The martingale property in the context of stochastic differential
        equations

Lions and Musiela give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert and Mijatović and Urusov give necessary and sufficient conditions in the case of perfect correlation (ρ = 1). For financial applications, such as checking the martingale property of the stock price process in correlated stochastic volatility models, we extend their work to the arbitrary correlation case (−1 ≤ ρ ≤ 1). We give a complete classification of the convergence properties of both perpetual and capped integral functionals of time-homogeneous diffusions and generalize results in Mijatović and Urusov with direct proofs avoiding the use of separating times (concept introduced by Cherny and Urusov and extensively used in the proofs of Mijatović and Urusov).KEY WORDS: Martingale property, local martingale, stochastic volatility, Engelbert Schmidt zero-one law.

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“…A similar result for the general setting of multidimensional diffusions appears in theorem 1 ofRuf (2013a).…”

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confidence: 62%

On the Martingale Property in Stochastic Volatility Models Based on Time‐homogeneous Diffusions

Bernard

Cui

McLeish

2014

Mathematical Finance

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“…The remainder of this note is devoted to the proofs of Theorems 1 and 2, which we now outline. The proof of Theorem 1 follows the classical argument (found already in the aforementioned [19,15,16], see also [3,18] for additional references) relating the martingale property of stochastic exponentials with explosions of a SDE (in our case, this will be a Volterra SDE). The martingale property (1) is then essentially immediate, while the proof of (2) follows from the fact that the Volterra SDE may blow up in arbitrarily short time with positive probability.…”

Section: Introduction and Main Resultsmentioning

confidence: 76%

On the martingale property in the rough Bergomi model

Gassiat¹

2019

Electron. Commun. Probab.

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We consider a class of fractional stochastic volatility models (including the so-called rough Bergomi model), where the volatility is a superlinear function of a fractional Gaussian process. We show that the stock price is a true martingale if and only if the correlation ρ between the driving Brownian motions of the stock and the volatility is nonpositive. We also show that for each ρ < 0 and m > 1 1−ρ 2 , the m-th moment of the stock price is infinite at each positive time.

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“…In each of these four cases, the local martingale M(·) is a true martingale with expectation equal to M(0) = 1 . This is obvious in the first three cases, and follows from the considerations of [41] or of [50] in the last case. Thus, we may define a probability measure Q (T ) on F(T ) via the recipe dQ (T ) /dP o ξ = M(T ).…”

Section: Proposition 44 (The Explosion Time Distribution Is Not Suppmentioning

confidence: 70%

Distribution of the time to explosion for one-dimensional diffusions

Karatzas

Ruf

2015

Probab. Theory Relat. Fields

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We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to the computation of expectations of suitable nonnegative local martingales. Moreover, we characterize the distribution function of the time to explosion as the minimal solution to a certain Cauchy problem for an appropriate parabolic differential equation; this leads to alternative characterizations of Feller's criterion for explosions. We discuss in detail several examples for which To the Memory of Marc Yor. for their insightful remarks that improved the manuscript. We thank Alex Mijatović for raising the issues of strict positivity and full support and for prompting us to think about them (Sects. 4.2 and 4.3, respectively). We are grateful to the referees for their meticulous reading of the paper and their many and incisive suggestions. J.R. acknowledges generous support from the Oxford-Man Institute of Quantitative Finance, University of Oxford, where a major part of this work was completed. The problem discussed here was suggested to us by Marc Yor, who also gave us generous expert advice on several of its aspects. We dedicate the paper to his memory with deep sorrow at his passing. I.

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The martingale property in the context of stochastic differential equations

Cited by 29 publications

References 25 publications

On the Martingale Property in Stochastic Volatility Models Based on Time‐homogeneous Diffusions

On the Martingale Property in Stochastic Volatility Models Based on Time‐homogeneous Diffusions

On the martingale property in the rough Bergomi model

Distribution of the time to explosion for one-dimensional diffusions

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