a stochastic calculus crash course for Financial Engineering.

Building on Kolmogorov’s axiomatic probability theory, define continuous time stochastic process, for time $\mathcal T = [0,\infty)$ a sample path is essentially $$ {X_t : t\ge 0} $$

Brownian Motion

History anecdote:

In 1828, British Botanist R. Brown observed the random movement exhibited by plant pollens suspended in water.

In 1905, A. Einstein proposed the first quantitative model of Brownian motion in terms of a Gaussian process, which explains some principles of molecular movements.

In 1923, N. Wiener developed rigorous theoretical system of Brownian motion and relevant functional structures. Wiener’s construction is basically equivalent with Einstein’s definition, and is quite close to our modern definition.

Formally, a stochastic process ${W_t: t\ge 0}$ is called Brownian motion or Wiener process, if satisfying

  • (almost WLOG) $W_0 =0$, and it’s continuous.

  • independent increments: for $0 = t_0 < t_1 < \cdots < t_m$, the increments $$ W_{t_i } - W_{t_i -1}, \forall i = 1, \ldots, m $$ are mutually independent.

  • stationary increments: for $0 \le s < t$: $$ W(t) - W(s) \sim \mathcal N(0, t - s)\ \text{ (normal distribution with \textbf{variance} $t - s$)} $$

properties

  • Non-differentiable everywhere: except for a null set (w.r.t. the probability measure), all sample paths of a Brownian motion $W_t$ are non-differentiable.

  • Unbounded variation: except for a null set, all sample paths of a Brownian motion $W_t$ are not of bounded variable. More specificallly, for any interval $[T_1, T_2]$ we’d have $$ \lim_{|\Pi|\to 0}\sum_{j = 1}^n|W_{t_j} - W_{t_{j-1}}| = +\infty $$ where $\Pi :={t_0, t_1, \ldots t_n} $ is a partition of $[T_1, T_2]$​ and $|\Pi|$ is its mesh: $$ |\Pi| :=\max_{j\in [n]}|t_j - t_{j-1}| $$

  • Bounded quadratic variation: similarly, for any interval $[T_1, T_2]$ of a Brownian motion $W_t$: $$ \lim_{|\Pi|\to 0}\sum_{j = 1}^n|W_{t_j} - W_{t_{j-1}}|^2 = T_2 - T_1 $$ Essentially, we can define quadratic variation process for any continuous random varaible of bounded quadratic variation ${X_t, t\ge 0}$: $$ \lang X, X\rang_t := \lim_{|\Pi|\to 0}\sum_{j = 1}^n|X_{t_j} - X_{t_{j-1}}|^2. $$ Specifically, for Brownian motion: $\lang W, W\rang_t = t$.

Because Brownian motion is not of unbounded variable, its sample path $W(t)$ is (almost) not (Riemann) integrable.

But mathematicians don’t stop from here. And now

Itô’s intergral

Recall that, for Riemann integral: $$ \int_{T_1}^{T_2}f(t),dt = \lim_{|\Pi|\to 0}\sum_{j = 1}^n f(s_j)(t_j - t_{j-1}), $$ $s_j$ can take any value in between $ [t_{j-1}, t_{j}]$.

As for Itô’s intergral, restricting $s_t = t_{j-1}$ for every partition in the definition can solve the integrability issue. Define Itô intergral of an $L^2$ process ${X_t}$: $$ I_X(T) = \int_0^T X(t) ,dW_t:=\lim_{|\Pi|\to 0}\sum_{j = 1}^n X(t_{j-1})(W_{t_j} - W_{t_{j-1}}) $$ And, the limit on the RHS exists, so that the integral is well-defined. This is a stochastic integral because, basically, every sample path $X$ is a realization of the random variable, say $X(\omega)$ for some $\omega\in \Omega$. So, $I_X(T)$ is a random variable as well.

Remark. $W$ is a Brownian motion but can be generalized to being a semimartingale and things will work as well.

properties | Itô’s Lemma

  • For a smooth function $f$, we have $$ df(X_t) = f’(X_t) dX_t + \frac12 f’’(X_t)d\lang X, X\rang_t $$

  • For bivariate smooth function $f$, the process $f(W_t, t)$ satisfies $$ df(X_t, t) = f_1(X_t, t)dX_t + f_2(X_t, t) dt+ \frac12 f_{11}(X_t, t), d\lang X, X\rang_t $$ Specifically, for Brownian motion $W_t$: $$ df(W_t, t) = f_1(W_t, t)dW_t + \left[f_2(W_t, t) + \frac12 f_{11}(W_t, t)\right],dt $$ because $\lang W, W\rang_t = t$, as we’ve solved before.