# Category Archives: Numerical Examples

## 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.

## Simple Numerical Exercise

Let $$\omega_i$$ , $$i=1,\cdots$$ be a collection of mutually independent, uniform on $$[0,1]$$ random variables. Define

$\eta_i(\omega)= \omega_i -\frac12$

and

$X_n(\omega) = \sum_{i=1}^n \eta_i(\omega)\,.$

1. What is $$\mathbf{E}\,X_n$$ ?
2. What is $$\mathrm{Var}(X_n)$$ ?
3. What is $$\mathbf{E}\,X_{n+k} | X_n$$ for $$n, k >0$$ ?
4. What is $$\mathbf{E}(\,X_5^2 \,|\, X_3)$$ ?
5. [optional] Write a computer program to simulate some realizations of this process viewing $$n$$ as time. Plot some plots of $$n$$ vs $$X_n$$.
6. [optional] How do you simulations agree with the first two parts ?