Stochastic process: $X(t), t\in[0,\infty)$

Brownian motion $B(t)$:

  • Continuous
  • Independent increments
  • Normal distribution of increments, i.e. $Pr[B(s+t)-B(s)\in A]=\int_A \frac{1}{\sqrt{2\pi t}}e^{-x^2/2t} dt$ for $s,t>0$

Symmetric random walk (SRW):

  • $X_i \in {-1,1}$ with probability $p$ and $q$ respectively
  • SRW is $M_n = \sum_{i=1}^n X_i$ and $M_0 = 0$
  • Scaled SRW: $B^{(n)}(k/n) = \frac{1}{\sqrt{n}} M_k$
  • Limit: Let $t=k/n$, then $B(t)=\lim_{n\to\infty} B^{(n)}(k/n)$

Covariance of BM, assumed $s<t$,

Therefore, we have

Geometric BM:

First passage time problem:

  • Let $M_t = \max_{0<s<t} X_s$ be the running maximum of stochastic process $X_t$
  • Let $T_a = \inf{t>0: X_t = a}$ be the first passage time
  • $T_a < t$ is equivalent to $M_t > a$

Reflection principle of Brownian motion $B_t$:

To obtain the p.d.f. of $T_a$, we can consider the m.g.f. $E[\exp(uT_a)]$.

  • Consider the trail of a Brownian motion $B_t$ that $B_0=0$, $B_s=a$, $B_t=b$ for $s<t$, $a>b$, then it has the same probability as $B’_t$ that $B’_0=0$, $B’_s=a$, $B’_t=2a-b$, just as the path of the B.M. reflected about $a$.
  • $\Pr[T_a < t, u < B_t < v] = \Pr[2a-v < B_t < 2a-u]$
  • $\Pr[T_a < t, B_t = u] = \Pr[B_t = 2a-u] = \Pr[M_t > a, B_t = u]$
  • $\Pr[M_t = a, B_t = u] = \frac{d}{da}\Pr[M_t > a, B_t = u]$

Stochastic integral: $\int_s^t f(\tau,w)dX_{\tau}(w)$

Ito integral: $I(f) = \int_0^t f(s,w)dB_s$

  • $f$ is a $\mathcal{F}_t$-adapted, square-integrable stochastic process
  • It is a martingale
  • $E[I^2(f)] = E[\int_0^t f^2(s,w)ds]$

Ito formula:

Example: Solving $\int_0^t B_s dB_s$

Ito formula in general form:

where $dX_t = \mu dt + \nu dB_t$, and $(dX_t)^2 = \nu^2 dt$

Multidimensional version: