Some usefull definitions for stochastic processes

Bayes’s Rule

$\begin{align*} P[A_j|B] = \frac{P[B|A_j]P[A_j]}{P[B]}\\ \\ P[B]=P[B|A_1]P[A_1]+ \ldots + P[B|A_n]P[A_n]\\ \\ f_y (y| X = x) = \frac{f_x (x | Y = y) f_y (y) }{f_x (x)} \\ \text{for random variable} \end{align*}$

Mean – Variance and r-moment about the mean

$\begin{align*} \mu = \int_{-\infty}^{+\infty} x f(x) dx\\ \sigma^2 = \int_{-\infty}^{+\infty} (x - \mu)^2 f(x) dx\\ \mu_r = \int_{-\infty}^{+\infty} (x - \mu)^r f(x) dx \\ \\ \mu_{kr} = E\{(X - \mu_x)^k (Y - \mu_y)^r \} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (x - \mu_x)^k (y - \mu_y)^r f(x,y) dx dy \\ \\ \mu^\prime _ {kr}= E\{ X^k Y^r\} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x ^k y^r f(x,y) dx dy \\ \end{align*}$

Laplace transform of pdf

$\begin{align*} f^L_s = \int_0 ^{\infty} e^{-sx} f(x) dx \end{align*}$

Information content

$\begin{align*} I = - \int_{-\infty}^{\infty} f(x) \log_2(f(x)) dx \end{align*}$

Conditional distribution

$\begin{align*} F_x (x|M) = P\{X \leq x | M \} = \frac{P\{ X \leq x , M \}}{P\{ M \}} \end{align*}$

Characteristic function

$\begin{align*} \Phi (\omega) = E\{ e^{j\omega X} \} = \int_{-\infty}^{\infty} e^{j \omega x} f(x) dx \\ \text{for continuous r.v.} \\ \Phi (\omega) = \sum_k e^{j \omega x_k} P\{ X = x_k \} \ \ \ \text{for discrete r.v} \\ \Phi(0) = 1, |\Phi (\omega)| < 1 \end{align*}$

Independent random variables

note:

$\begin{align*} F_{xy} (x, y) = F_x (x) F_y (y) \\ f_{xy} (x,y) = f(x) f(y) \\ f_y (y|x) = f_y (y) \\ f_x (x|y) = f_x (x) \\ \\ \\ E\{g(X) h(X)\} = E\{ g(X)\} E\{h(X)\} \\ \text{(If X, Y are independent, their functions are independent too)} \end{align*}$

Expected value

$\begin{align*} E\{ X \} = \int_{-\infty} ^{\infty} x f(x) dx \ \ \ \ \text{continuous r.v.} \\ \\ E\{ X \} = \sum_n x_n P \{ X = x_n\} = \sum_n x_n p_n \ \ \ \text{discrete r.v. } \\ \\ E\{ Y = g(X) \} = \int_{-\infty} ^{\infty} y f_Y (y) dy = \int_{-\infty} ^{\infty} g(x) f_X (x) dx \ \ \ \ \\\text{for function g(X)} \\ \\ E\{ g(X) | Y = y \} = \int_{-\infty} ^{\infty} g(x) f_x (x| Y = y) dx \\ = \frac{\int_{-\infty}^\infty g(x) f_{xy} (x,y) dx}{ \int_{-\infty}^\infty f_{xy} (x,y) dy} \ \ \ \text{conditional expected value} \\ \ \\ \\ E\{ g(X,Y |M)\} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x,y) f(x,y|M) dx dy \end{align*}$

Jointly Normal Distribution density function

$\begin{align*} f(x,y) = \frac{1}{2 \pi \sigma_1 \sigma_2 \sqrt{1-r^2}} \text{exp} \left[-\frac{1}{2(1-r^2)} \left[ \frac{(x-\mu_1)^2}{\sigma_1^2} - \frac{2r(x - \mu_1)(y - \mu_2)}{\sigma_1 \sigma_2} + \frac{(y-\mu_2)^2}{\sigma_2^2} \right] \right] \\ \ \\ \\ |r| < 1 \ \ \ \ \text{correlation coefficient} \end{align*}$

Orthogonal r.v.’s

$\begin{align*} E\{XY\} = 0 \end{align*}$

Correlation coefficient

$\begin{align*} r = \frac{E\{ (X - \mu_x) (Y - \mu_y)\}}{ \sqrt{E\{(X - \mu_x)^2\} E\{(Y - \mu_y)^2\} }} = \frac{\mu_{11} ^\prime }{\sigma_x \sigma_y} \end{align*}$

