# Notes on Stochastic Calculus

### 2019-02-21

This is a brief selection of my notes on the stochastic calculus course. Content may be updated at times. The general topics range from martingale, Brownian motion and its variants, option pricing, etc.
`$\newcommand{\E}{\text{E}}\newcommand{\P}{\text{P}}\newcommand{\Q}{\text{Q}}\newcommand{\F}{\mathcal{F}}\newcommand{\d}{\text{d}}\newcommand{\N}{\mathcal{N}}\newcommand{\sgn}{\text{sgn}}\newcommand{\tr}{\text{tr}}\newcommand{\bs}{\boldsymbol}\newcommand{\eeq}{\ \!=\mathrel{\mkern-3mu}=\ \!}\newcommand{\eeeq}{\ \!=\mathrel{\mkern-3mu}=\mathrel{\mkern-3mu}=\ \!}\newcommand{\R}{\mathbb{R}}\newcommand{\MGF}{\text{MGF}}$`

# MGF of Normal Distribution

For `$X\sim\N(\mu,\sigma^2)$`

, we have `$\MGF(\theta)=\exp(\theta\mu + \theta^2\sigma^2/2)$`

. We have `$\E(X^k) = \MGF^{\ (k)}(0)$`

.

# Truncated Normal Distribution

Consider a two-sided truncation `$(a,b)$`

on `$\N(\mu,\sigma^2)$`

, then

$$ \E[X\mid a < X < b] = \mu - \sigma\frac{\phi(\alpha) - \phi(\beta)}{\Phi(\alpha) - \Phi(\beta)} $$

where `$\alpha:=(a-\mu)/\sigma$`

and `$\beta:=(b-\mu)/\sigma$`

.

# Doob’s Identity

Let `$X$`

be a MG and `$T$`

a stopping time, then `$\E X_{T\wedge n} = \E X_0$`

for any `$n$`

.

# Matingale Transform

Define `$(Z\cdot X)_n:=\sum_{i=1}^n Z_i(X_i - X_{i-1})$`

where `$X$`

is MG with `$X_0=0$`

and `$Z_n$`

is predictable and bounded, then `$(Z\cdot X)$`

is MG. If `$X$`

is sub-MG, then also is `$(Z\cdot X)$`

. Furthermore, if `$Z\in[0,1]$`

, then `$\E(Z\cdot X)\le \E X$`

.

# Common MGs

`$S_n:=\sum_{i=1}^n X_i$`

for`$X\sim(0,\sigma^2)$`

(this is called symmetric RW)`$S_n^2 - n\sigma^2$`

for symmetric RW`$S_n$`

`$\exp(\theta S_n)\ /\ \MGF_X(\theta)$`

for symmetric RW`$S_n$`

`$(q/p)^{S_n}$`

for assymetric RW`$S_n$`

(`$P(X=1)=p$`

,`$P(X=-1)=q=1-p$`

)`$S_n - n (p-q)$`

for assymetric RW`$S_n$`

`$B_t^2 - t$`

for standard BM`$B_t$`

`$\exp(\theta B_t - \theta^2 t / 2)$`

for standard BM`$B_t$`

`$\exp(-2\mu B_t)$`

for`$B_t = W_t + \mu t$`

where`$W_t$`

is standard BM

# Convex Mapping

If `$X$`

is MG and `$\phi(\cdot)$`

is a convex function, then `$\phi(X)$`

is sub-MG.

`$L^p$`

and `$L^p$`

Boundedness

`$X$`

is an`$L^p$`

MG if`$\E X_n^p$`

is finite for all`$n$`

.`$X$`

is an`$L^p$`

-bounded MG if`$\sup_{n\ge 0}\E X_n^2$`

is finite.

