SERVING THE QUANTITATIVE FINANCE COMMUNITY

frame
Topic Author
Posts: 81
Joined: September 27th, 2009, 3:55 pm

### martingale, novikov and time dependent diffusion

Hi all,I would understand how to deal with a martingale process with diffusion depending on time.Assume a process of the form:x(t) = exp( -t B(t) - .5 t^3 )where B(t) is a Brownian motion and t is the time index.Does x(t) is a well defined martingale?Does E_t [ int_t^\infty e^{-r u} x(u) du | F(t)], where F(t) is the filtration generated by B(t), u>t and r>0, converge?thanks

Antonio
Posts: 630
Joined: June 30th, 2004, 3:13 pm
Location: Imperial College London
Contact:

### martingale, novikov and time dependent diffusion

Hello,I am not sure why you want to answer these questions. Anyhow, (i) Does x(t) is a well defined martingale? Check (in a book, lecture notes,...) the definition of a martingale. Since you know that B(t) is Gaussian, this should give you the answer straight away.(ii) Since you have a closed-form representation for x(t), just compute the expectation and check whether it converges or not.These seem to be problem classes to me. The point is that you should try to do them and then ask a precise question about what you do not understand.Best,

frame
Topic Author
Posts: 81
Joined: September 27th, 2009, 3:55 pm

### martingale, novikov and time dependent diffusion

the problem is that for t-->inf, then the diffusion explodes and intuitively i would say that novikov is not satisfied and then martingale property does not obtain.but i'm not sure. because if this is true, then diffusion can only depend on time if their absolute value is decreasing in time, isn't it?this is not a homework and i'm not a student: i don't remember so much about stoch processes since i only use them in very simple cases.i have no time to study this stuff in detail and i have supposed/hoped that some one here was expert enough to give me an answer with minimum effort.

croot
Posts: 104
Joined: July 23rd, 2006, 8:30 pm

### martingale, novikov and time dependent diffusion

I'll try to provide some pointers:1) Your process x(t) has continuous paths2) it is an Ito process which coefficients you can write down3) its expectation is constant in time.4) it's well defined since Bt and exp are well defined5) don't worry about the diffusion exploding, forget novikov, in your case you have strong solution6) there are arbitrage reasons why the process must tend to 0. In all rigor this a well documented argument, confirmed by empirical evidence and stylized facts. For memory, think of dividends. 7)actually, absolute value of a price process can increase in expectation, cf the technical computations of Dupire formulaIt's amazing that you use such sophisticated mathematics in your work. May I ask what kind of finance you are involved in? If you want to go deeper, I can solve this kind of problem all day for a very reasonable wage.

bearish
Posts: 5615
Joined: February 3rd, 2011, 2:19 pm

### martingale, novikov and time dependent diffusion

This process is no more a martingale than any process of the form f(t)B(t) is, for non-constant functions f. In your case E[x(s)| F(t)] is not equal to x(t) for s>t and B(t) not equal to zero. To put it another way, you have B(t)x(t)dt term in the Ito expansion, and that term is not zero most of the time.