for Weiner Process
\[X_1=2.057+\left(\frac{5-2.057}{7-0.1}\right)0.1+\left(\frac{1}{2}-\frac{1}{2}\right)=2.0996\]