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 orthogonal projection of X onto , such that:

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