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## Getting your feet wet numerically

Simulate the following stochastic differential equations:

• $dX(t) = – \lambda X(t) dt + dW(t)$
• $dY(t) = – \lambda Y(t) dt +Y(t) dW(t)$

by using the following Euler type numerical approximation

• $X_{n+1} = X_n – \lambda X_n h + \sqrt{h} \eta_n$
• $Y_{n+1} = Y_n – \lambda Y_n h + \sqrt{h} Y_n\eta_n$

where $$n=0,1,2,\dots$$ and $$h >0$$ is a small number that give the numerical step side.  That is to say that we consider $$X_n$$ as an approximation of $$X( t)$$ and $$Y_n$$ as an approximation of $$Y( t)$$ each with $$t=h n$$.  Here $$\eta_n$$ are a collection of mutually independent random variables each with a Gaussian distribution with mean zero and variance one. (That is $$N(0,1)$$.)

Write code to simulate the two equations using the numerically methods suggested.  Plot some trajectories. Describe how the behavior changes for different choices of $$\lambda$$. Can you conjecture where it changes ? Compare and contrast the behavior of the two equations.