## Stochastic Jump Diffusions (Ch1-Ch3)

Overall I find the writing of this book to be sloopy and imprecise at times, and some crucial concepts could have been more carefully explained, This can be quite annoying if you are teaching yourself with this. While it certainly is much simpler than Oksendal-Sulem and avoids semimartingales all together, I really cannot recommend teaching yourself with this book. So I decided to stop using this book after reading the first 3 chapters. My summaries are below.

### Ch1 Stochastic Jump and Diffusion Processes

**Markov Process**

SP
is a Markov process if the conditional probability
satisfies:
(domain of state space), we have
.

**Wiener Process**

the standard Wiener process has:

continuous path:

independent increments: are mutually independent for all on non-overlapping time intervals

is a

**stationary process**: the distro of is independent of . Note that it is really difference stationary, should say Brownian motion has “stationary increments” to be precise.is Markov

, so the density of is

with prob 1:

So, if we think of Brownian increments of equal time steps, , where . These are iid with normal distro:

The book then refer to as “differential process”, and when , it has the same distro as , which is normal with mean 0 and variance .

Non-differentiability of sample path:

**Poisson Processes**

has unit jumps: if jumps occurs at , then

is right-continuous

has independent increments: are mutually independent for all on non-overlapping time intervals

is a stationary process: distro of is independent of . Again, the terminology should be “stationary increments”.

is Markov:

is Poisson distributed with mean and variance : . Here denotes the probability of the Poisson RV being equal to , not some parameter.

with probability 1:

is a martingale

Thus as for BM, are iid, and has the same discrete Poisson distribution as : .

As with BM, define , and this has the same discrete distro as , i.e., .

If we are to simulate , usually simulate time between jumps as we can show that has exponential distro: .

**Poisson 0-1 Jump law**

As
,
with probability
, otherwise no jumps. Other possibilities have
probabilities that vanish quicker than these two. To be “precise”, the
book states it as:

- As
- As
- As

Then using this 0-1 jump law, we formally write

**temporal/non-stationary Poisson process**

time dependent jump rate:
.

given the rate process , define , or in differential form, .

The **temporal Poisson process* has the following analogous results:

up to order , with prob we have , otherwise

inter-jump times is again exponentially distributed:

### Ch2 Stochastic Integration for Diffusions

Jump diffusion SDE with initial conditions has the form:

This is a symbolic equation, it has no meaning until we specify the
**methods of integration** for the **3 types of
integrals**:

**Riemann Integration**

use to denote

the partition of the interval is index by . A total of intervals. Denote as mesh size

on each subinterval, take an “approximation point” , where .

Define (constructively) , where

Because BM is continuous with prob1, the integral of wrt can be defined via Riemann: . Here we chose but any is fine.

Even for as solutions to jump-diffusion SDEs, can define this way.

Stieltjes integral refers to a deterministic integration wrt the position on the path of . Define it (constructively) as: . This makes sense if is continuous and BV.

**Ito integration wrt
**

start with trying to define . You would expect it may mimic deterministic case where . But turns out to be not the case.

So we go back to a discrete approximation first, and we use Ito’s choice of approximation () to preserve independent increments:

After some algebra, we can show that

Then the book calculated the expectation of this expression, which is 0, and claims that this suggest a reasonable form of stochastic integral to be: . I don’t see why it seems natural, but certainly the sum of the squares converge to in L2 sense, and in the end the conjecture is correct.

**convergence in mean square** The RV
converges in mean square to RV
if
, and we write it as
, or we use the notation
which stands for **Ito mean square equals
to**$.

It would seem here that whenever this appears, to its right it should be a discrete approximation, for example,