Local martingale and pathwise solutions for an abstract fluids model

We consider stochastic suppression and stabilization for nonlinear delay differential system. The system is assumed to satisfy local Lipschitz condition and one-side polynomial growth condition. Since the system may explode in a finite time, we stochastically perturb this system by introducing independent Brownian noises and Lévy noise feedbacks. The contributions of this paper are as follows. (a) We show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. (b) Using the exponential martingale inequality with jumps, we discuss the fact that the sample Lyapunov exponent is nonpositive. (c) Considering linear Lévy processes, by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stability are investigated in Theorem 13.

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INEFFICIENT BUBBLES AND EFFICIENT DRAWDOWNS IN FINANCIAL MARKETS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024920500478 ◽

2020 ◽

Vol 23 (07) ◽

pp. 2050047 ◽

Cited By ~ 1

Author(s):

MICHAEL SCHATZ ◽

DIDIER SORNETTE

Keyword(s):

Rational Expectations ◽

Rational Expectation ◽

Asset Price ◽

Theoretical Background ◽

Local Martingale ◽

Management Problem ◽

Market Failures ◽

Asset Price Bubbles ◽

Price Bubbles ◽

Hyman Minsky

At odds with the common “rational expectations” framework for bubbles, economists like Hyman Minsky, Charles Kindleberger and Robert Shiller have documented that irrational behavior, ambiguous information or certain limits to arbitrage are essential drivers for bubble phenomena and financial crises. Following this understanding that asset price bubbles are generated by market failures, we present a framework for explosive semimartingales that is based on the antagonistic combination of (i) an excessive, unstable pre-crash process and (ii) a drawdown starting at some random time. This unifying framework allows one to accommodate and compare many discrete and continuous time bubble models in the literature that feature such market inefficiencies. Moreover, it significantly extends the range of feasible asset price processes during times of financial speculation and frenzy and provides a strong theoretical background for future model design in financial and risk management problem settings. This conception of bubbles also allows us to elucidate the status of rational expectation bubbles, which, by design, suffer from the paradox that a rational market should not allow for misvaluation. While the discrete time case has been extensively discussed in the literature and is most criticized for its failure to comply with rational expectations equilibria, we argue that this carries over to the finite time “strict local martingale”-approach to bubbles.

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The identification problem for BSDEs driven by possibly non-quasi-left-continuous random measures

Stochastics and Dynamics ◽

10.1142/s0219493720400110 ◽

2020 ◽

Vol 20 (06) ◽

pp. 2040011

Author(s):

Elena Bandini ◽

Francesco Russo

Keyword(s):

Random Measure ◽

Jump Process ◽

Identification Problem ◽

Local Martingale ◽

Random Measures ◽

Underlying Process ◽

Process Solution ◽

Deterministic Function ◽

Singular Coefficients

In this paper, we focus on the so-called identification problem for a BSDE driven by a continuous local martingale and a possibly non-quasi-left-continuous random measure. Supposing that a solution [Formula: see text] of a BSDE is such that [Formula: see text] where [Formula: see text] is an underlying process and [Formula: see text] is a deterministic function, solving the identification problem consists in determining [Formula: see text] and [Formula: see text] in terms of [Formula: see text]. We study the over-mentioned identification problem under various sets of assumptions and we provide a family of examples including the case when [Formula: see text] is a non-semimartingale jump process solution of an SDE with singular coefficients.

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Optimal estimation for semimartingales

Journal of Applied Probability ◽

10.1017/s0021900200029703 ◽

1986 ◽

Vol 23 (02) ◽

pp. 409-417 ◽

Cited By ~ 1

Author(s):

A. Thavaneswaran ◽

M. E. Thompson

Keyword(s):

Continuous Time ◽

Asymptotic Properties ◽

Optimal Estimation ◽

Parametric Estimation ◽

Local Martingale ◽

Regularity Conditions ◽

Probability Measures ◽

Simple Process ◽

Special Cases ◽

Least Squares Estimates

This paper extends a result of Godambe's theory of parametric estimation for discrete-time stochastic processes to the continuous-time case. LetP={P} be a family of probability measures such that (Ω,F, P) is complete, (Ft, t≧0) is a standard filtration, andX = (XtFt, t ≧ 0)is a semimartingale for everyP ∈ P. For a parameterθ(Ρ), supposeXt=Vt,θ+Ht,θwhere theVθprocess is predictable and locally of bounded variation and theHθprocess is a local martingale. Consider estimating equations forθof the formprocess is predictable. Under regularity conditions, an optimal form forαθin the sense of Godambe (1960) is determined. WhenVt,θis linear inθthe optimal, corresponds to certain maximum likelihood or least squares estimates derived previously in special cases. Asymptotic properties of, are discussed.

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Comparison Theorem for any Solutions of Backward Stochastic Differential Equations and its Application

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.524-527.3801 ◽

2012 ◽

Vol 524-527 ◽

pp. 3801-3804

Author(s):

Shi Yu Li ◽

Wu Jun Gao ◽

Jin Hui Wang

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Comparison Theorem ◽

Backward Stochastic Differential Equations ◽

Stochastic Equations ◽

Local Martingale ◽

One Dimensional ◽

Lipschitz Continuous ◽

The One ◽

Uniformly Lipschitz

ƒIn this paper, we study the one-dimensional backward stochastic equations driven by continuous local martingale. We establish a generalized the comparison theorem for any solutions where the coefficient is uniformly Lipschitz continuous in z and is equi-continuous in y.

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Complications with stochastic volatility models

Advances in Applied Probability ◽

10.1017/s0001867800008193 ◽

1998 ◽

Vol 30 (01) ◽

pp. 256-268 ◽

Cited By ~ 12

Author(s):

Carlos A. Sin

Keyword(s):

Stochastic Volatility ◽

Stock Price ◽

Local Martingale ◽

Stochastic Volatility Models ◽

Martingale Measures ◽

Equivalent Martingale Measures ◽

Volatility Models ◽

Equivalent Martingale

We show a class of stock price models with stochastic volatility for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of equivalent martingale measures, and provide one explicit example.

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