MHB Integral Notation of an exponential Brownian motion

AI Thread Summary
The discussion focuses on the integral notation of exponential Brownian motion, specifically comparing two forms of the integral: one with the differential element $ds$ preceding the exponential term and the other with it following. It is clarified that both notations are equivalent in meaning. References to quantum mechanics texts are provided as examples where similar notation is used. Additionally, the scaling property of Brownian motion is mentioned, emphasizing that the transformation maintains the standard properties of Brownian motion. Understanding these notations is essential for grasping the mathematical framework in stochastic processes.
gnob
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Good day!

I am reading the paper of Marc Yor (www.jstor.org/stable/1427477). equation (1.a) seems unfamiliar to me since the $ds$ comes first before the exponential part;
$$
\int_0^t ds \exp(aB_s + bs).
$$
Can you please help me clarify if there is a difference with the above notation as compared to if I write it this way:
$$
\int_0^t \exp(aB_s + bs) ds.
$$
Please give me some reference (books) on this. thanks

Secondly, how does the scaling property applied to (1.a) to become
$$
\int_0^t ds \exp 2(B_s + vs).
$$
Thanks a lot for your response. I know that the Brownian scaling states that if $B_s$ is a standard Brownian motion, then $\sqrt{c}B_{cs}$ is also a standard Brownian motion.
 
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gnob said:
Good day!

I am reading the paper of Marc Yor (www.jstor.org/stable/1427477). equation (1.a) seems unfamiliar to me since the $ds$ comes first before the exponential part;
$$
\int_0^t ds \exp(aB_s + bs).
$$
Can you please help me clarify if there is a difference with the above notation as compared to if I write it this way:
$$
\int_0^t \exp(aB_s + bs) ds.
$$
Please give me some reference (books) on this. thanks


Hi gnob, :)

Yes they do mean the same thing. I have seen this notation used in quantum mechanics books such as,

1) Modern Quantum Mechanics by J. Sakurai

2) Quantum Physics by S. Gasiorowicz

3) Quantum Mechanics by C.C. Tannoudji

Also a brief description about the two notations can be found >>here<<.

Kind Regards,
Sudharaka.
 
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