# Doob’s Maximal Ineq.

`$L^1$`

: if`$X$`

is a non-negative sub-MG with`$X_0=0$`

, then`$\P(\max_{i\le n} X_i\ge \alpha)\le\E X_n / \alpha$`

.`$L^2$`

: if`$X$`

is an`$L^2$`

MG with`$X_0=0$`

, then`$\P(\max_{i\le n} X_i\ge \alpha)\le \E X_n^2 / \alpha^2$`

.`$L^2$`

-bounded: if`$X$`

is`$L^2$`

-bounded, then`$\P(\sup_{n\ge 0}|X_n|\ge \alpha) \le \sup_{n\ge 0}\E X_n^2 / \alpha^2$`

.

# MG Convergence Theorem

`$L^1$`

-bounded:`$\exists X_{\infty}\in L^1$`

s.t.`$\lim_{n\to\infty}X_n \overset{\text{a.s.}}{\eeq} X_{\infty}$`

.`$L^2$`

-bounded:`$\exists X_{\infty}\in L^2$`

s.t.`$\lim_{n\to\infty}X_n \overset{\text{a.s.}}{\eeq} X_{\infty}$`

,`$\lim_{n\to\infty}\E(X_n - X_{\infty})^2=0$`

and`$\lim_{n\to\infty}\E X_n^2 = E X_{\infty}^2$`

.

# Change of Measure

Given `$\P$`

-measure, we define the likelihood ratio `$Z:=\d\Q / \d\P$`

for another measure `$\Q$`

. Then we have

`$\E_{\P} Z = 1$`

.`$\E_{\Q} Y = \E_{\P} (ZY)$`

for all`$Y$`

. Specifically, for`$Y=\textbf{1}_{\omega}$`

we have`$\Q(\omega) = \E_{\P}(Z\textbf{1}_{\omega}) \overset{\text{discr.}}{\eeeq} \P(\omega)Z(\omega)$`

.- Example (changing numeraire from
`CASH`

`$\P$`

- to`STOCK`

`$\Q$`

-measure):`$Z(\omega) = (\d\Q/\d\P)(\omega) = S_N(\omega) / S_0$`

. - Example (importance sampling):
`$\P_{.5}(S_{100} > 80) = \E_{.8}\left(\textbf{1}_{S_{100} > 80}\cdot \frac{.5^{100}}{.8^{S}.2^{100-S}}\right)$`

. - Example (foreign exchange):
`$\d\Q_A/\d\Q_B = A_TY_T/B_TY_0$`

where`$Y_t$`

is the exchange rate, i.e. the number of currency`$B$`

in exchange for`$1$`

unit of currency`$A$`

.

# Cameron-Martin

**Theorem**: given`$B$`

is standard BM under`$\P_0$`

measure, then`$\exists \P_{\theta}$`

s.t.`$Z_T^{\theta} := \d\P_{\theta}/\d\P_0 = \exp(\theta B_T - \theta^2 T / 2)$`

, under which`$\{B_t\}_{t\le T}$`

is a BM with drift`$\theta$`

and variance`$1$`

.**Corollary**: given any two drift`$\theta$`

and`$\eta$`

, define similarly`$Z_T:=\d\P_{\theta}/\d\P_{\eta}$`

and then for any stopping time`$\tau$`

, we have`$\P_{\theta}(\tau\le T) = \E_{\eta}(\textbf{1}_{\tau\le T}Z_T)$`

.**Example**: given`$B$`

is a BM with drift`$-b < 0$`

, now define`$T=\tau(a)$`

for some`$a > 0$`

, then we have`$\P_{-b}(T < \infty) = \E_{+b}[\exp(-2b B_T) \textbf{1}_{T < \infty}] = \exp(-2ab)$`

.

# Strong Markov Property

If `$B$`

is a BM and `$T=\tau(\cdot)$`

is a stopping time, then `$\{B_{t+T} - B_T\}_{t\ge T}$`

is a BM indep. of `$\{B_t\}_{t\le T}$`

.

# Orthogonal Transform

If `$B$`

is a standard `$k$`

-BM and `$U\in\mathbb{R}^{k\times k}$`

is orthogonal, then `$UB$`

is also a standard `$k$`

-BM.

