# 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

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

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 - \(u(\mathbf{W}_t)\)

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

\[ \frac{S_t}{M_t} = \frac{S_0}{M_0}\exp\left\{\sigma W_t - \frac{\sigma^2t}{2}\right\} \]

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 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

\[ \begin{align*} V_0 &= \exp(-rT)\P\!\left(\max_{t\le T} S_t \ge AS_0\right) = \exp(-rT)\P\!\left(\max_{t\le T} W_t + \mu t \ge a\right)\\ &= \exp(-rT)\P_{\mu}\!\left(\max_{t\le T} W_t \ge a\right) \end{align*} \]

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 \(\{\mathbf{X}_t\}_{t\ge 0}\) 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 \(\R^M\).