# Notes on Stochastic Calculus

# Discrete Time Martingales

## Conditional expectation

** Definition** A

**Borel**set is any set in a topological space that can be formed from open sets through the operations of:

- complement
- countable union
- countable intersection

** Definition** Let be a random vector and be a integrable random variable with . The

**conditional expectation**of given is the unique measureable function such that for every Borel set :

We denote as

** Example 1** Suppose random variable and are discrete.

** Example 2** Suppose random variable and are continuous, with joint probability density function and marginal density and .

Here are some basic properties of conditional expectation:

- Linearity:
- Constant: if , then
- Independence: if is independent of , then
- Tower Property: if then
- Factorization Property: if Z is -measurable then
- Monotonicity: if , then a.s.

## Theory

** Definition** A

**-algebra**is a collection of subsets of a Borel set , that is closed under:

- complement, e.g. if , then
- countable unions, e.g. if , then

** Definition** is the set of all -measurable square-integrable random variable , with finite 2nd moment .

** Definition** A real

**Hilbert space**is a real vector space with an inner product , such that is a complete metric space w.r.t. to the metric , where:

Hilbert space examples: , with inner product . Or, , with inner product . The reason we are interested at rather than for other is that the innner product give rise of orthogonality.

** Proposition** If , then for any -algebra , the conditional expectation is the

**of X onto , such that:**

*orthogonal projection*Also, can be interpreted as a -measurable random variable that minimizes the mean square error .

## Martingales

** Definition** A

**filtration**is an increasing sequence of -algebra , where is the -algebra of all events.

** Definition** A

**martingale**is a sequence of measurable integrable random variable such that:

The tower property implies that .

** Example 1** Given I.I.D. random variable with and variance .

- Sequence , and
- Sequence

are both martingales.

** Example 2** Let be any random variable and be any filtration. Then the sequence is a

**closed martingales**.

Note that the St. Petersburg martingale is not closed, where and and . This is because .

** Example 3** Given I.I.D. random variable with moment generating function . Then the

**exponential martingales**is a positive martingale with definition:

## Doob’s Indentity

** Definition** A sequence of random variables is

**predictable**with respect to filtration if is measurable with respect to

** Definition** A sequence of random variables is

**adapted**to filtration if is measurable with respect to

** Proposition** If is a martingale with and is a predictable sequence of bounded random variables, then the

**martingale transform**is a martingale:

** Definition** A

**stopping time**with respect to filtration is a random variable such that

** Lemma** Let be a stopping time, then the sequence is predictable.

** Theorem** Let be a martingale and be a stopping time. For all , the

**Doob’s Identity**states that . Note that if is bounded for all , DCT shows that .

Proof. is a martingale:

** Theorem** Let be a sequence functions on measure space that converge point-wise to a function f. For ,

The

**Dominated Convergence Theroem**(DCT) requires to be dominated by an integrable function :The

**Monotone Convergence Theroem**(MCT) requires to be monotone (increasing or decreasing): or

** Example 1** Let be a simple random walk with . Let stopping time , where .

We know that is a martingale and . Apply Doobs’s Identity and DCT we have:

We know that is a martingale. Apply Doobs’s Identity we have . Since is bounded by and is monotone, apply DCT on the RHS and MCT on LHS we get:

Combine both results we can get some interesting result for the **Gambler’s Ruin** problem:

** Example 2** Let be a simple random walk. Let stopping time , where . Note that now DCT fails as is not bounded. Hence .

In fact, because :

## Doob’s Maximal Inequality

** Definition** An adapted sequence of random variable is a:

- sub-martingale if
- super-martingale if

** Proposition** If is a convex function and is a martingale, then:

- The
**Jensen’s Inequality**holds: - the sequence is a sub-martingale.