# Doob’s Decomposition

For any sub-MG `$X$`

, we have unique decomposition `$X=M+A$`

where `$M_n:=X_0 + \sum_{i=1}^n [X_i - \E(X_i\mid \F_{i-1})]$`

is a martingale and `$A_n:=\sum_{i=1}^n[\E(X_i\mid \F_{i-1}) - X_{i-1}]$`

is a non-decreasing predictable sequence.

# Gambler’s Ruin

- Symmetric: for (discrete- or continuous-time) MG
`$S$`

with`$S_0=0$`

, define stopping time`$T=\min\{\tau(-A), \tau(B)\}$`

then`$\P(S_T=B)=\frac{A}{A+B}$`

,`$\P(\tau(B)<\infty)=\lim_{A\to\infty}\P(S_T=B)=1$`

and`$\E T = AB$`

. - Assymetric (
`$p<q$`

): for RW`$S_n:=S_0 + \sum_{i=1}^{n}X_i$`

with`$S_0=0$`

and`$\P(X=+1)=p$`

,`$\P(X=-1)=q=1-p$`

, define similarly`$T$`

, then`$\P(S_T=B)=\frac{1-(q/p)^{-A}}{(q/p)^{B} - (q/p)^{-A}}$`

,`$\P(\tau(B)<\infty)=\lim_{A\to\infty}\P(S_T=B)=\left(p/q\right)^{B}$`

and`$\E T = \frac{\E S_T}{q-p}$`

.

# Reflection Principle

For BM `$B$`

and stopping time `$T=\tau(a)$`

, define `$B^*$`

s.t. `$B_t^*=B_t$`

for all `$t\le T$`

and `$B_t^* = 2a - B_t$`

for all `$t>T$`

, then `$B^*$`

is also a BM.

# First Passage Time `$T:=\tau(a)$`

- CDF:
`$\P(T \le t) = 2\P(B_t > a) = 2\Phi(-a / \sqrt{t})$`

. - PDF: follows from CDF
`$\E T$`

:`$X_t:=\exp(\theta B_t - \theta^2 t / 2)$`

is a MG, we know`$\E(X_T)=X_0 = 1$`

, from which expectation is calculated.

# Joint Distribution of BM and its Maximum

`$\P(\max_{s\le t}B_s > x\text{ and }B_t < y) = \Phi\!\left(\frac{y-2x}{\sqrt{t}}\right)$`

.

`$2$`

-BM Stopped on 1 Boundary

Let `$X$`

and `$Y$`

be indep. BM. Note that for all `$t\ge 0$`

, from exponential MG we know `$\E[\exp(i\theta X_t)]=\exp(-\theta^2 t/2)$`

. Now define `$T=\tau(a)$`

for `$Y$`

and we have `$\E[\exp(i\theta X_T)] = \E[\exp(-\theta^2 T /2)]=\exp(-|\theta| a)$`

, which is the Fourier transform of the Cauchy density `$f_a(x)=\frac{1}{\pi}\frac{a}{a^2+x^2}$`

.

# Itô Integral

We define Itô integral `$I_t(X) := \int_0^t\! X_s\d W_s$`

where `$W_t$`

is a standard Brownian process and `$X_t$`

is adapted.

# Martingality of Itô Integral

`$I_t(X)$`

is a martingale`$I_t(X)^2 - [I(X), I(X)]_t$`

is a martingale, where`$[I(X), I(X)]_t := \int_0^t\! X_s^2\d s$`

# Itô Isometry

This is the direct result from the second martingality property above. Let `$X_t$`

be nonrandom and continuously differentiable, then

$$ \E!\left[!\left(\int_0^t X_t\d W_t\right)^{!!2}\right] = \E!\left[\int_0^t X_t^2\d t\right]. $$

# Itô Formula - `$f(W_t)$`

Let `$W_t$`

be a standard Brownian motion and let `$f:\R\mapsto\R$`

be a twice-continously differentiable function s.t. `$f$`

, `$f'$`

and `$f''$`

are all bounded, then for all `$t>0$`

we have

$$ \d f(W_t) = f’(W_t)\d W_t + \frac{1}{2}f’’(W_t) \d t. $$

# Itô Formula - `$f(t,W_t)$`

Let `$W_t$`

be a standard Brownian motion and let `$f:[0,\infty)\times\R\mapsto\R$`

be a twice-continously differentiable function s.t. its partial derivatives are all bounded, then for all `$t>0$`

we have

$$ \d f(t, W_t) = f_x\d W_t + \left(f_t + \frac{1}{2}f_{xx}\right) \d t. $$

# Wiener Integral

The Wiener integral is a special case of Itô integral where `$f(t)$`

is here a nonrandom function of `$t$`

. Variance of a Wiener integral can be derived using Itô isometry.

