scholarly journals Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Wenqing Hu
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Closed Form Expression ◽  
Itô’S Formula ◽  
Change Of Measure ◽  
Quantum Time ◽  
Itô's Formula ◽  
Ito’S Formula ◽  
General Time ◽  
Ito's Formula

We provide an Itô formula for stochastic dynamical equation on general time scales. Based on this Itô’s formula we give a closed-form expression for stochastic exponential on general time scales. We then demonstrate Girsanov’s change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale (q-time scale).

A remark on the proof of Itô's formula for C2 functions of continuous semimartingales

1992 ◽  
Vol 29 (01) ◽  
pp. 216-221
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Fundamental Theorem ◽  
Short Note ◽  
Stochastic Calculus ◽  
Itô’S Formula ◽  
Itô Formula ◽  
Ito Formula ◽  
Multivariate Case ◽  
Itô's Formula ◽  
Ito’S Formula ◽  
Ito's Formula

The Itô formula is the fundamental theorem of stochastic calculus. This short note presents a new proof of Itô's formula for the case of continuous semimartingales. The new proof is more geometric than previous approaches, and has the particular advantage of generalizing immediately to the multivariate case without extra notational complexity.

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