MHB Brownian Motion: Martingale Property

AI Thread Summary
The discussion revolves around proving that the process defined as M_t = φ(X_t) is a martingale, where X_t is a Brownian motion with drift. Participants clarify the role of T_r, which is not essential for the initial proof but may be relevant later. The conversation includes steps to demonstrate that M_t is adapted to the filtration, that its expected value is finite, and how to establish the martingale property through conditional expectations. Key points involve the treatment of the exponential terms and the independence of certain components in the expectation calculations. The thread concludes with a collaborative effort to understand the mathematical properties necessary for the proof.
mathmari
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Hi!
I need some help at the following exercise...
Let $$B$$ be a typical brownian motion with $$μ>0$$ and $$x$$ ε $$R$$. $$ X_{t}:=x+B_{t}+μt$$, for each $$t>=0$$, a brownian motion with velocity $$μ$$ that starts at $$x$$. For $$r$$ ε $$R$$, $$T_{r}$$:=inf{$$s>=0:X_{s}=r$$} and $$φ(r):=exp(-2μr)$$. Show that $$M_{t}:=φ(X_{t})$$ for t>=0 is martingale.

Could you tell me the purpose of $$T_{r}$$??
 
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Hi!

The definition of $T_r$ will be probably used latter, but it seems it's not needed to solve the first question. Did you manage to do it?
 
To show that Mt is martingale, I have to show that:
1. Mt is adapted to the filtration {Ft}t>=0
2. For every t>=0, E(|Mt|)<oo
3. E(Mt|Fs)=Ms, for every 0<=s<=t
Right?

For the property 2. what I've done until now is:

E(|e-2μ(x+Bt+μt|)=E(|e-2μ(x+Bt+Bs-Bs+μt)|)=E(|e-2μ(x+Bs)e-2μ(Bt-Bs)e-2$$μ^2$$t|)=e-2μ(x+Bs)E(|e-2μ(Bt-Bs)|)E(|e-2$$μ^2$$t|).

Since B is a typical brownian motion, is the mean value E(|e-2μ(Bt-Bs)|) equal to e0=1?
 
I don't understand why you put $e^{-2\mu(x+B_s)}$ outside the expectation.

To see that $E(|M_t|)$ is finite for each $t$, it's enough to show that $E(e^{-2\mu B_t})$ is finite. Note that $B_t$ is normally distributed, hence the term $e^{-2\mu x}$ in the computation of the expectation should not be problematic.
 
I put $$e^{-2μ(x+B_s)}$$ outside the expectation, because I thought that since it has no t, it's like a constant...
And how can I show the property 3? :confused:
 
mathmari said:
I put $$e^{-2μ(x+B_s)}$$ outside the expectation, because I thought that since it has no t, it's like a constant...
:confused:

The problem is that what you get is a random variable, whereas an expectation should be a (deterministic number).

mathmari said:
And how can I show the property 3?
You can use the idea you used for point 2 (it was not needed there, but it is now). Indeed, one can write $M_t=\exp(-2\mu(x+B_t+\mu t))=e^{-2\mu(x+\mu t) }\exp(-2\mu(B_t-B_s))\exp(-2\mu B_s)$. Computing the conditional expectation of this term goes like that: $\exp(-2\mu(B_t-B_s))$ is independent of $\mathcal F_s$ and $\exp(-2\mu B_s)$ is $\mathcal F_s$-measurable.
 
Ok... Thank you very much! ;)
 
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