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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.
Tell your story with pictures.
Transforming Brownian Motion
Let \(W(t)\) be standard Brownian motion on \([0,\infty)\).
- Show that \(-W(t)\) is also a Brownian motion.
- For any \(c>0\), show that \(Y(t)= c W(t/c^2)\) is again a standard Brownian motion.
- Fix any \(s>0\) and define \(Z(t)=W(t+s)-W(s)\). Show that \(Z(t)\) is a standard Brownian motion.
- * 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.
- Find a function \(f(t)\) so that \(W_t^2 -f(t)\) is a Martingale.
- * 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:
- \[ \mathbf{E} \Big(\sum_{k=1}^n W_{t_k} \big[ W_{t_{k}} – W_{t_{k-1}} \big] \Big)\]
Hint: you might want to do the second part of the problem first and then return to this question and write
\[W_{t_k} \big[ W_{t_{k}} – W_{t_{k-1}} \big]= W_{t_{k-1}} \big[ W_{t_{k}} – W_{t_{k-1}} \big]+ \big[W_{t_{k}} -W_{t_{k-1}}\big]\big[ W_{t_{k}} – W_{t_{k-1}}\big]\] - \[ \mathbf{E} \Big(\sum_{k=1}^n W_{t_{k-1}} \big[ W_{t_{k}} – W_{t_{k-1}} \big] \Big)\]
