Math Material for Algo- & HFT, Book by Cartea-Jaimungal-Penalva

This is just mathematical background needed to understand the book.It contains the Math Appendix and Ch5 (the Math tools chapter). I will collect substantive results from the book in a separate post.

For further reference:

  1. Bertsekas-Shreve 1978: Stochastic Optimal Control, The Discrete Time Case (fairly advanced due to its generality and abstractness, to my surprise)
  2. Yong-Zhou 1999: SMP & HJB
  3. Fleming-Soner 2006: Controlled Markov Processes and Viscosity Solutions
  4. Oksendal-Sulem 2007: Applied Stochastic Control of Jump Diffusions
  5. Pham 2010: Cont-time Stochastic Control and Optimization with Fin Applications
  6. Touzi 2013: Optimal Stochastic Control, Stochastic Target Problems, and Backard SDE (advanced)

Appendix: Stochastic Calculus

definition of Brownian Motion;

definition of stochastic (Ito) integral. Shorter way:

  1. for adapted, L2 proecss , define

  2. this is called an Ito process, and often written as . Can show that this SP is a martingale.

  3. More generally, Ito process can be written as

  4. generally, we just need and to be adapted and satisfy certain integrability conditions. But in the special case where , the equation is called SDE. But namewise, book also mentioned that Ito processes are stochastic processes satisfying SDEs with Brownian noise terms.

definition of stochastic (Ito) integral. Rigorous way:

  1. define Ito integral for simple functions
  2. prove that any can be approximated, in , by sequence of simple functions
  3. define Ito integral for as the limiting value of the Ito integral of the sequence of simple functions

Ito isometry: for adapted ,

infinitesimal generator: the generalization of derivative of a function, to make it applicable to stochastic process.

This is the generator of an Ito process satisfying a certain SDE, e.g., .

Jump Processes

Poisson process
, valued in , with intensity param , is a SP s.t.:

  1. , a.s.
  2. has Poisson distro with param :
  3. has independent increments: is independent of
  4. has stationary increments:

Classic result 1: time between successive jumps of are independent, and exponentially distributed.

compensated Poisson process:
with . Note that this is a martingale.

as with BM, we can define stochastic integrals wrt compensated Poisson processes in a way that the resulting integral process is a martingale.

let be adapted, define stochastic integral of wrt by:

, where the ’s are jump times.

  1. need , not , to make integral a martingale.
  2. alt. def: replace first term with , where , which in this case is either 0 or 1. sum over a continuum of , what is the formal def?

Ito formula for Poisson process

recall we can write

Ito’s formula for such process is:
suppose satisfies for differentiable . Then:

or written in compensated poisson process.

we also see from this (compensated version) formula that the generator of the process is

jump diffusion

Ito formula for jump diffusion for the above , let be defined by , then:

again, common to write it using compensated poisson process .

compound Poisson process is built out of:

  1. a Poisson process with intensity
  2. a collection of iid RVs , with common distro

. The process jumps when Poisson event arrives, but the jump size is drawn from .

We can show, as before:

  1. defined by is a martingale

  2. we can define stochastic integral wrt compound Poisson too.

  3. note that

  4. a Ito’s formula for where

Doubly Stochastic Poisson Processes (Cox process)

these are jump processes which has stochastic intensity

  1. given counting process , we want its intensity process be stochastic

  2. the approach is to give a way to compute the probability that an event arrives at , given info we have at time : to define , where is the natural filtration generated by .

  3. this means:

  4. the driver of the intensity process can be diverse, leading to Feller/OU/Hawkes processes.

  5. as before, can define its compensated version which is a martingale; can define stochastic integral wrt the compensated doubly stochastic Poisson process, and can derive a Ito’s formula for such integral processes, and from which we can derive an expression for the generator of the joint process .

Feynman-Kac

certain linear PDEs are related to SDEs.

Let be an Ito process satisfying: .

The generator of is then where

Now suppose we try to solve PDE: , with terminal condition ,

then we have a probabilistic representation of solution :

(note there is a typo in the book)

Consider the simplest example:

  1. Now introduce a BM , and define

  2. is a martingale, and Markov: for some

  3. use Ito’s lemma to write out

  4. devide the above by and take limit, we get

  5. by definition, . Thus, satisfies the PDE. Recall .

Ch5 Stochastic Optimal Control and Stopping

A few motivating examples (just to be familiar with notation)

Merton Problem

value function: , where:

  1. at , place dollars in risky asset
  2. wealth level is

state dynamics follow:

is the admissible set, the set of -predictable, self-financing strategies satisfying . (to prevent doubling strategies)

Optimal Liquidation

state dynamics follow:

  1. (note the sign)

is the set of -predictable, non-negative bounded strategies (excluding repurchasing of shares, and keep liquidation rate finite)

optimal Limit Order placement identical value function expression, just change the to , which means that agent posts a LO at when current stock price is .

state dynamics:

  1. denotes market orders