- \(\mathbf{B}_t \) is continuous in time and \(\mathbf{B}_0=0\).
- \(\mathbf{B}_t-\mathbf{B}_s\) has a normal distribution with a mean of zero and a variance of \(t-s\) for \(0 \leq s < t \).
- \(\mathbf{B}_t \) is a time homogeneous process with the increment \(\mathbf{B}_{t+\delta t}-\mathbf{B}_{t} \) is independent of \(t\) and follows the distribution of \(\mathbf{B}_{\delta t}\)
Exercise Let \(\mathbf{B}_t\) be a standard Brownian motion, show that the \(Cov(\mathbf{B}_t,\mathbf{B}_s)=min(t,s)\).
Revision on stochastic integration
A brief introduction is given on some properties of stochastic integrations.
Ito isometry lemma.
If \(f(t,\mathbf{X}_t)\) is a bounded function such that \(\mathrm{E}[\int_0^T f^2(t,\mathbf{X}_t) ~\mathrm{d}t ]<\infty\), then $$ \mathrm{E}\left[ \left(\int_0^T f(t,\mathbf{X}_t) ~\mathrm{d} \mathbf{B}_t\right)^2\right]=\mathrm{E} \left[\int_0^T f^2(t,\mathbf{X}_t) ~\mathrm{d}t \right]. $$ For the function defined above, \(f(t,\mathbf{X}_t)\), and a standard Brownian motion \(\mathbf{B}_t\) defined over the probability space \( (\Omega, \mathcal{F}, \mathbf{P}) \), we transform the following stochastic integral into a probabilistic representation $$ I=\int_0^T f(t,\mathbf{X}_t)~ \mathrm{d} \mathbf{B}_t. $$ \(I\) is considered as a random variable that is normally distributed with a mean of \(\mathrm{E}(I)=0\) and a variance of $$ \mathrm{Var} (I) = \mathrm{E}(I^2)- [\mathrm{E}(I)]^2 = \mathrm{E} \left[ \left( \int_0^T f(t,\mathbf{X}_t) \mathrm{d} \mathbf{B}_t \right)^2 \right] $$ $$ =\int_0^T f^2(t,\mathbf{X}_t) \mathrm{d} t. $$ The above was derived using Ito isometry lemma and we conclude that the stochastic integral, \(I\), follows a normal distribution with the above mean and variance. i.e. $$ I\sim N\left(0,\int_0^T f^2(t,\mathbf{X}_t) \mathrm{d} t\right). $$ Definition Let \(\mathbf{B}_t\) be a Brownian motion on \((\Omega, \mathcal{F}, \mathbf{P})\), then \(\mathbf{X}_t\) is a stochastic process on the probability space \((\Omega, \mathcal{F}, \mathbf{P})\) that has the form $$ \mathrm{d} \mathbf{X}_t=b(t,\mathbf{X}_t) \mathrm{d} t + v(t,\mathbf{X}_t) \mathrm{d} \mathbf{B}_t, $$ such that $$ \mathbf{P} \left[\int_0^t v^2(s,\mathbf{X}_s)~ \mathrm{d} s < \infty , ~\forall~ t \geq 0 \right]=1 , ~~~~ \mathbf{P} \left[\int_0^t |b(s,\mathbf{X}_s)| \mathrm{d} s < \infty , ~\forall t \geq 0 \right]=1. $$ Heuristic rules: The following heuristic rules are very important in stochastic differentials $$ (\mathrm{d} t)^2=0 , ~~\mathrm{d} t \cdot \mathrm{d} \mathbf{B}_t =0 ,~~ (\mathrm{d} \mathbf{B}_t)^2=\mathrm{d} t, $$ and are often used with Ito's lemma to solve SDEs.
References
- Glasserman P., 2004,
Monte Carlo Methods in Financial Engineering
, Springer. - Oksendal B.,
Stochastic Differential Equations: An Introduction With Applications
, 5th ed. Springer, 2000.
No comments:
Post a Comment