1 Random Process

Consider a probability space \((\Omega,\mathcal{F},\mathbb{P}).\)

1. A random process (or stochastic process) is a collection of random variables \(\left\{X_t\right\}_{t\in I};\)

• \(I\) is called the index set (or parameter set) of the (random) process, and each \(t\in I\) is called a time;
• For each \(\omega\in\Omega,X_t(\omega)\) is a function relative to \(t,\) called the sample path;
• The set \(S=\left\{X_t(\omega):t\in I,\omega\in\Omega\right\}\) is called the state space of the process.

2. A filtration is an increasing collection of \(\sigma-\)algebras \(\left\{\mathcal{F}_t\right\}_{t\geq0}\) satisfying \[\mathcal{F}_t\subset\mathcal{F},\forall t\geq0\textrm{ and }\mathcal{F}_s\subset\mathcal{F}_t,\forall 0\leq s\leq t.\] The quadruple \((\Omega,\mathcal{F},\left\{\mathcal{F}_t\right\},\mathbb{P})\) is called a ;

3. Given a process \(\left\{X_t\right\}_{t\in I},\) the collection \[\sigma(\left\{X_t\right\})=\sigma\left(\left\{X_t^{-1}(A):t\in I,A\in\mathcal{B}(\mathbb{R})\right\}\right)\] is a \(\sigma-\)algebra on \(\Omega,\) called the \(\left\{X_t\right\}.\)

2 Martingale

3 Poisson Process

A Poisson process with intensity (or rate) \(\lambda\) is a process \(\left\{N_t\right\}_{t\geq0}\) satisfying:

1. \(N_0=0;\)
2. \(N_t\sim\textrm{Pois}(\lambda t),\forall t\geq0;\)
3. Stationary Increments: \(N_{s+t}-N_s\sim N_t,\forall s,t>0;\)
4. Independent Increments: if \(s,t,u,v>0\) and \(s+t<u,\) then \(N_{s+t}-N_s\) and \(N_{u+v}-N_u\) are independent.

\(N_t\) is called the arrival count up to time \(t.\)

4 Markov Chain

1. A process \(\left\{X_n\right\}_{n\in\mathbb{N}}\) is called a Markov chain if it satisfy the Markov property: \[X_{n+1}|(X_0,...,X_n)=X_{n+1}|X_n,\forall n\in\mathbb{N}.\]
2. Associated with a Markov chain \(\left\{X_n\right\}_{n\in\mathbb{N}}\) is the transition probability \[p_{ij}=\mathbb{P}(X_{n+1}=j|X_n=i),\forall n\in\mathbb{N}\] and the transition matrix \[P=\begin{pmatrix}p_{11} & p_{12} & ...\\p_{21} & p_{22} & ...\\... & ... & ...\end{pmatrix}.\]
3. For each \(n\in\mathbb{N},\) define the transition probability after \(n\) steps as \[r_{ij}(n)=\mathbb{P}(X_{n+k}=j|X_k=i),\forall k\in\mathbb{N}.\] Then \[P^n=\begin{pmatrix}p_{11} & p_{12} & ...\\p_{21} & p_{22} & ...\\... & ... & ...\end{pmatrix}^n=\begin{pmatrix}r_{11}(n) & r_{12}(n) & ...\\r_{21}(n) & r_{22}(n) & ...\\... & ... & ...\end{pmatrix}.\]
4. For each \(n\in\mathbb{N},\) define the unconditional distribution of \(X_n\) as \[\pi^{(n)}=(\pi_1^{(n)},\pi_2^{(n)},...)\textrm{ where }\pi_i^{(n)}=\mathbb{P}(X_n=i)\] then \(\pi^{(n)}=\pi^{(0)}\cdot P^n.\) In the long term, \(\pi^{(n)}\) approaches the stationary distribution \(\pi=(\pi_1,\pi_2,...)\) given by \[\pi\cdot P=\pi\textrm{ and }\sum_i\pi_i=1.\] Furthermore, \(r_{ij}(n)\) approaches \(\pi_j\) as \(n\rightarrow\infty.\)

5 Random Walk

1. Consider a process \(\left\{X_n\right\}_{n\in\mathbb{N}}\) of independent, identically distributed random variables with distribution \[\mathbb{P}(X_n=-1)=\mathbb{P}(X_n=1)=\frac{1}{2}\] then the process \(\left\{M_n\right\}_{n\in\mathbb{N}}\) defined by \[M_0=0\textrm{ and }M_n=\sum_{i=1}^nX_i,\forall n\geq1\] is called a symmetric random walk.

