# Notes on Stochastic Calculus

This is a brief selection of my notes on the stochastic calculus course. Content may be updated at times. \(\newcommand{\E}{\text{E}} \newcommand{\P}{\text{P}} \newcommand{\Q}{\text{Q}} \newcommand{\F}{\mathcal{F}} \newcommand{\d}{\text{d}} \newcommand{\N}{\mathcal{N}} \newcommand{\sgn}{\text{sgn}} \newcommand{\tr}{\text{tr}} \newcommand{\bs}{\boldsymbol} \newcommand{\eeq}{\ \!=\mathrel{\mkern-3mu}=\ \!} \newcommand{\eeeq}{\ \!=\mathrel{\mkern-3mu}=\mathrel{\mkern-3mu}=\ \!} \newcommand{\R}{\mathbb{R}} \newcommand{\MGF}{\text{MGF}}\)

# 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 \[ f'(x)\mu(x) + f''(x)\sigma^2(x)/2 = 0\Rightarrow f(x) = \int_A^x C_1\exp\left\{-\!\int_A^z\frac{2\mu(y)}{\sigma^2(y)}\d y\right\}\d z + C_2 \] 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 - \(u(\bs{W}_t)\)

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

# 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 \(\bs{x}\in\mathcal{U}\), let \(\E^{\bs{x}}\) be the expectation under measure \(\P^{\bs{x}}\) s.t. \(\bs{W}_t - \bs{x}\) is a \(K\)-dimensional standard BM. Then

- \(u(\bs{W}_{T\wedge t})\) is a MT.
- \(\E_{\bs{x}}[u(\bs{W}_T)] = u(\bs{x})\).

# Multivariate Itô Process

A multivariate Itô process is a continuous-time stochastic process \(X_t\in\R\) of the form \[ X_t = X_0 + \int_0^t\! M_s \d s + \int_0^t\! \bs{N}_s\cdot \d \bs{W}_s \] where \(\bs{N}_t\) is an adapted \(\R^K\)−valued process and \(\bs{W}_t\) is a \(K\)−dimensional standard BM.

# General Multivariable Itô Formula - \(u(\bs{X}_t)\)

Let \(\bs{W}_t\in\R^K\) be a standard \(K\)−dimensional BM, and let \(\bs{X}_t\in\R^m\) be a vector of \(m\) multivariate Itô processes satisfying \[ \d X_t^i = M_t^i\d t + \bs{N}_t^i\cdot \d \bs{W}_t. \] Then for any \(C^2\) function \(u:\R^m\mapsto\R\) with bounded first and second partial derivatives \[ \d u(\bs{X}_t) = \nabla u(\bs{X}_t)\cdot \d \bs{X}_t + \frac{1}{2}\tr[\Delta u(\bs{X}_t)\cdot \d [\bs{X},\bs{X}]_t]. \]

# Knight's Theorem

Let \(\bs{W}_t\) be a standard \(K\)−dimensional BM, and let \(\bs{U}_t\) be an adapted \(K\)−dimensional process satisfying \[ |{\bs{U}_t}| = 1\quad\forall t\ge 0. \] Then we know the following \(1\)-dimensional Itô process is a standard BM: \[ X_t := \int_0^t\!\! \bs{U}_s\cdot \d W_s. \]

# Radial Process

Let \(\bs{W}_t\) be a standard \(K\)−dimensional BM, and let \(R_t=|\bs{W}_t|\) 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 \(\d W_t^{\sgn} := \sgn(\bs{W}_t)\cdot \d \bs{W}_t\).

