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Existence and stability of measure solutions for BSDE with generators of quadratic growth

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Alexander Fromm, Peter Imkeller, Jianing Zhang

Probability · Submitted 15 Oct 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We derive the existence of measure solutions in scenarios in which the generating functions are just continuous, of at most linear growth in the control variable (corresponding to generators of at most quadratic growth in the usual sense), and with a random bound in the time parameter whose stochastic integral is a BMO martingale.

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arXiv ID1310.4041 (version 1)
Pith IDZDNUBAC6, Pith's permanent record of this paper and its review
CitationAlexander Fromm, Jianing Zhang, Peter Imkeller. "Existence and stability of measure solutions for BSDE with generators of quadratic growth." arXiv:1310.4041 (2013). https://pith.science/paper/ZDNUBAC6
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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.45 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.16 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; .

With an emphasis on generators with quadratic growth in the control variable we consider measure solutions of BSDE, a solution concept corresponding to the notion of risk neutral measure in mathematical finance. In terms of measure solutions, solving a BSDE reduces to martingale representation with respect to an underlying filtration. Measure solutions related to measures equivalent to the historical one provide classical solutions. We derive the existence of measure solutions in scenarios in which the generating functions are just continuous, of at most linear growth in the control variable (corresponding to generators of at most quadratic growth in the usual sense), and with a random bound in the time parameter whose stochastic integral is a BMO martingale. Our main tools include a stability property of sequences of measure solutions, for which a limiting solution is obtained by means of the weak convergence of measures.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 With an emphasis on generators with quadratic growth in the control variable we consider measure solutions of BSDE, a solution concept corresponding to the notion of risk neutral measure in mathematical finance. contribution 0.55 0.64 0.06 asserted 0.81
s1 In terms of measure solutions, solving a BSDE reduces to martingale representation with respect to an underlying filtration. background 0.77 0.16 0.06 asserted 1.00
s2 Measure solutions related to measures equivalent to the historical one provide classical solutions. background 0.57 0.14 0.05 asserted 1.00
s3 We derive the existence of measure solutions in scenarios in which the generating functions are just continuous, of at most linear growth in the control variable (corresponding to generators of at most quadratic growth in the usual sense), and with a random bound in the time parameter whose stochastic integral is a BMO martingale. result 0.92 0.95 0.60 asserted 1.00
s4 Our main tools include a stability property of sequences of measure solutions, for which a limiting solution is obtained by means of the weak convergence of measures. method 0.99 0.92 0.11 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original abstract

With an emphasis on generators with quadratic growth in the control variable we consider measure solutions of BSDE, a solution concept corresponding to the notion of risk neutral measure in mathematical finance. In terms of measure solutions, solving a BSDE reduces to martingale representation with respect to an underlying filtration. Measure solutions related to measures equivalent to the historical one provide classical solutions. We derive the existence of measure solutions in scenarios in which the generating functions are just continuous, of at most linear growth in the control variable (corresponding to generators of at most quadratic growth in the usual sense), and with a random bound in the time parameter whose stochastic integral is a BMO martingale. Our main tools include a stability property of sequences of measure solutions, for which a limiting solution is obtained by means of the weak convergence of measures.

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