Stochastic Differential Equations (SDE)

Definition A standard Brownian motion $\mathbf{B}_t$ is a stochastic process and is often called Wiener process with the following properties

• $\mathbf{B}_t$ 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.

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References

1. Glasserman P., 2004, Monte Carlo Methods in Financial Engineering, Springer.
2. Oksendal B., Stochastic Differential Equations: An Introduction With Applications, 5th ed. Springer, 2000.