# Itô Process

We say `$X_t$`

is an Itô process if it satisfies

$$ \d X_t = Y_t\d W_t + Z_t\d t $$

where `$Y_t$`

and `$Z_t$`

are adapted and `$\forall t$`

$$ \int_0^t! \E Y_s^2\d s < \infty\quad\text{and}\quad\int_0^t! \E|Z_s|\d s < \infty. $$

The quadratic variation of `$X_t$`

is

$$ [X,X]_t = \int_0^t! Y_s^2\d s. $$

# Itô Product and Quotient

Assume `$X_t$`

and `$Y_t$`

are two Itô processes, then

$$ \frac{\d (XY)}{XY} = \frac{\d X}{X} + \frac{\d Y}{Y} + \frac{\d X\d Y}{XY} $$

and

$$ \frac{\d (X/Y)}{X/Y} = \frac{\d X}{X} - \frac{\d Y}{Y} + \left(\frac{\d Y}{Y}\right)^{!2} - \frac{\d X\d Y}{XY}. $$

# Brownian Bridge

A Brownian bridge is a continuous-time stochastic process `$X_t$`

with both ends pinned: `$X_0=X_T=0$`

. The SDE is

$$ \d X_t = -\frac{X_t}{1-t}\d t + \d W_t $$

which solves to

$$ X_t = W_t - \frac{t}{T}W_T. $$

# Itô Formula - `$u(t, X_t)$`

Let `$X_t$`

be an Itô process. Let `$u(t,x)$`

be a twice-continuously differentiable function with `$u$`

and its partial derivatives bounded, then

$$ \d u(t, X_t) = \frac{\partial u}{\partial t}(t, X_t)\d t + \frac{\partial u}{\partial x}(t, X_t)\d X_t + \frac{1}{2}\frac{\partial^2 u}{\partial x^2}(t, X_t)\d [X,X]_t. $$

# The Ornstein-Uhlenbeck Process

The OU process describes a stochastic process that has a tendency to return to an “equilibrium” position `$0$`

, with returning velocity proportional to its distance from the origin. It’s given by SDE

$$ \d X_t = -\alpha X_t \d t + \d W_t \Rightarrow \d [\exp(\alpha t)X_t] = \exp(\alpha t)\d W_t $$

which solves to

$$ X_t = \exp(-\alpha t)\left[X_0 + \int_0^t! \exp(as)\d W_s\right]. $$

**Remark**: In finance, the OU process is often called the Vasicek model.

# Diffusion Process

The SDE for general diffusion process is `$\d X_t = \mu(X_t)\d t + \sigma(X_t)\d W_t$`

.

# Hitting Probability for Diffusion Processes

In order to find `$\P(X_T=B)$`

where we define `$T=\inf\{t\ge 0: X_t=A\text{ or }B\}$`

, we consider a harmonic function `$f(x)$`

s.t. `$f(X_t)$`

is a MG. This gives ODE

where `$C_{1,2}$`

are constants. Then since `$f(X_{T\wedge t})$`

is a bounded MG, by Doob’s identity we have

$$ \P(X_T=B) = \frac{f(X_0) - f(A)}{f(B) - f(A)}. $$

# Multivariable Itô Formula - `$u(\mathbf{W}_t)$`

Let `$\bs{W_t}$`

be a `$K$`

-dimensional standard Brownian motion. Let `$u:\R^K\mapsto \R$`

be a `$C^2$`

function with bounded first and second partial derivatives. Then

$$ \d u(\mathbf{W}_t) = \nabla u(\mathbf{W}_t)\cdot \d \mathbf{W}_t + \frac{1}{2}\tr[\Delta u(\mathbf{W}_t)] \d t $$

where the gradient operator `$\nabla$`

gives the vector of all first order partial derivatives, and the Laplace operator (or Laplacian) `$\Delta\equiv\nabla^2$`

gives the vector of all second order partial derivatives.

