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Omar
Posts: 1552
Joined: August 27th, 2001, 12:17 pm

### What are the forward and backward equations?

FP now applies??? You've changed your mind again??? Stop wasting my time and everyone else's with your garbage. This is the very last time that I will respond to anything you say.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### What are the forward and backward equations?

The forward evolution of the pdf is no problem (forward FP). However, assuming aWiener process and forming a PDE (and calling it FP) is presumptuous at best.

WaaghBakri
Posts: 732
Joined: March 21st, 2002, 4:07 am

### What are the forward and backward equations?

Omar, Above, I started off by stating that GIVEN, dX = m(X,t) * dt + v(X,t) * dBthe Ito & Stratonovich interpretations of this sde will result in different FP eqns. The "culprit" is this statement. Ito and Stratonovich interpretations of the above equation will result in two different processes, which differ in the drift rate. And, therefore it should be no surprise if the FP eqns are different, after all the underlying processes are different. So if we want to discuss Ito & Stratonovich interpretations of a GIVEN process (as opposed to a sde), then one should state & consider sde's with drift adjusted: Stratonovich interpretation of: dX = m * dt + v * dBand Ito interpretation of: dX = [m - v v'/2] * dt + v * dBwill result in the SAME FP eqn. Above v' = dv/dx. Now we are comparing the same process, apples & apples, and not apples & oranges. My apologies for the confusion...

HA
Posts: 117
Joined: February 1st, 2003, 7:11 pm

### What are the forward and backward equations?

So the issue is similar to whether you want to write a PDE in a cannoical form or not?If you want to talk about regularity conditions, existence, uniqueness, ... , it is better off starting from theIto convention, foresaking the intuition of differential geometry. The solution exists for a larger class of coefficients for Ito convention. For example, in Stratonovich convention, you'll need a regularity on v' as well.

HA
Posts: 117
Joined: February 1st, 2003, 7:11 pm

### What are the forward and backward equations?

I guess I stepped in.A be a continuous semigroup: A(t+s) = < A(t), A(s) > for each non-negative s & t 1) A(t+s) - A(t) = < A(s) - I, A(t) > yields the forward equation (d/dt) A(t) = L * A(t), where L is the infinitesmal generator of A 2) 0 = (d/dt) < A(t), A(T-t) > : being (d/dt) A(T) = < (d/dt) A(t), A(T-t) > + < A(t), (d/dt) A(T-t) > : Leibnitz = < L * A(t), A(T-t) > + < A(t), (d/dt) A(T-t) > : by forward eq = < A(t), R * A(T-t) > + < A(t), (d/dt) A(T-t) > : integration by parts, where R is the adjoint of L, and conclude the backward eq (d/dt) A(T-t) + R * A(T-t) = 03) If the limit exists when t tends to infinity, the forward eq becomes time stationary: 0 = L * A(oo) which is often called a steady state eq.For Markov process (Feller type?), the regular conditional expectation form a continuoussemigroup. L can be either uniformly elliptic differential operator or an integro-differentialoperator.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### What are the forward and backward equations?

HA,Log-normal assumptions always lead to Levy distributions. FP doesn't. Can you explain this?Is the noise in log-normal process independent? Why is this not true.Your pal,N

HA
Posts: 117
Joined: February 1st, 2003, 7:11 pm

### What are the forward and backward equations?

newton, you are a master of disguise indeed.Elaborate your question on FP.The noise (returns) in plain BS is iid. The log-normal model in general isnot necessarily of iid returns. For example, you can model a spot as exponential ofa general Gaussian process rather than exponential of a standard BM.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### What are the forward and backward equations?

HA,Your answer allowed me to get the final piece of the puzzle: The optimum filter for any noise. Itmust have two constraints, no more, no less, just like spin.I also found that all the comments regarding forward and backward equations are equivalent.On to the next chapter, Fixing Interest Rate Models.

HA
Posts: 117
Joined: February 1st, 2003, 7:11 pm

### What are the forward and backward equations?

Newton, How about fixing your blender first, so that I can switch on.

Nonius
Posts: 5115
Joined: January 22nd, 2003, 6:48 am

### What are the forward and backward equations?

QuoteOriginally posted by: HAI guess I stepped in.A be a continuous semigroup: A(t+s) = < A(t), A(s) > for each non-negative s & t 1) A(t+s) - A(t) = < A(s) - I, A(t) > yields the forward equation (d/dt) A(t) = L * A(t), where L is the infinitesmal generator of A 2) 0 = (d/dt) < A(t), A(T-t) > : being (d/dt) A(T) = < (d/dt) A(t), A(T-t) > + < A(t), (d/dt) A(T-t) > : Leibnitz = < L * A(t), A(T-t) > + < A(t), (d/dt) A(T-t) > : by forward eq = < A(t), R * A(T-t) > + < A(t), (d/dt) A(T-t) > : integration by parts, where R is the adjoint of L, and conclude the backward eq (d/dt) A(T-t) + R * A(T-t) = 03) If the limit exists when t tends to infinity, the forward eq becomes time stationary: 0 = L * A(oo) which is often called a steady state eq.For Markov process (Feller type?), the regular conditional expectation form a continuoussemigroup. L can be either uniformly elliptic differential operator or an integro-differentialoperator.your fourth equality under 2 follows by defintion of "adjointness", but thanks for the info.