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A martingale approach for fractional Brownian motions and related path dependent PDEs. (English) Zbl 1441.60031

Summary: In this paper, we study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the seminal work of B. Dupire [Quant. Finance 19, No. 5, 721–729 (2019; Zbl 1420.91458)] to our more general framework. In particular, unlike in [loc. cit.] where one needs only to consider the stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths. This new feature is due to the time inconsistency involved in this paper. We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.

MSC:

60G22 Fractional processes, including fractional Brownian motion
60H20 Stochastic integral equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K10 Second-order parabolic equations
91G20 Derivative securities (option pricing, hedging, etc.)

Citations:

Zbl 1420.91458
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References:

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