# Dynkin’s Formula

If `$T$`

is a stopping time for `$\bs{W_t}$`

, then for any fixed `$t$`

we have

$$ \E[u(\mathbf{W}_{T\wedge t})] = u(\bs{0}) + \frac{1}{2}\E!\left[\int_0^{T\wedge t}!!\Delta u(\mathbf{W}_s)\d s\right]. $$

# Harmonic Functions

A `$C^2$`

function `$u:\R^k\mapsto\R$`

is said to be harmonic in a region `$\mathcal{U}$`

if `$\Delta u(x) = 0$`

for all `$x\in \mathcal{U}$`

. Examples are `$u(x,y)=2\log(r)$`

and `$u(x,y,z)=1/r$`

where `$r$`

is defined as the norm.

**Remark**: `$f$`

being a harmonic function is equivalent to `$f(X_t)$`

being a MG, i.e. `$f'(x)\mu(x) + f''(x)\sigma^2(x)/2 = 0$`

for a diffusion process `$X_t$`

.

# Harmonic Corollary of Dynkin

Let `$u$`

be harmonic in the an open region `$\mathcal{U}$`

with compact support, and assume that `$u$`

and its partials extend continuously to the boundary `$\partial \mathcal{U}$`

. Define `$T$`

to be the first exit time of Brownian motion from `$\mathcal{U}$`

. for any `$\mathbf{x}\in\mathcal{U}$`

, let `$\E^{\mathbf{x}}$`

be the expectation under measure `$\P^{\mathbf{x}}$`

s.t. `$\mathbf{W}_t - \mathbf{x}$`

is a `$K$`

-dimensional standard BM. Then

`$u(\mathbf{W}_{T\wedge t})$`

is a MT.`$\E_{\mathbf{x}}[u(\mathbf{W}_T)] = u(\mathbf{x})$`

.

# Multivariate Itô Process

A multivariate Itô process is a continuous-time stochastic process `$X_t\in\R$`

of the form

$$ X_t = X_0 + \int_0^t! M_s \d s + \int_0^t! \mathbf{N}_s\cdot \d \mathbf{W}_s $$

where `$\mathbf{N}_t$`

is an adapted `$\R^K$`

−valued process and `$\mathbf{W}_t$`

is a
`$K$`

−dimensional standard BM.

# General Multivariable Itô Formula - `$ u(\mathbf{X}_t)$`

Let `$\mathbf{W}_t\in\R^K$`

be a standard `$K$`

−dimensional BM, and let `$\mathbf{X}_t\in\R^m$`

be a vector of `$m$`

multivariate Itô processes satisfying

$$ \d X_t^i = M_t^i\d t + \mathbf{N}_t^i\cdot \d \mathbf{W}_t. $$

Then for any `$C^2$`

function `$u:\R^m\mapsto\R$`

with bounded first and second partial derivatives

$$ \d u(\mathbf{X}_t) = \nabla u(\mathbf{X}_t)\cdot \d \mathbf{X}_t + \frac{1}{2}\tr[\Delta u(\mathbf{X}_t)\cdot \d [\mathbf{X},\mathbf{X}]_t]. $$

# Knight’s Theorem

Let `$\mathbf{W}_t$`

be a standard `$K$`

−dimensional BM, and let `$\mathbf{U}_t$`

be an adapted `$K$`

−dimensional process satisfying

$$ |{\mathbf{U}_t}| = 1\quad\forall t\ge 0. $$

Then we know the following `$1$`

-dimensional Itô process is a standard BM:

$$ X_t := \int_0^t!! \mathbf{U}_s\cdot \d W_s. $$

# Radial Process

Let `$\mathbf{W}_t$`

be a standard `$K$`

−dimensional BM, and let `$R_t=|\mathbf{W}_t|$`

be the corresponding radial process, then `$R_t$`

is a Bessel process with parameter `$(K-1)$`

given by

$$ \d R_t = \frac{K-1}{R_t}\d t + \d W_t^{\sgn} $$

where we define `$\d W_t^{\sgn} := \sgn(\mathbf{W}_t)\cdot \d \mathbf{W}_t$`

.

