The frozen operator has been used to develop Dyson-series like representations for random variables generated by classical Brownian motion, Lévy processes and fractional Brownian with Hurst index greater than 1/2.The relationship between the conditional expectation of a random variable (or fractional conditional expectation in the case of fractional Brownian motion)and that variable's Dyson-series like representation is the exponential formula. These results had not yet been extended to either fractional Brownian motion with Hurst index less than 1/2, or d-dimensional Brownian motion. The former is still out of reach, but we hope our review of stochastic integration for fractional Brownian motion and our results for the later will provide a framework. Examining the case of d-dimensional Brownian motion in general, and two-dimensional Brownian motions in specific, have led to a number of new insights.The first of which, is realizing the component operators in the Dyson-series expansion can be written concisely as iterated applications of the Gross Laplacian.The original choice of \"Dyson-series\" as nomenclature was to suggest some connection between the original results and expressions which occur in quantum field theory. There have always been connections between financial and mathematical physics, and expressing the exponential formula in terms of a foundational operator originally used to study the theory of potentials on Hilbert space suggests another. The second major insight: is to realize the natural domain of the freezing operator on stochastic integrals is asan operator over Stratonovich integrals.Freezing a Skorokhod integral has always been a challenge. The naïve assumption of how they interact is wrong, and there was little intuition in the ensuing calculations.However, the naïve assumption does work for Stratonovich integrals.In retrospect, this is understandable. The mental model for the frozen operator is in terms of realizations of paths. One would expect the integral which emphasizes the geometric nature of stochastic processes to be a more natural fit. These two developments have allowed us to prove an exponential formula for random variables generated by two Brownian motions, and apply those results to the SABR model.There has been work in this field, but the extension from one Brownian motion to two Brownian motions was ad hoc. Our development is more systematic and more readily extended to d-dimensional Brownian motion.
","abstract_html":"<p>The frozen operator has been used to develop Dyson-series like representations for random variables generated by classical Brownian motion, Lévy processes and fractional Brownian with Hurst index greater than 1/2.The relationship between the conditional expectation of a random variable (or fractional conditional expectation in the case of fractional Brownian motion)and that variable's Dyson-series like representation is the exponential formula. These results had not yet been extended to either fractional Brownian motion with Hurst index less than 1/2, or d-dimensional Brownian motion. The former is still out of reach, but we hope our review of stochastic integration for fractional Brownian motion and our results for the later will provide a framework. Examining the case of d-dimensional Brownian motion in general, and two-dimensional Brownian motions in specific, have led to a number of new insights.The first of which, is realizing the component operators in the Dyson-series expansion can be written concisely as iterated applications of the Gross Laplacian.The original choice of "Dyson-series" as nomenclature was to suggest some connection between the original results and expressions which occur in quantum field theory. There have always been connections between financial and mathematical physics, and expressing the exponential formula in terms of a foundational operator originally used to study the theory of potentials on Hilbert space suggests another. The second major insight: is to realize the natural domain of the freezing operator on stochastic integrals is asan operator over Stratonovich integrals.Freezing a Skorokhod integral has always been a challenge. The naïve assumption of how they interact is wrong, and there was little intuition in the ensuing calculations.However, the naïve assumption does work for Stratonovich integrals.In retrospect, this is understandable. The mental model for the frozen operator is in terms of realizations of paths. One would expect the integral which emphasizes the geometric nature of stochastic processes to be a more natural fit. These two developments have allowed us to prove an exponential formula for random variables generated by two Brownian motions, and apply those results to the SABR model.There has been work in this field, but the extension from one Brownian motion to two Brownian motions was ad hoc. Our development is more systematic and more readily extended to d-dimensional Brownian motion.</p>","abstract_has_math":false,"creators":["Baroi, Maximilian Lawrence"],"institution":null,"degree_name":"Mathematics, PhD","degree_level":"Open Access Dissertation","degree_discipline":"Institute of Mathematical Sciences","degree_department":null,"school":null,"contributors":["John Angus","Qidi Peng"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-01T08:00:00Z","date_published":"2022-01-01T08:00:00Z","updated_at":"2026-07-24T01:39:59Z","subjects":["Brownian Motion","Exponential Formula","Isonormal Gaussian Processes","Malliavin Calculus","SABR","Stochastic Analysis","Applied Mathematics","Finance","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarship.claremont.edu/cgu_etd/437","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["John Angus","Qidi Peng"]},{"key":"dc:creator","label":"Author","values":["Baroi, Maximilian Lawrence"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2023-04-19T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Institute of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Mathematics, PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Brownian