# 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 \[ \frac{S_t}{M_t} = \frac{S_0}{M_0}\exp\left\{\sigma W_t - \frac{\sigma^2t}{2}\right\} \] 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 \[
\E[f(S_T)/M_T\mid \F_t] = \frac{1}{M_T}\E\!\left[f\!\left(S_t\exp\!\left\{\sigma W_{T-t}^* - \frac{\sigma^2(T-t)}{2} + R_T - R_t\right\}\right)\middle|\F_t\right]
\] 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 \(\$1\) at time \(T\) if \(\max_{t\le T} S_t \ge AS_0\) and \(\$0\) 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 \[\begin{align*} V_0 &= \exp(-rT)\P\!\left(\max_{t\le T} S_t \ge AS_0\right) = \exp(-rT)\P\!\left(\max_{t\le T} W_t + \mu t \ge a\right)\\ &= \exp(-rT)\P_{\mu}\!\left(\max_{t\le T} W_t \ge a\right) \end{align*}\] where \(\mu=r\sigma^{-1} - \sigma/2\) and \(a = \sigma^{-1}\log A\). Now, by Cameron-Martin we know \[\begin{align*} \P_{\mu}\!\left(\max_{t\le T} W_t \ge a\right) &= \E_0[Z_T\cdot \mathbf{1}_{\{\max_{t\le T} W_t\ge a\}}] = \E_0[\exp(\mu W_T - \mu^2 T / 2)\cdot \mathbf{1}_{\{\max_{t\le T} W_t\ge a\}}] \\ &= \exp(- \mu^2 T / 2)\cdot \E_0[\exp(\mu W_T)\cdot \mathbf{1}_{\{\max_{t\le T} W_t\ge a\}}] \end{align*}\] and by reflection principle we have \[\begin{align*} \E_0[\exp(\mu W_T)\cdot \mathbf{1}_{\{\max_{t\le T} W_t\ge a\}}] &= e^{\mu a}\int_0^{\infty} (e^{\mu y} + e^{-\mu y}) \P(W_T - a \in \d y) \\&= \Phi(\mu\sqrt{T} - a/\sqrt{T}) + e^{2\mu a}\Phi(-\mu\sqrt{T}-a/\sqrt{T}). \end{align*}\]

# Exponential Process

The exponential process \[ Z_t = \exp\!\left\{\int_0^t\! Y_s\d W_s - \frac{1}{2}\int_0^t\! Y_s^2\d s\right\} \] 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 \[ Z_t = \exp\!\left\{\int_0^t\! Y_s \d W_s - \frac{1}{2}\!\int_0^t\! Y_s^2 \d s\right\} \] is a MG given \[ \E\left[\exp\!\left\{\frac{1}{2}\!\int_0^t\! Y_s^2\d s\right\}\right] < \infty. \] 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 \[ \left(\frac{\d\Q^x}{\d\P^x}\right)_{\!\F_T} \!= \frac{\mathbf{1}_{\{W_T=A\}}}{\P^x\{W_T=x\}} \Rightarrow \left(\frac{\d\Q^x}{\d\P^x}\right)_{\!\F_{T\wedge t}} \!= \E\!\left[\left(\frac{\d\Q^x}{\d\P^x}\right)_{\!\F_T}\middle|\F_{T\wedge t}\right] = \frac{W_{T\wedge t}}{x}. \] Notice \[\begin{align*} \frac{W_{T\wedge t}}{x} &= \exp\left\{\log W_{T\wedge t}\right\} / x \overset{\text{Itô}}{\eeq} \exp\left\{\log W_0 + \int_0^{T\wedge t}W_s^{-1}\d W_s - \frac{1}{2}\int_0^{T\wedge t} W_s^{-2}\d s\right\} / x \\&= \exp\left\{\int_0^{T\wedge t}W_s^{-1}\d W_s - \frac{1}{2}\int_0^{T\wedge t} W_s^{-2}\d s\right\} \end{align*}\] 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 \(\{\bs{X}_t\}_{t\ge 0}\) with \(\bs{X}_0=\bs{0}\) 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 \(\{\bs{X}_t\}_{t\ge 0}\), then \(\{F(\bs{X}_t)\}_{t\ge 0}\) is a Lévy process on \(\R^M\).