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- W4313372246 abstract "Abstract We introduce some families of generalized Black–Scholes equations which involve the Riemann–Liouville and Weyl space-fractional derivatives. We prove that these generalized Black–Scholes equations are well-posed in $$(L^1-L^infty )$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -interpolation spaces. More precisely, we show that the elliptic-type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial-like operators and sectorial operators in an abstract setting. Such a connection extends the scaling property of sectorial operators to a wider family of both operators and the functions involved." @default.
- W4313372246 created "2023-01-06" @default.
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- W4313372246 date "2022-12-30" @default.
- W4313372246 modified "2023-10-13" @default.
- W4313372246 title "Introducing and solving generalized Black–Scholes PDEs through the use of functional calculus" @default.
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- W4313372246 doi "https://doi.org/10.1007/s00028-022-00858-8" @default.
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