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Martin Schweizer - One of the best experts on this subject based on the ideXlab platform.
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strong bubbles and strict local Martingales
International Journal of Theoretical and Applied Finance, 2016Co-Authors: Martin Herdegen, Martin SchweizerAbstract:In a numeraire-independent framework, we study a financial market with N assets which are all treated in a symmetric way. We define the fundamental value ∗S of an asset S as its super-replication price and say that the market has a strong bubble if ∗S and S deviate from each other. None of these concepts needs any mention of Martingales. Our main result then shows that under a weak absence-of-arbitrage assumption (basically NUPBR), a market has a strong bubble if and only if in all numeraire s for which there is an equivalent local martingale measure (ELMM), asset prices are strict local Martingales under all possible ELMMs. We show by an example that our bubble concept lies strictly between the existing notions from the literature. We also give an example where asset prices are strict local Martingales under one ELMM, but true Martingales under another, and we show how our approach can lead naturally to endogenous bubble birth.
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a minimality property of the minimal martingale measure
Statistics & Probability Letters, 1999Co-Authors: Martin SchweizerAbstract:Let X be a continuous adapted process for which there exists an equivalent local martingale measure (ELMM). The minimal martingale measure P is the unique ELMM for X with the property that local P-Martingales strongly orthogonal to the P-martingale part of X are also local P-Martingales. We prove that if P exists, it minimizes the reverse relative entropy H(P|Q) over all ELMMs Q for X. A counterexample shows that the assumption of continuity cannot be dropped.
Martin Herdegen - One of the best experts on this subject based on the ideXlab platform.
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strict local Martingales and optimal investment in a black scholes model with a bubble
2017Co-Authors: Martin Herdegen, Sebastian HerrmannAbstract:There are two major streams of literature on the modeling of financial bubbles: the strict local martingale framework and the Johansen-Ledoit-Sornette (JLS) financial bubble model. Based on a class of models that embeds the JLS model and can exhibit strict local martingale behavior, we clarify the connection between these previously disconnected approaches. While the original JLS model is never a strict local martingale, there are relaxations which can be strict local Martingales and which preserve the key assumption of a log-periodic power law for the hazard rate of the time of the crash. We then study the optimal investment problem for an investor with constant relative risk aversion in this model. We show that for positive instantaneous expected returns, investors with relative risk aversion above one always ride the bubble.
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strong bubbles and strict local Martingales
International Journal of Theoretical and Applied Finance, 2016Co-Authors: Martin Herdegen, Martin SchweizerAbstract:In a numeraire-independent framework, we study a financial market with N assets which are all treated in a symmetric way. We define the fundamental value ∗S of an asset S as its super-replication price and say that the market has a strong bubble if ∗S and S deviate from each other. None of these concepts needs any mention of Martingales. Our main result then shows that under a weak absence-of-arbitrage assumption (basically NUPBR), a market has a strong bubble if and only if in all numeraire s for which there is an equivalent local martingale measure (ELMM), asset prices are strict local Martingales under all possible ELMMs. We show by an example that our bubble concept lies strictly between the existing notions from the literature. We also give an example where asset prices are strict local Martingales under one ELMM, but true Martingales under another, and we show how our approach can lead naturally to endogenous bubble birth.
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Single jump processes and strict local Martingales
Stochastic Processes and their Applications, 2016Co-Authors: Martin Herdegen, Sebastian HerrmannAbstract:Abstract Many results in stochastic analysis and mathematical finance involve local Martingales. However, specific examples of strict local Martingales are rare and analytically often rather unhandy. We study local Martingales that follow a given deterministic function up to a random time γ at which they jump and stay constant afterwards. The (local) martingale properties of these single jump local Martingales are characterised in terms of conditions on the input parameters. This classification allows an easy construction of strict local Martingales, uniformly integrable Martingales that are not in H 1 , etc. As an application, we provide a construction of a (uniformly integrable) martingale M and a bounded (deterministic) integrand H such that the stochastic integral H • M is a strict local martingale.
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A class of strict local Martingales
SSRN Electronic Journal, 2014Co-Authors: Martin Herdegen, Sebastian HerrmannAbstract:Many results in stochastic analysis and mathematical finance involve local Martingales. However, specific examples of strict local Martingales are rare and analytically often rather unhandy. We study local Martingales that follow a given deterministic function up to a random time γ at which they jump and stay constant afterwards. The (local) martingale properties of these single jump local Martingales are characterised in terms of conditions on the input parameters. This classification allows an easy construction of strict local Martingales, uniformly integrable Martingales that are not in H¹, etc. As an application, we provide a construction of a (uniformly integrable) martingale M and a bounded (deterministic) integrand H such that the stochastic integral H • M is a strict local martingale. Moreover, we characterise all local martingale deflators and all equivalent local martingale measures for a given special semimartingale with respect to the smallest filtration that turns γ into a stopping time. Two new counter-examples show, using direct arguments only, that neither of the no-arbitrage conditions NA and NUPBR implies the other. The structural simplicity of these examples allows to understand the difference between NA and NUPBR on an intuitive level.
Khalid Tahri - One of the best experts on this subject based on the ideXlab platform.
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Representation theorem of set valued regular martingale: Application to the convergence of set valued martingale
Statistics & Probability Letters, 2019Co-Authors: Fatima Ezzaki, Khalid TahriAbstract:Abstract In this paper, we prove a new theorem concerning a representation of set valued regular Martingales, the proof is based on martingale selectors approach. As applications, various convergence results of set valued Martingales are provided.