# Bessel Process

A Bessel process with parameter `$a$`

is a stochastic process `$X_t$`

given by

$$ \d X_t = \frac{a}{X_t}\d t+ \d W_t. $$

Since this is just a special case of diffusion processes, we know the corresponding harmonic function is `$f(x)=C_1x^{-2a+1} + C_2$`

, and the hitting probability is

$$ \P(X_T=B) = \frac{f(X_0) - f(A)}{f(B) - f(A)} = \begin{cases} 1 & \text{if }a > 1/2,\ (x/B)^{1-2a} & \text{otherwise}. \end{cases} $$

# Itô’s Representation Theorem

Let `$W_t$`

be a standard `$1$`

-dimensional Brownian motion and let `$\F_t$`

be the `$\sigma$`

−algebra of all events determined by the path `$\{W_s\}_{s\le t}$`

. If `$Y$`

is any r.v. with mean `$0$`

and finite variance that is measurable with respect to `$\F_t$`

, then for some `$t > 0$`

$$ Y = \int_0^t! A_s\d W_s $$

for some adapted process `$A_t$`

that satisfies

$$ \E(Y^2) = \int_0^t! \E(A_s^2)\d s. $$

This theorem is of importance in finance because it implies that in the Black-Sholes setting, every contingent `CLAIM`

can be hedged.

**Special case:** let `$Y_t=f(W_t)$`

be any mean `$0$`

r.v. with `$f\in C^2$`

. Let `$u(s,x):=\E[f(W_t)\mid W_s = x]$`

, then

$$ Y_t = f(W_t) = \int_0^t! u_x(s,W_s)\d W_s. $$

# Assumptions of the Black-Scholes Model

- Continuous-time trading
- No arbitrage
- Riskless asset
`CASH`

with non-random rate of return`$r_t$`

- Risky asset
`STOCK`

with share price`$S_t$`

such that`$\d S_t = S_t(\mu_t \d t + \sigma \d W_t)$`

# Black-Scholes Model

Under a risk-neutral measure `$\P$`

, the discounted share price `$S_t / M_t$`

is a martingale and thus

where we used the fact that `$\mu_t = r_t$`

by the Fundamental Theorem.

# Contingent Claims

A European contingent `CLAIM`

with expiration date `$T > 0$`

and payoff function `$f:\R\mapsto\R$`

is a tradeable asset that pays `$f(S_T)$`

at time `$T$`

. By the Fundamental Theorem we know the discounted share price of this `CLAIM`

at any `$t\le T$`

is `$\E[f(S_T)/M_T\mid \F_t]$`

. In order to calculate this conditional expectation, let `$g(W_t):= f(S_t)/M_t$`

, then by the Markov property of BM we know `$\E[g(W_T)\mid \F_t] = \E[g(W_t + W_{T-t}^*)\mid \F_t]$`

where `$W_t$`

is adapted in `$\F_t$`

and independent of `$W_t^*$`

.

# Black-Scholes Formula

The discounted time−`$t$`

price of a European contingent `CLAIM`

with
expiration date `$T$`

and payoff function `$f$`

is

where `$S_t$`

is adapted in `$\F_t$`

and independent of `$W_t^*$`

. The expectation is calculated using normal. Note here `$R_t = \int_0^t r_s\d s$`

is the log-compound interest rate.

# Black-Scholes PDE

Under risk-neutral probability measure, the discounted share price of `CLAIM`

is a martingale, i.e. it has no drift term. So we can differentiate `$M_t^{-1}u(t,S_t)$`

by Itô and derive the following PDE

$$ u_t(t,S_t) + r_t S_tu_x(t,S_t) + \frac{\sigma^2S_t^2}{2}u_{xx}(t,S_t) = r_t u(t,S_t) $$

with terminal condition `$u(T,S_T)=f(S_T)$`

. Note here everything is under the BS model.

