Itô's stochastic calculus: Its surprising power for applications

By Hiroshi Kunita. 2010

152 citations recorded by the source corpus.

We trace Ito's early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Ito's formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Ito's formula in mathematical finance in the 1970s. Throughout the paper, we treat Ito's jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/591c3dacfb6432ab813c2a5887c8f8cde9468bbb

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In Economics and Econometrics

1. St'ephane Cr'epey
2. Tomasz R. Bielecki
3. Damiano Brigo
4. Ke Wang
5. Meng Liu
6. Daniel Sevcovic

In Finance

1. St'ephane Cr'epey
2. Damiano Brigo
3. Tomasz R. Bielecki
4. Ke Wang
5. Meng Liu
6. Agostino Capponi
7. Andrea Pallavicini
8. Monique Jeanblanc

Citations in the corpus

• Counterparty Risk and Funding: A Tale of Two Puzzles (2014, methodology): “Defining pathwise by (13.61) the Qψ-stochastic integral of η in P2loc against the random measure (dnt − ctdt), one then extends in the standard way (see e.g. Kunita (2010)) this notion of a stochastic integral such that ηt · (dnt − ctdt) yields, for any η ∈ Ploc, a welldefined Qψ-local martingale.”
• Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps (2013, background): “, Kunita [21]) that x(t) = eu(t), y(t) = ev(t) is the unique positive local solution to system (3) with initial data x0 > 0, y0 > 0.”