Normal random variables

$\begin{align*} E\{ (X - \mu_x) (Y - \mu_y)\} = r_{xy} \sigma_x \sigma_y\\ \\ \text{When X, Y are jointly normal with zero mean}\\ E\{ (aX + bY)^2 \} = a^2 \sigma_x ^2 + b^2 \sigma_y ^2 + 2ab r_{xy} \sigma_x \sigma_y \\ \\ E\{ (aX + bY) (cX + dY)\} = ac \sigma_x ^2 + bd \sigma_y ^2 + (ad +bc) r_{xy} \sigma_x \sigma_y \end{align*}$

Conditional densities

$\begin{align*} f(x_1, ..., x_k | x_{k+1} , ..., x_n) = \frac{ f(x_1,...,x_n)}{ f(x_{k+1}, ... ,x_n)} \end{align*}$

Chain rule for sequence of random variables

$\begin{align*} f(x_1,...,x_n) = f(x_n|,x_{n-1},...,x_1)...f(x_2|x_1)f(x_1) \end{align*}$

Sample mean

Note that the variance of sample mean is n-time smaller than the one of a single sample.

$\begin{align*} &\overline X = (X_1 + \cdots + X_n) \slash n \\ \\ &\text{Density function of the sample mean}\\ &\text{(zero-mean, Gaussian case)}\\ &f_{\overline x}(x) = \frac{1}{\sqrt{2\pi \sigma^2 \slash n}} e^{-nx^2 \slash 2 \sigma^2} \\ \end{align*}$

Sample variance

note:Sample variance

$\begin{align*} \overline S = [(X_1 - \overline X)^2 + \cdots + (X_n - \overline X)^2] \slash n = \sum_{i=1}^n \frac{X_i ^2}{n} - \overline X^2 \end{align*}$

Random process X(t) and basis

note:Random process X(t)

$\begin{align*} &\text{Random process}\\ &X(t) = X(t,\zeta)\\ \\ &\text{First order distribution function}\\ &F(x;t) = P\{ X(t) \le x \}\\ \\ &\text{n-order distribution function}\\ &F(x_1,\cdots,x_n; t_1,\cdots,t_n) = P\{ X(t_1) \le x_1, \cdots , X(t_n) \le x_n\} \\ \\ &\text{ n-order Density function or pdf}\\ &f(x_1,\cdots,x_n; t_1,\cdots, t_n) = \frac{ \partial F(x_1,\cdots, x_n ; t_1,\cdots,t_n)}{ \partial x_1, \cdots, \partial x_n}\\ \\ &\text{Autocorrelation}\\ &R(t_1,t_2) = E\{ X(t_1) X(t_2)\} = \int_{-\infty}^{\infty} x_1 x_2 f(x_1,x_2;t_1,t_2)dx_1 dx_2\\ \\ &\text{Autocovariance} \\ &C(t_1,t_2) = E\{[X(t_1) - \mu(t_1) ] [X(t_2) - \mu(t_2)]\} = R(t_1,t_2) - \mu(t_1) \mu(t_2) \end{align*}$

Simulating the Schwartz type 1 stochastic process

The Schwartz type 1 model is a log price Ornstein-Uhlenbeck stochastic process. Monte Carlo simulation of the model can be done using the equation above. The above equation is an exact solution of the model, this means that the distribution of the simulation is exact, and that time steps can be any size.

$\begin{align*} \ln S_t \sim&\; e^{-\kappa t}\ln S_0+\left(1-e^{-\kappa t}\right)\left(\mu-\frac{\sigma^2}{2\kappa}\right)\\ &+\sigma \sqrt{\frac{1-e^{-2\kappa t}}{2\kappa}}N_{0,1} \end{align*}$

Calibrating the Schwartz type 1 model

The Schwartz type 1 model is a log price Ornstein-Uhlenbeck stochastic process. The calibration can be done through a regression of the logprices as described in the above equation.

$\begin{align*} &\text{the Schwartz type 1 model is defined as:}\\ &dS_t =\; \kappa (\mu-\ln S_t) S_t dt + \sigma S_tdW \\ \\ &\text{Calibration can be done through a linear }\\ &\text{regression which recover a,b,c from:}\\ &\ln S_{t+\delta t}=a\ln S_t+b+cN_{0,1}\\ \\ &\text{The model parameters are then given by:}\\ &\kappa = -\frac{\ln a}{\delta t}\\ &\sigma = c \sqrt{\frac{2\kappa}{1-e^{-2\kappa \delta t}}}\\ & \mu = \frac{b}{1-e^{-\kappa \delta}}+\frac{\sigma^2}{2\kappa} \end{align*}$

Complex process