Peter Auer - One of the best experts on this subject based on the ideXlab platform.
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pac bayesian inequalities for Martingales
IEEE Transactions on Information Theory, 2012Co-Authors: Yevgeny Seldin, Francois Laviolette, Nicolo Cesabianchi, John Shawetaylor, Peter AuerAbstract:We present a set of high-probability inequalities that control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent Martingales. Our results extend the PAC-Bayesian (probably approximately correct) analysis in learning theory from the i.i.d. setting to Martingales opening the way for its application to importance weighted sampling, reinforcement learning, and other interactive learning domains, as well as many other domains in probability theory and statistics, where Martingales are encountered. We also present a comparison inequality that bounds the expectation of a convex function of a martingale difference sequence shifted to the [0, 1] interval by the expectation of the same function of independent Bernoulli random variables. This inequality is applied to derive a tighter analog of Hoeffding-Azuma's inequality.
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pac bayesian inequalities for Martingales
Uncertainty in Artificial Intelligence, 2012Co-Authors: Yevgeny Seldin, Francois Laviolette, Nicolo Cesabianchi, John Shawetaylor, Peter AuerAbstract:We present a set of high-probability inequalities that control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent Martingales. Our results extend the PAC-Bayesian analysis in learning theory from the i.i.d. setting to Martingales opening the way for its application in reinforcement learning and other interactive learning domains, as well as many other domains in probability theory and statistics, where Martingales are encountered. We also present a comparison inequality that bounds the expectation of a convex function of a martingale difference sequence shifted to the [0,1] interval by the expectation of the same function of independent Bernoulli variables. This inequality is applied to derive a tighter analog of Hoeffding-Azuma's inequality. For the complete paper see Seldin et al. (2012).
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pac bayesian inequalities for Martingales
arXiv: Learning, 2011Co-Authors: Yevgeny Seldin, Francois Laviolette, Nicolo Cesabianchi, John Shawetaylor, Peter AuerAbstract:We present a set of high-probability inequalities that control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent Martingales. Our results extend the PAC-Bayesian analysis in learning theory from the i.i.d. setting to Martingales opening the way for its application to importance weighted sampling, reinforcement learning, and other interactive learning domains, as well as many other domains in probability theory and statistics, where Martingales are encountered. We also present a comparison inequality that bounds the expectation of a convex function of a martingale difference sequence shifted to the [0,1] interval by the expectation of the same function of independent Bernoulli variables. This inequality is applied to derive a tighter analog of Hoeffding-Azuma's inequality.
Sebastian Herrmann - One of the best experts on this subject based on the ideXlab platform.
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strict local Martingales and optimal investment in a black scholes model with a bubble
2017Co-Authors: Martin Herdegen, Sebastian HerrmannAbstract:There are two major streams of literature on the modeling of financial bubbles: the strict local martingale framework and the Johansen-Ledoit-Sornette (JLS) financial bubble model. Based on a class of models that embeds the JLS model and can exhibit strict local martingale behavior, we clarify the connection between these previously disconnected approaches. While the original JLS model is never a strict local martingale, there are relaxations which can be strict local Martingales and which preserve the key assumption of a log-periodic power law for the hazard rate of the time of the crash. We then study the optimal investment problem for an investor with constant relative risk aversion in this model. We show that for positive instantaneous expected returns, investors with relative risk aversion above one always ride the bubble.
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Single jump processes and strict local Martingales
Stochastic Processes and their Applications, 2016Co-Authors: Martin Herdegen, Sebastian HerrmannAbstract:Abstract Many results in stochastic analysis and mathematical finance involve local Martingales. However, specific examples of strict local Martingales are rare and analytically often rather unhandy. We study local Martingales that follow a given deterministic function up to a random time γ at which they jump and stay constant afterwards. The (local) martingale properties of these single jump local Martingales are characterised in terms of conditions on the input parameters. This classification allows an easy construction of strict local Martingales, uniformly integrable Martingales that are not in H 1 , etc. As an application, we provide a construction of a (uniformly integrable) martingale M and a bounded (deterministic) integrand H such that the stochastic integral H • M is a strict local martingale.
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A class of strict local Martingales
SSRN Electronic Journal, 2014Co-Authors: Martin Herdegen, Sebastian HerrmannAbstract:Many results in stochastic analysis and mathematical finance involve local Martingales. However, specific examples of strict local Martingales are rare and analytically often rather unhandy. We study local Martingales that follow a given deterministic function up to a random time γ at which they jump and stay constant afterwards. The (local) martingale properties of these single jump local Martingales are characterised in terms of conditions on the input parameters. This classification allows an easy construction of strict local Martingales, uniformly integrable Martingales that are not in H¹, etc. As an application, we provide a construction of a (uniformly integrable) martingale M and a bounded (deterministic) integrand H such that the stochastic integral H • M is a strict local martingale. Moreover, we characterise all local martingale deflators and all equivalent local martingale measures for a given special semimartingale with respect to the smallest filtration that turns γ into a stopping time. Two new counter-examples show, using direct arguments only, that neither of the no-arbitrage conditions NA and NUPBR implies the other. The structural simplicity of these examples allows to understand the difference between NA and NUPBR on an intuitive level.