** Proposition** If is a martingale with and is a predictable sequence of boundedm

**non-negative**random variables, then the martingale transform is a

**sub-martingale**:

** Proposition** If is a martingale with and is a predictable sequence of random variables such that , then

** Corollary** If is a non-negative sub-martingale with initial term , then

**Doob’s Maximal Inequality**claims that for any :

and that:

Note that this is a big improvement on the **Chebyshev Inequality**, which claims that given -bounded random variable and for any :

## Martingale Convergence Theorem

** Definition** a sequence of real numbers is called a

**Cauchy sequence**if for every positive real number , there is a positive integer such that for all natural numbers such that

** Definition** martingales have

**orthogonal increments**. Given a martingale with increments and , then:

- , , and

** Theorem** Suppose is -bounded martingale, then there exists a -bounded random variable such that:

** Theorem** Suppose is -bounded martingale, then there exists a -bounded random variable such that:

(1)

(2)

## Change Of Measure

** Proposition** Given a probability measure and is a non-negative random variable satisfying , then there exist a probability measure such that for any bounded or non-negative random variable that . Z is called the

**likelihood ratio**of probability measure w.r.t. , written as and that:

** Proposition** If the outcome space is finite, then for each outcome ,

** Example 1** In a -period market with finite set of outcomes and tradable assets. Let denote the risk-neutural measure for USD and EUR investors. Let denote the USD and EUR price of the risk-less (w.r.t. its own measure) asset at time t. Then

Proof. By fundamental theorem, , and , so:

** Theorem** Let and be two probability measure on the same measurable space, and let be a filtration such that for all n is absolutely continuous w.r.t. on . Then the sequence of likelihood ratio is a martingale:

# Brownian Motion

## Standard Bronwian Motion

** Definition** A standard

**Brownian motion**(SBM) is a continuous-time random process such that and:

(a) has stationary increments.

(b) has independent increments.

(c) The sample path are continuous.

Note that (a), (b), and (c) imply that for some constant the distribution of is

** Definition** Given a SBM , is a Brownian motion with drift and variance .

** Proposition** Given a SBM , its reflection is also a SBM.

** Proposition** Given a SBM , then for any , is a SBM

## Quadratic Variation

** Definition** The

**nth level quadratic variation**of a function is the sum of squares of the increments across intervals of length :

** Theorem** Given a SBM with drift and variance , then for all with probability :

## Strong Markov Property

** Definition** Given a SBM , a

**stoping time**is a non-negative random variable such that for every fixed , the event depends only on the path

** Theorem** If is a Brownian motion and is a stopping time then the

**strong Markov property**holds:

(a) the process is a Brownian motion, and

(b) the process is independent of the path

** Theorem** Run Brownian motion , at the first time that , reflect the path in the line , by the

**reflection principle**the new process is another Brownian motion:

- for ,
- for ,

*Corollary*

** Corollary** has the same distribution as

** Corollary** has the same distribution as . Hence . Consequently, for every with probability 1 adn . Therefore for every , the Brownian path crosses the t-axis infinitely many times by time

## Martingales In Continuous Times

** Definition** A

**filtration**is a nested family of -algebra indexed by time .

** Definition** The

**natural filtration**for a Brownian motion is the filtration with -the collection of all events determined by Brownian path up to time .

** Definition** A continuous-time stohastic process X_t is a martingale relative to a filtration if:

(a) each random variable is measurable w.r.t. and

(b) for any ,

** Proposition** Given a SBM then each of these is a martingale relative to the natural filtration:

(a)

(b)

(c)

** Theorem** Define to be the probability measure with likehood ratio . The

**Cameron-Martin**theorem states that the SBM under is a Brownian motion with drift and variance under .

** Corollary** For any real value and

** Corollary** For any stopping time and ,

# Ito Calculus

## Ito Integral

** Definition** If is an uniformally bounded process with continuous paths adapted to then we can define an

**Ito Integral**, where is truncted at :

** Property** The Ito Integral satisfy the following properties:

(1) Linearity: .

(2) Continuity: the paths are continuous.

(3) Mean Zero:

(4) Variance， a.k.a.

**Ito Isometry**:

** Defintion** Define the

**quadratic variation**of the Ito Itegral:

*Proposition*

(a) The process is a martingale

(b) The process is a martingale

*Example*

** Example** For any stopping time and any :

** Theorem** Let be a SBM and let be the −algebra of all events determined by the path . If is any random variable with mean 0 and finite variance that is measurable with respect to , for some , then the

**Ito representation theorem**claims that adapted process such that:

This theorem is of importance in finance because it implies that in the Black-Scholes setting, every contingent claim can be hedged.