# Hedging in Continuous Time

A replicating portfolio for a contingent `CLAIM`

in `STOCK`

and `CASH`

is given by

$$ V_t = \alpha_t M_t + \beta_t S_t $$

where `$\alpha_t = [u(t,S_t) - S_t u_x(t,S_t)]/M_t$`

and `$\beta_t = u_x(t,S_t)$`

.

# Barrier Option

A barrier option pays 1USD at time `$T$`

if `$\max_{t\le T} S_t \ge AS_0$`

and 0USD otherwise. This is a simple example of a path-dependent option. Other commonly used examples are knock-ins, knock-outs, lookbacks and Asian options.

The time-`$0$`

price of such barrier options is calculated from

where `$\mu=r\sigma^{-1} - \sigma/2$`

and `$a = \sigma^{-1}\log A$`

. Now, by Cameron-Martin we know

and by reflection principle we have

# Exponential Process

The exponential process

is a positive MG given

$$ \E!\left[\int_0^t! Z_s^2Y_s^2\d s\right] < \infty. $$

Specifically, the exponential martingale is given by the SDE `$\d X_t = \theta X_t \d W_t$`

.

# Girsanov’s Theorem

Assume that under the probability measure `$\P$`

the **exponential process** `$Z_t(Y)$`

is a MG and `$W_t$`

is a standard BM. Define the absolutely continuous probability measure `$Q$`

on `$\F_t$`

with likelihood ratio `$Z_t$`

, i.e. `$(\d\Q/\d\P)_{\F_t} = Z_t$`

, then under `$Q$`

the process

$$ W_t^* := W_t - \int_0^t! Y_s\d s $$

is a standard BM. Girsanov’s Theorem shows that drift can be added or removed by change of measure.

# Novikov’s Theorem

The exponential process

is a MG given

This theorem gives another way to show whether an exponential process is a MG.

# Standard BM to OU Process

Assume `$W_t$`

is a standard BM under `$\P$`

, define likelihood ratio `$Z_t = (\d\Q/\d\P)_{\F_t}$`

as above where `$Y_t = -\alpha W_t$`

, then by Girsanov `$W_t$`

under `$\Q$`

is an OU process.

# Fundamental Principle of Statistical Mechanics

If a system can be in one of a collection of states `$\{\omega_i\}_{i\in\mathcal{I}}$`

, the probability of finding it in a particular state `$\omega_i$`

is proportional to `$\exp\{-H(\omega_i)/kT\}$`

where `$k$`

is Boltzmann’s constant, `$T$`

is temperature and `$H(\cdot)$`

is energy.

# Conditioned Brownian Motion

If `$W_t$`

is standard BM with `$W_0 = x \in (0, A)$`

, how does `$W_t$`

behave conditional on the event that it hits `$A$`

before `$0$`

? Define

`$\P^x$`

is a measure under which`$W_0=x$`

`$\Q^x$`

is a measure under which`$W_0=x$`

and`$W_T=A$`

where`$T=\inf\{t\ge 0: W_t=A\text{ or }0\}$`

Then the likelihood ratios are

Notice

which is a Girsanov likelihood ratio, so we conclude `$W_t$`

is a BM under `$\Q^x$`

with drift `$W_t^{-1}$`

, or equivalently

$$ W_t^* = W_t - \int_0^{T\wedge t}W_s^{-1}\d s $$

is a standard BM with initial point `$W_0^* = x$`

.

# Lévy Process

A one-dimensional Lévy process is a continuous-time random process `$\{X_t\}_{t\ge 0}$`

with `$X_0=0$`

and i.i.d. increments. Lévy processes are defined to be a.s. right continuous with left limits.

