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EdisonCruise
Topic Author
Posts: 118
Joined: September 15th, 2012, 4:22 am

### How to make ito's forumula for jump-diffusion a martingale

$dY_t=Z_{N_t} dN_t$ is a compound Poisson process with intensity $\lambda$ and $Z_{N_t}$ is a random variable for the jump size. $dW_t$ is Brownian motion.
The jump diffusion process $X_t$ is defined as
$$dX_t=\nu_t dt+u_t dW_t + \eta_t dY_t$$
So the Ito's lemma for this jump diffusion process is
$$df(X_t)=\nu_t f'(X_t)dt+u_t f'(X_t) dW_t+ \frac{1}{2}u^2_t f''(X_t)+(f(X_t)-f(X_t-))dN_t$$
The compensated Poisson process of $dN_t$ is $d\hat{N_t}=dN_t-\lambda dt$, which is a martingale. Then how to use $d\hat{N_t}$ to make $df(X_t)$ a martingale?
A simple subsitution as below seems somehow not correct:
$$df(X_t)=\nu_t f'(X_t)dt+u_t f'(X_t) dW_t+ \frac{1}{2}u^2_t f''(X_t)+\lambda(f(X_t)-f(X_t-))dt+(f(X_t)-f(X_t-))d\hat{N_t}$$

According to page 26 of aybe it should be something like this:
$$df(X_t)=\nu_t f'(X_t)dt+u_t f'(X_t) dW_t+ \frac{1}{2}u^2_t f''(X_t)+\lambda (E[(f(X_t)-f(X_t-)]-E[Z_{N_t}]\eta_tf'(X_t))dt+(f(X_t)-f(X_t-))d\hat{N_t}$$
but I can't understand why.

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: How to make ito's forumula for jump-diffusion a martingale

I think you're making this harder than it needs to be.

If you want $V_t \equiv f(t,X_t)$ to be a martingale, first add a $f_t dt$ term to the 2nd line. (I added a t-dependence because in applications, typically V will be the value of a derivative security with a known value formula at $t=T$, some expiration/maturity). Then, take the $E_{x,t}[ \cdots]$ expectation of the amended second line and set the resulting coefficient of dt to be zero. Here $E_{x,t}[ \cdots]$ is the time-t expectation, conditional on $X_{t-}=x$. Doing so yields

$f_t + \frac{1}{2} u^2 f_{xx} + \nu f_x + \lambda \int (f(t,x+\xi) - f(t,x)) \, q(\xi) \, d \xi = 0$,

where $q(\xi)$ is the jump-size density and subscripts (with the exception of the one on V) denote partial derivatives. Also, to avoid confusion about the meaning of subscripts, here $u = u(t,x)$ and $\nu = \nu(t,x)$, two given functions. If the jump-size density also depends upon $t$, you can write $q(t,\xi)$ instead.

If $f(t,x)$ satisfies this PIDE, then  $V_t \equiv f(t,X_t)$  is a martingale.

As a check, take $dX_t = \nu \, dt + c \, dN_t$, where $\nu$ and $c$ are constants. Suppose $f(t,x)=x$. Since $q(\xi) = \delta(\xi-c)$, using the Dirac delta, the PIDE implies that $\nu + \lambda \, c = 0$, which means that $dX_t = c \, (dN_t - \lambda \, dt)$, as expected.

EdisonCruise
Topic Author
Posts: 118
Joined: September 15th, 2012, 4:22 am

### Re: How to make ito's forumula for jump-diffusion a martingale

Thank you so much Alan. That's my also my understanding of this problem before reading page 26 of http://people.ucalgary.ca/~aswish/JumpProcesses.pdf, based on which after vanishing the $dt$ term, there should be an additional $-E [Z_{N_t}] \eta_t f'(X_t)$  term in the $f(t,x)$ PIDE.

FaridMoussaoui
Posts: 507
Joined: June 20th, 2008, 10:05 am
Location: Genève, Genf, Ginevra, Geneva

### Re: How to make ito's forumula for jump-diffusion a martingale

Go to the basics. There is a chapter on Jump Processes in Shreve book "Stochastic Calculus for Finance II". Very well explained.
For advanced material, check Rama & Tankov book.