## Ito Formula

** Theorem** Let be a SBM, and let be a twice-continuously differentiable function such that are all bounded (or at most have exponential growth). Then for any :

** Theorem** Let be a SBM, and let be a twice-continuously differentiable function whose partial derivatives are all bounded. Then for any :

** Proposition** Assume is nonrandom and continuously differentiable. Then:

## Ito Process

** Definition** An

**Ito process**is a stochastic process that satisfies a stochastic differential equation of the form:

Equivalently, satisfies the stochastic integral equation:

** Definition** For any adapted process define:

** Theorem** Let be an Ito process, and let be a twice-continuously differentiable function whose partial derivatives are all bounded. Then:

### The Ornstein-Uhlenbeck Process

** Definition** The Ornstein-Uhlenbeck SDE:

(a) This SDE describes a process Xt that has a proportional tendency to return to an “equilibrium” position 0.

(b) In finance, the OU process is often called the Vasicek model.

(c) Solving the SDE:

(d) The Ornstein-Uhlenbeck process is Gaussian.

### The Exponential Martingale

** Definition** The Exponential Martingale SDE:

(a) Solving the SDE:

### The Diffusion Process

** Definition** The Diffusion SDE:

** Definition** The

**Harmonic Function**is a function that satisfies the ODE:

** Example** Let be a solution of the diffusion SDE with initial value , and for any real numbers let . Find

We first apply the Ito Formula to and observe that a harmonic function will force the term to vanish. Therefore is a martingale and that

We can solve for :

### The Diffusion Process - Bassel Process

** Definition** The Diffusion SDE:

** Example** Similar problem as above:

Note that if and then will never reach .

## Ito Formula - Multi-Variable

** Theorem** Let be a K−dimensional SBM, and let be a function with bounded first and second partial derivatives. Then the

**Ito Formula**states:

Where:

** Corollary** If is a stopping time for the SBM then

**Dynkin’s Formula**shows that for any fixed time :

And that is a martingale

** Definition** A function is said to be a

**Harmonic Function**in a region if

(a) 2D Harmonic Function Exmaple:

(b) 3D Harmonic Function Example:

** Corollary** Let be harmonic in the an open region with compact support, and assume that and its partials extend continuously to the boundary . Define to be the first exit time of Brownian motion from , then:

(a) the process is a martingale, and

(b) for every ,

** Example** If a

**2D**SBM starts at a point on the circle of radius 1, find out the probability that it hits concentric circles before .

Let be harmonic. Then is a martingale and that .

** Example** If a

**3D**SBM starts at a point on the sphere of radius 1, find out the probability that it hits concentric sphere before .

Let be harmonic. Then is a martingale and that .

## Ito Process - Multi-Variable

** Definition** An

**Ito process**is a continuous-time stochastic process of the form:

Where the quadratic variation

Let be a vector of Ito processes. For any function with bounded first and second partial derivatives, then:

** Theorem** Let be a K −dimensional SBM, and let be an adapted, K−dimensional process satisfying . Then the

**Knight’s Theorem**states that the 1-dimensional Ito process is a SBM:

** Proposition** Let be a K −dimensional SBM. Define be the

**radial part**of . Then is a

**Bessel process**with parameter :

# Barrier Option

## Pricing

** Definition** A

**barrier option**at time pays:

(a) 1max_{0 \leq t \leq T}\;S_t \geq AS_0$,

(b) 0$ otherwise.

Assume that follows GBM:

The no-arbitrage price of the barrier option at is the expected payoff:

At time , there are two possibilities:

(a) if , then

(b) if , then is the same as the time- value of a barrier option with time-to-maturity and

## Hedging

Let be the value of the barrier option at time . The Fundamental Theorem and Ito Formula show that v(t, S_t satisfy the Black-Scholes PDE:

A **replicating portfolio** for the barrier option holds

(a) share of stock

(b) share of cash

provided that . Once the portfolio convert all holdings to cash and hold till maturity.

