Alternative Forms of Compound Fractional Poisson Processes

Table 2

Main results in asymptotic domain.

Process	Hypothesis on jumps	Limiting process	Limiting equation
TFCPP 𝑌𝛽	𝑋^∗𝑗	𝒮𝛼(ℒ𝛽(𝑡))	𝜕^𝛽𝑢𝜕𝑡^𝛽𝜕=𝜆^𝛼𝑢𝜕|𝑦|^𝛼
"	𝑋𝑗^(𝛼)	𝒜𝛼(ℒ𝛽(𝑡))	𝜕^𝛽𝑢𝜕𝑡^𝛽𝜕=−𝜆^𝛼𝑢𝜕𝑦^𝛼
Altern. TFCPP 𝑌𝛽	𝑋^∗𝑗	𝒮^𝛽𝛼(𝑡)	𝜕𝑢=𝜆𝜕𝑡𝛽^1/𝛽𝜕^𝛼𝑢𝜕|𝑦|^𝛼
"	𝑋𝑗^(𝛼)	𝒜𝛼^𝜆,𝜉(𝑡)	𝜕𝑢𝜆𝜕𝑡=−𝜉𝜕^𝛼𝑢𝜕𝑦^𝛼
ΔFCPP 𝑌Δ	𝑋^∗𝑗	𝒮𝛼𝛾(𝑡)	𝜕𝑢𝜕𝑡=𝜆^𝛾𝜕^𝛼𝛾𝑢𝜕|𝑦|^𝛼𝛾
"	𝑋𝑗^(𝛼)	𝒜^𝜆,𝜉𝛼𝛾(𝑡)	𝜕𝑢𝜆𝜕𝑡=−^𝛾𝜉𝜕^𝛼𝛾𝑢𝜕𝑦^𝛼𝛾