# Tag Archives: JCM_math545_HW1_S23

## 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.

## Transforming Brownian Motion

Let $$W(t)$$ be standard Brownian motion on $$[0,\infty)$$.

1.  Show that $$-W(t)$$ is also a Brownian motion.
2. For any $$c>0$$, show that $$Y(t)= c W(t/c^2)$$ is again a standard Brownian motion.
3. Fix any $$s>0$$ and define $$Z(t)=W(t+s)-W(s)$$. Show that $$Z(t)$$ is a standard Brownian motion.
4. * Define $$X(t)= t W(1/t)$$ for $$t>0$$ and $$X(0)=0$$. Show that $$X(t)$$ is a standard Brownian Motion. Do this by arguing that $$X(t)$$  is continuous almost surely, that for each $$t \geq 0$$ it is a Gaussian random variable with mean zero and variance $$t$$. Instead of continuity, one can rather show that $$\text{Cov}(t,s)=\mathbf E X(t) X(s)$$ equals $$\min(t,s)$$. To prove continuity, notice that
$\lim_{t \rightarrow 0+} t W(1/t) = \lim_{s \rightarrow \infty} \frac{W(s)}{s}$

## Martingale Brownian Squared

Let $$W_t$$ be standard Brownian Motion.

1. Find a function  $$f(t)$$ so that $$W_t^2 -f(t)$$ is a Martingale.
2. * Argue that in some sense this $$f(t)$$is unique among increasing functions with finite variation. Compare this with the problem here.

## Calculating with Brownian Motion

Let $$W_t$$ be a standard brownian motion. Fixing an integer $$n$$ and a terminal time $$T >0$$, let $$\{t_i\}_{i=1}^n$$ be a partition of the interval $$[0,T]$$ with

$0=t_0 < t_1< \cdots< t_{n-1} < t_n=T$

Calculate the following two expressions:

1. $\mathbf{E} \Big(\sum_{k=1}^n W_{t_k} \big[ W_{t_{k}} – W_{t_{k-1}} \big] \Big)$
Hint: you might want to do the second part of the problem first and then return to this question and write
$W_{t_k} \big[ W_{t_{k}} – W_{t_{k-1}} \big]= W_{t_{k-1}} \big[ W_{t_{k}} – W_{t_{k-1}} \big]+ \big[W_{t_{k}} -W_{t_{k-1}}\big]\big[ W_{t_{k}} – W_{t_{k-1}}\big]$
2. $\mathbf{E} \Big(\sum_{k=1}^n W_{t_{k-1}} \big[ W_{t_{k}} – W_{t_{k-1}} \big] \Big)$