Continuous¶
Challenge¶
How to describe probability distribution
Brownian Motion/Wiener Process¶
Basically a continuous version of simple RW: ‘Limit’ of simple RW
Denoted using \(y_t = B(t)\)
Properties¶
- Always starts at 0: \(P( y_0 = 0 ) = 1\)
- Stationary \(\forall s \in [0, t)\)
- \(y_t - y_s \sim N(0, t-s)\)
- where \((t-s)\) is the length of the interval
- Independent increment: If intervals \([s_i, t_i]\) are non-overlapping, then \(y_{t_i} - y_{s_i}\) are independent
Characteristics¶
- Cross the independent axis indefinitely-often
- Does not deviate too much from \(y_t = \sqrt{t}\)
- Not differentiable
- Standard calculus cannot be applied
- Requires Ito calculus
- Max series
- Quadratic variation
$$ t = \frac{i}{n} T \ \implies \lim_{n \to \infty} \sum_{i=1}^n (y_{t} - y_{t-1})^2 = T \ \forall T > 0 \
\implies (dB)^2 = dt $$
Implications¶
This is because \(\dfrac{d B_t}{dt}\) is not defined since \(B_t\) is not differentiable
Ito’s Lemma¶
Consider \(y_t = f(B_t)\), where \(f\) is a smooth function $$ \begin{aligned} y_t &= f(B_t) \ \implies df & \ne f'(B_t) \cdot d B_t \quad [\because (dB)^2 = dt] \ \implies df &= f'(B_t) \cdot d B_t + \dfrac{1}{2} f''(B_t) \cdot dt \end{aligned} $$
IDK¶
Assuming \(\mu, \sigma\) are constant $$ \begin{aligned} dy_t &= \underbrace{\mu dt}_\text{Drift} + \sigma d B_t \ \implies y_t &= \mu t + \sigma B_t \end{aligned} $$ Using Ito’s Lemma (Basically Taylor’s expansion) $$ d f(t, x) = \dfrac{\partial f}{\partial t} + \mu \dfrac{\partial f}{\partial x} + \dfrac{1}{2} \sigma^2 \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial f}{\partial x} d B_t $$
Integration¶
Ito integral is the limit of Riemanian sums when we always take leftmost point of each integral
Intuitively, it only uses the data you have seen so far
Adapted Process¶
A strategy/decision \(D_t\) is said to be adapted to \(y_t\), if \(D_t\) only depends on \(y_s, s \le t, \forall t\)
If \(D_t\) only depends on \(t\) and not on \(B_t\), then \(y_t = \int D_t \cdot d B_t\) is normally-distributed at all times
Ito Isometry¶
Used to calculate variance of Brownian motion $$ \begin{aligned} D_t &\text{ adapted to } B_t \ \implies V(B_t) &= E \left[ (\int_0^t D_s \cdot dB_s)^2 \right] \ &= E \left[ \int_0^t D^2_t \cdot ds \right] \end{aligned} $$ Due to quadratic variance
Martingale¶
If \(g(t, B_t)\) is adapted to \(B_t\) then \(\int g(t, B_t) \cdot dB_t\), as long as \(g\) is “reasonable”
\(g\) is reasonable if \(\int \int g^2 \cdot dt \cdot dB_t < \infty\)
If a stochastic differential equation does not have a drift term, then it is a martingale $$ d y_t = \sigma \cdot dB_t \qquad [\mu = 0] $$ Defining stock price as brownian motion, as it is a martingale process $$ \begin{aligned} S_t &= \exp(\frac{-\sigma^2 t}{2} + \sigma B_t) \ \implies \dfrac{d S_t}{S_t} &= \sigma \cdot d B_t \end{aligned} $$
Stochastic Differential Equation¶
Change of measure¶
Consider
- \(B\) is brownian process w/ drift and pdf \(P\)
- \(\tilde B\) is brownian process w/o drift and pdf \(\tilde P\)
\(z\) is called the Radon-Nikodym derivative