# 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:

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

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

• 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 $$\R^M$$.