Motion","Exponential Formula","Isonormal Gaussian Processes","Malliavin Calculus","SABR","Stochastic Analysis","Applied Mathematics","Finance","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarship.claremont.edu/cgu_etd/437"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The frozen operator has been used to develop Dyson-series like representations for random variables generated by classical Brownian motion, Lévy processes and fractional Brownian with Hurst index greater than 1/2.The relationship between the conditional expectation of a random variable (or fractional conditional expectation in the case of fractional Brownian motion)and that variable's Dyson-series like representation is the exponential formula. These results had not yet been extended to either fractional Brownian motion with Hurst index less than 1/2, or d-dimensional Brownian motion. The former is still out of reach, but we hope our review of stochastic integration for fractional Brownian motion and our results for the later will provide a framework. Examining the case of d-dimensional Brownian motion in general, and two-dimensional Brownian motions in specific, have led to a number of new insights.The first of which, is realizing the component operators in the Dyson-series expansion can be written concisely as iterated applications of the Gross Laplacian.The original choice of \"Dyson-series\" as nomenclature was to suggest some connection between the original results and expressions which occur in quantum field theory. There have always been connections between financial and mathematical physics, and expressing the exponential formula in terms of a foundational operator originally used to study the theory of potentials on Hilbert space suggests another. The second major insight: is to realize the natural domain of the freezing operator on stochastic integrals is asan operator over Stratonovich integrals.Freezing a Skorokhod integral has always been a challenge. The naïve assumption of how they interact is wrong, and there was little intuition in the ensuing calculations.However, the naïve assumption does work for Stratonovich integrals.In retrospect, this is understandable. The mental model for the frozen operator is in terms of realizations of paths. One would expect the integral which emphasizes the geometric nature of stochastic processes to be a more natural fit. These two developments have allowed us to prove an exponential formula for random variables generated by two Brownian motions, and apply those results to the SABR model.There has been work in this field, but the extension from one Brownian motion to two Brownian motions was ad hoc. Our development is more systematic and more readily extended to d-dimensional Brownian motion.
"]},{"key":"dc:title","label":"Title","values":["An Exponential Formula for Random Variables Generated by Multiple Brownian Motions"]}]}],"canonical_facts":{"dc:contributor":["John Angus","Qidi Peng"],"dc:creator":["Baroi, Maximilian Lawrence"],"dc:date.available":["2023-04-19T07:00:00Z"],"dc:description.abstract":["The frozen operator has been used to develop Dyson-series like representations for random variables generated by classical Brownian motion, Lévy processes and fractional Brownian with Hurst index greater than 1/2.The relationship between the conditional expectation of a random variable (or fractional conditional expectation in the case of fractional Brownian motion)and that variable's Dyson-series like representation is the exponential formula. These results had not yet been extended to either fractional Brownian motion with Hurst index less than 1/2, or d-dimensional Brownian motion. The former is still out of reach, but we hope our review of stochastic integration for fractional Brownian motion and our results for the later will provide a framework. Examining the case of d-dimensional Brownian motion in general, and two-dimensional Brownian motions in specific, have led to a number of new insights.The first of which, is realizing the component operators in the Dyson-series expansion can be written concisely as iterated applications of the Gross Laplacian.The original choice of \"Dyson-series\" as nomenclature was to suggest some connection between the original results and expressions which occur in quantum field theory. There have always been connections between financial and mathematical physics, and expressing the exponential formula in terms of a foundational operator originally used to study the theory of potentials on Hilbert space suggests another. The second major insight: is to realize the natural domain of the freezing operator on stochastic integrals is asan operator over Stratonovich integrals.Freezing a Skorokhod integral has always been a challenge. The naïve assumption of how they interact is wrong, and there was little intuition in the ensuing calculations.However, the naïve assumption does work for Stratonovich integrals.In retrospect, this is understandable. The mental model for the frozen operator is in terms of realizations of paths. One would expect the integral which emphasizes the geometric nature of stochastic processes to be a more natural fit. These two developments have allowed us to prove an exponential formula for random variables generated by two Brownian motions, and apply those results to the SABR model.There has been work in this field, but the extension from one Brownian motion to two Brownian motions was ad hoc. Our development is more systematic and more readily extended to d-dimensional Brownian motion.
"],"dc:identifier":["https://scholarship.claremont.edu/cgu_etd/437"],"dc:subject":["Brownian Motion","Exponential Formula","Isonormal Gaussian Processes","Malliavin Calculus","SABR","Stochastic Analysis","Applied Mathematics","Finance","Mathematics"],"dc:title":["An Exponential Formula for Random Variables Generated by Multiple Brownian Motions"],"thesis:degree_discipline":["Institute of Mathematical Sciences"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Mathematics, PhD"]},"updated_at":"2026-07-24T01:39:59Z"}