Convergence of integral functionals of one-dimensional
        diffusions

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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“…Next, we note that ∞ 0 h Y (θ ) dθ = ∞ holds almost surely, by Theorem 2.10(ii) in [40]; to apply this result, use…”

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

confidence: 99%

Distribution of the time to explosion for one-dimensional diffusions

Karatzas

Ruf

2015

Probab. Theory Relat. Fields

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“…Engelbert and Tittel () obtain a generalized Engelbert–Schmidt type zero‐one law for the integral functional

\int_{0}^{t} f false(X_{s} false) d s

, where f is a nonnegative Borel function and X is a strong Markov continuous local martingale. In an expository paper, Mijatović and Urusov () consider the case of a one‐dimensional time‐homogeneous diffusion and the zero‐one law is given in their theorem 2.11. They provide two proofs that circumvent the use of Jeulin's lemma .…”

Section: Classification Of Convergence Properties Of Integral Functiomentioning

confidence: 99%

“…Define

\tilde{s} false(\cdot false)

\tilde{v} false(\cdot false)

, and

{\overset{v}{true}}_{b} false(\cdot false)

similarly based on the SDE under

\overset{P}{true}

. Throughout this section, we assume that

λ (x \in false(ℓ, r false) : b^{2} false(x false) > 0) > 0

, which is assumed in Mijatović and Urusov ().…”

Section: Classification Of Convergence Properties Of Integral Functiomentioning

confidence: 99%

“…We have the following Engelbert–Schmidt type zero‐one law for the SDE under P , which is theorem 2.11 of Mijatović and Urusov () with

f false(\cdot false) = b^{2} false(\cdot false)

using our notation. Proposition (Engelbert–Schmidt type zero‐one law for a time‐homogeneous diffusion, theorem 2.11 of Mijatović and Urusov ) . Assume conditions , and

s (r) < \infty

.…”

Section: Classification Of Convergence Properties Of Integral Functiomentioning

confidence: 99%

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

Bernard

Cui

McLeish

2014

Mathematical Finance

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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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“…where Q x is the law of a 3-dimensional Bessel process starting at x. Mijatovic and Urusov show in Theorem 2.11 [19] that Q x (A ∞ < ∞) = 1 (resp. = 0) if…”

mentioning

confidence: 99%

Path transformations for local times of one-dimensional diffusions

Çetın

2018

Stochastic Processes and their Applications

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Abstract. Let X be a regular one-dimensional transient diffusion and L y be its local time at y. The stochastic differential equation (SDE) whose solution corresponds to the process X conditioned on [L y ∞ = a] for a given a ≥ 0 is constructed and a new path decomposition result for transient diffusions is given. In the course of the construction Bessel-type motions as well as their SDE representations are studied. Moreover, the Engelbert-Schmidt theory for the weak solutions of one dimensional SDEs is extended to the case when the initial condition is an entrance boundary for the diffusion. This extension was necessary for the construction of the Bessel-type motion which played an essential part in the SDE representation of X conditioned on [L y ∞ = a].

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Convergence of integral functionals of one-dimensional diffusions

Cited by 31 publications

References 21 publications

Distribution of the time to explosion for one-dimensional diffusions

Distribution of the time to explosion for one-dimensional diffusions

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

Path transformations for local times of one-dimensional diffusions

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