2. Properties of \(\left\{M_n\right\}:\)

• It is a martingale and \(\mathbb{E}(M_n)=0,\textrm{Var}(M_n)=n;\)
• The first passage time \(\tau_1=\inf\left\{n:M_n=1\right\}\) has distribution \[\mathbb{P}(\tau_1=2j-1)=\frac{1}{2^{2j-1}}\cdot\frac{(2j-2)!}{j!\cdot(j-1)!};\]
• Stationary Increments: \(M_{s+t}-M_s\sim M_t,\forall t,s\in\mathbb{N};\)
• Independent Increments: if \(s,t,u,v\in\mathbb{N}\) and \(s+t<u,\) then \(M_{s+t}-M_s\) and \(M_{u+v}-M_u\) are independent.
• Quadratic Variation: \[\left<M,M\right>_k=\sum_{j=1}^k(M_j-M_{j-1})^2=k,\forall k\in\mathbb{N}.\]
• The process \(\left\{W_t^{(n)}\right\}\) defined by \[W_t^{(n)}=\frac{M_{nt}}{\sqrt{n}}\textrm{ whenever }nt\textrm{ is an integer}\] is called a scaled symmetric random walk. It preserves all properties of \(\left\{M_n\right\}:\) martigale property, stationary & independent increments, quadratic variation.
• By the Central Limit Theorem, the process \(\left\{W_t^{(n)}\right\}\) converges in distribution to a process \(\left\{B_t\right\}\) called the Brownian motion as \(n\rightarrow\infty.\)

6 Brownian Motion

1. A process \(\left\{B_t\right\}_{t\geq0}\) is called a Brownian motion if it has the following properties:

• For each \(\omega\in\Omega,B_t(\omega)\) is a continuous function of \(t;\)
• \(B_0=0\) and \(B_t\sim\mathcal{N}(0,t),\forall t>0;\)
• Stationary Increments: \(B_{s+t}-B_s\sim B_t,\forall t,s>0;\)
• Independent Increments: if \(s,t,u,v>0\) and \(s+t<u,\) then \(B_{s+t}-B_s\) and \(B_{u+v}-B_u\) are independent.

2. Properties of \(\left\{B_t\right\}:\)

• \(\textrm{cov}(B_{t+s},B_s)=s,\forall s,t\geq0;\)
• It is a martingale;
• The first passage time \(\tau_m=\inf\left\{t:B_t=m\right\}\) has distribution \[\mathbb{P}(\tau_m\leq t)=2\cdot\mathbb{P}(B_t\geq m)\] and density \[f_{\tau_m}(t)=\frac{|m|}{t\sqrt{2\pi t}}\cdot e^{-\frac{m^2}{2t}},t\geq0.\] Also, \(\mathbb{P}(\tau_m<\infty)=1\) and \(\mathbb{E}(\tau_m)=\infty;\)
• The maximum to date \(\left\{M_t\right\}_{t\geq0}\) where \(M_t=\max\left\{B_s:0\leq s\leq t\right\}\) has distribution \[\mathbb{P}(M_t\geq x)=2\cdot\mathbb{P}(B_t\geq x);\]
• Quadratic Variation: \(\left<B\right>(T)=T\) almost surely, i.e.  \[(dB_t)^2=dt,\hspace{3mm}dB_tdt=dtdt=0.\]

7 Ito Integral

1. An Ito integral has the form \[I_t=\int_0^t\delta_sdB_s\] where the integrator \(\left\{B_s\right\}\) is the Brownian motion and the integrand \(\left\{\delta_s\right\}\) is a square-integrable, adapted process.
2. Properties of \(I_t:\)

• Series representation: \[\int_0^Tf(s)dB_s=\lim_{n\rightarrow\infty}\sum_{i=0}^{n-1}f\left(\frac{iT}{n}\right)\left[B_{(i+1)T/n}-B_{iT/n}\right];\]
• It is a continuous function of \(t;\)
• Linearity: \[\int_0^t(\alpha\cdot\delta_s\pm\beta\cdot\gamma_s)dB_s=\alpha\int_0^t\delta_sdB_s\pm\beta\int_0^t\gamma_sdB_s;\]
• Quadratic Variation: \[\left<I,I\right>(t)=\int_0^t\delta_s^2ds;\]
• Isometry: \(\mathbb{E}(I_t^2)=\mathbb{E}(\left<I,I\right>(t)).\)
• Integration by Parts: if \(f\) is continuous and deterministic then \[\int_0^tf(s)dB_s=f(t)\cdot B_t-\int_0^tB_sdf(s);\]
• If \(\delta_s\) is deterministic then \(I_t\sim\mathcal{N}(0,\left<I,I\right>(t)).\)

8 Ito-Doeblin Formula

Let \(f(t,x)\) be twice continuously differentiable and \(\left\{B_t\right\}\) be the Brownian motion.

1. The simple case: \[df(t,B_t)=f_t(t,B_t)dt+f_x(t,B_t)dB_t+\frac{1}{2}f_{xx}(t,B_t)dt.\]
2. Consider an Ito process \(\left\{X_t\right\}_{t\geq0}\) satisfying \[dX_t=\mu_tdt+\sigma_tdB_t\] where the drift term \(\mu_t\) and the diffusion term \(\sigma_t\) are adapted processes, then \[df(t,X_t)=f_t(t,X_t)dt+f_x(t,X_t)dX_t+\frac{1}{2}f_{xx}(t,X_t)\sigma_t^2dt.\]

9 Stochastic Differential Equation (SDE)

All SDEs considered in this section is of the form \[dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dB_t\] with \(X_0=x_0.\)