**Remark**: Brownian motion is the only Lévy process with continuous paths.

# First-Passage-Time Process

Let `$B_t$`

be a standard BM. Define the FPT process as `$\tau_x = \inf\{t\ge 0: B_t \ge x\}$`

. Then `$\{\tau_{x}\}_{x\ge 0}$`

is a Lévy process called the one-sided stable-`$1/2$`

process. Specifically, the sample paths `$x\mapsto \tau_x$`

is non-decreasing in `$x$`

. Such Lévy processes with non-decreasing paths are called subordinators.

# Poisson Process

A Poisson process with rate (or intensity) `$\lambda > 0$`

is a Lévy process `$N_t$`

such that for any `$t\ge 0$`

the distribution of the random variable `$N_t$`

is the Poisson distribution with mean `$\lambda t$`

. Thus, for any `$k=0,1,2,\cdots$`

we have `$\P(N_t=k) = (\lambda t)^k\exp(-\lambda t)\ /\ k!$`

for all `$t > 0$`

.

**Remark 1:** (Superposition Theorem) If `$N_t$`

and `$M_t$`

are independent Poisson processes of rates `$\lambda$`

and `$\mu$`

respectively, then the superposition `$N_t + M_t$`

is a Poisson process of rate `$\lambda+\mu$`

.

**Remark 2:** (Exponential Interval) Successive intervals are i.i.d. exponential r.v.s. with common mean `$1/\lambda$`

.

**Remark 3:** (Thinning Property) Bernoulli-`$p$`

r.v.s. by Poisson-`$\lambda$`

compounding is again Poisson with rate `$\lambda p$`

.

**Remark 4:** (Compounding) Every compound Poisson process is a Lévy process. We call the `$\lambda F$`

the Lévy measure where `$F$`

is the compounding distribution.

# MGF of Poisson

For `$N\sim\text{Pois}(\lambda)$`

, we have `$\MGF(\theta)=\exp[\lambda (e^{\theta}-1)]$`

.

For `$X_t=\sum_{i=1}^{N_t}\!Y_i$`

where `$N_t\sim\text{Pois}(\lambda t)$`

and `$\MGF_Y(\theta) = \psi(\theta) < \infty$`

, then `$\MGF_{X_t}(\theta)=\exp[\lambda t (\psi(\theta) - 1)]$`

.

# Law of Small Numbers

Binomial-`$(n,p_n)$`

distribution, where `$n\to\infty$`

and `$p_n\to 0$`

s.t. `$np_n\to\lambda > 0$`

, converges to Poisson-`$\lambda$`

distribution.

# Poisson-Exponential Martingale

If `$N_t$`

is a Poisson process with rate `$\lambda$`

, then `$Z_t=\exp[\theta N_t - (e^{\theta} - 1) \lambda t]$`

is a martingale for any `$\theta\in\R$`

.

**Remark:** Similar to **Cameron-Martin**, let `$N_t$`

be a Poisson process with rate `$\lambda$`

under `$\P$`

, let `$\Q$`

be the measure s.t. the likelihood ratio `$(\d\Q/\d\P)_{\F_t}=Z_t$`

is defined as above, then `$N_t$`

under `$\Q$`

is a Poisson process with rate `$\lambda e^{\theta}$`

.

If `$X_t$`

is a compound Poisson process with Lévy measure `$\lambda F$`

. Let the MGF of compounding distribution `$F$`

be `$\psi(\theta)$`

, then `$Z_t=\exp[\theta X_t - (\psi(\theta) - 1)\lambda t]$`

is a martingale for any `$\theta\in\R$`

.

# Vector Lévy Process

A `$K$`

-dimensional Lévy process is a continuous-time random process `$\{\mathbf{X}_t\}_{t\ge 0}$`

with `$\mathbf{X}_0=\bs{0}$`

and i.i.d. increments. Like the one-dimensional version, vector Lévy processes are defined to be a.s. right continuous with left limits.

**Remark:** Given non-random linear transform `$F:\R^K\mapsto \R^M$`

and a `$K$`

-dimensional Lévy process `$\{\mathbf{X}_t\}_{t\ge 0}$`

, then `$\{F(\mathbf{X}_t)\}_{t\ge 0}$`

is a Lévy process on `$\R^M$`

.