# The Black-Scholes

## The Black-Scholes Formula

** Theorem** Under a risk-neutral , the

**Fundamental Theorem**asserts that discounted share price is a martingale, where:

Therefore :

** Definition** A

**European contingent claim**with expiration date and payoff function is a tradeable asset with:

(a) share price at time :

(b) discounted share price at time :

** Proposition** Let be a standard Brownian motion and is a function such that . Then for every :

** Corollary** Given , the

**Black Scholes Formula**shows:

Under risk-neutral , the time option price is a martingale. With the Ito Formula we can set the drift of to be zero and therefore derive the **Black Scholes PDE**:

## Hedging In Continuous Times

** Definition** A portfolio is

**self-financing**if for all

** Proposition** A portfolio is self-financing if and only if its discounted value is a martingale and satisfies:

** Definition** A

**replicating portfolio**for a payoff function is a self-financing portfolio such that

** Theorem** A replicating portfolio for contingent claims is given by:

(a) cash, and

(b) shares of stock

where u is the solution of the Black Scholes PDE satisfying

# The Girsanov Theorem

** Proposition** The exponential process is a positive martingale.

Applying Ito Formula and therefore

** Therorem** Given a SBM under -measure and the likelihood ratio , define the -measure where . Then the

**Girsanov’s Theorem**states that under the -measure:

(a) is a SBM

(b) is a BM with time-dependent drift

** Example 1** Given a brownian motion with , define measure be the

**conditional probability measure**on event . Therefore is a BM with drift .

Proof. We know that , therefore by change of measure:

Therefore Girsanov’s Theorem implies that under , is a SBM.

** Example 2** Given currency and their respective bank account and . Define exchange rate (# B per A) that

** Theorem** If is a SBM under measure then .

Proof. is a martingale only if

*Theorem*

# Levy Process

## Poisson Process

** Definition** A

**Levy process**is a continuous-time random process such that and:

(a) has stationary increments;

(b) has independent increments;

(c) the sample paths X_t$ are right-continuous.

Note that Brownian motion and Poisson process are both Levy processes and the basic building blocks of Levy processes. Brownian motion is the only Levy process with continuous paths.

** Example** Let be a SBM and for , the random variable is a Levy process.

Note that:

(a) has stationary, independent increments

(b) has the same distribution as

** Definition** A

**Poisson process**with rate is a Levy process such that for all the random variable follows Poisson distribution with mean :

** Proposition** If are independent Poisson distributions with mean , then .

Proof.

** Corollary** IF are independent Poisson processes with rates then the

**superposition**is a Poisson process with rate

** Proposition** Every discontinuity of a Poisson process is of size

** Proposition** Let be a Poisson process of rate , and let be an independent sequence of i.i.d. Bernoulli− random variables. Then the

**Thinning Theorem**states that are independent Poisson processes with rates :

** Theorem** If and in such a way that , then the

**Law of Small Numbers**states that the distribution converges to the distribution.

** Proposition** If is a rate− Poisson process, then for any real number the process is a martingale.

** Theorem** Define with likelihood ratio such that . Then under the process is a rate- Poisson process.

## Compound Poisson Process

** Definition** A

**compound Poisson process**is a Levy process of the form:

Where is rate- Poisson process and are i.i.d. random variable independent of . The distribution is the **compounding distribution** and the measure is the **Levy measure**.

At each , a random is draw from . is the sum of all draws made by time

** Proposition** If , then , and , is an

**exponential martingale**.

## Poisson Point Process

** Definition** Let be a −finite Borel measure on . A

**Poisson point process**with intensity measure is a collection of extended nonnegative integer-valued random variables such that

(A) If then a.s.

(B) If then

(C) If are pairwise disjoint, then the r.v.s are independent, and

** Proposition** The point process associated with a CPP is a

**Poisson point process**with intensity measure , where is the Levy measure for the CPP.

** Theorem** Let be any Levy process, and let be the random set of points such that the Levy process has a jump discontinuity of size at time , i.e.,

Then is a Poisson point process with intensity measure where is a −finite measure called the Levy measure of the process.

Reference:

- Stochastic Calculus: An Introduction with Applications, Gregory F. Lawler
- FINM 34500 Lecture Notes, Steve Lalley, the University of Chicago