scholarly journals Continuous-time gambling problems

1974 ◽  
Vol 6 (4) ◽  
pp. 651-665 ◽  
Author(s):  
David C. Heath ◽  
William D. Sudderth
Keyword(s):  
Brownian Motion ◽  
Discrete Time ◽  
Continuous Time ◽  
Gambling Problems ◽  
Brownian Motion Process ◽  
Motion Process

An abstract gambler's problem is formulated in a continuous-time setting and analogues are proved for some of the discrete-time results of Dubins and Savage in their book How to Gamble if You Must. Applications are made to problems of controlling a Brownian motion process.

scholarly journals Continuous-time gambling problems

1974 ◽  
Vol 6 (04) ◽  
pp. 651-665 ◽  
Author(s):  
David C. Heath ◽  
William D. Sudderth
Keyword(s):  
Brownian Motion ◽  
Discrete Time ◽  
Continuous Time ◽  
Gambling Problems ◽  
Brownian Motion Process ◽  
Motion Process

An abstract gambler's problem is formulated in a continuous-time setting and analogues are proved for some of the discrete-time results of Dubins and Savage in their book How to Gamble if You Must. Applications are made to problems of controlling a Brownian motion process.

scholarly journals Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Caishi Wang ◽  
Jinshu Chen
Keyword(s):  
Brownian Motion ◽  
Growth Condition ◽  
Discrete Time ◽  
Continuous Time ◽  
Orthonormal Basis ◽  
Continuous Time Processes ◽  
Characterization Theorems ◽  
Normal Martingale

We aim at characterizing generalized functionals of discrete-time normal martingales. LetM=(Mn)n∈Nbe a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals ofMwith an appropriate orthonormal basis forM’s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems. Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes.

L'intégrale du mouvement brownien

1993 ◽  
Vol 30 (01) ◽  
pp. 17-27
Author(s):  
Aimé Lachal
Keyword(s):  
Fourier Transform ◽  
Brownian Motion ◽  
Hitting Time ◽  
First Hitting Time ◽  
Brownian Motion Process ◽  
Classical Technique ◽  
Motion Process ◽  
Mouvement Brownien ◽  
The Law ◽  
The Right

Let be the Brownian motion process starting at the origin, its primitive and Ut = (Xt+x + ty, Bt + y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa ), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab ), where σ α and σ ab denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process . Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb ).

scholarly journals Conditional generalized analytic Feynman integrals and a generalized integral equation

2000 ◽  
Vol 23 (11) ◽  
pp. 759-776 ◽  
Author(s):  
Seung Jun Chang ◽  
Soon Ja Kang ◽  
David Skoug
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Feynman Integral ◽  
Brownian Motion Process ◽  
Feynman Integrals ◽  
Motion Process ◽  
Generalized Brownian Motion Process

We use a generalized Brownian motion process to define a generalized Feynman integral and a conditional generalized Feynman integral. We then establish the existence of these integrals for various functionals. Finally we use the conditional generalized Feynman integral to derive a Schrödinger integral equation.

scholarly journals Asymptotic Behavior of the Stochastic Rayleigh-van der Pol Equations with Jumps

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
ZaiTang Huang ◽  
ChunTao Chen
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Stochastic Bifurcation ◽  
Brownian Motion Process ◽  
Van Der Pol ◽  
Limit Circle ◽  
Random Attractors ◽  
Motion Process ◽  
Van Der Pol Equations ◽  
The Stability

We study the stability, attractors, and bifurcation of stochastic Rayleigh-van der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleigh-van der Pol equations. We then examine the existence limit circle and obtain some new random attractors. We further establish stochastic bifurcation of random attractors. Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system which is significantly different from the classical Brownian motion process.

L'intégrale du mouvement brownien

1993 ◽  
Vol 30 (1) ◽  
pp. 17-27 ◽  
Author(s):  
Aimé Lachal
Keyword(s):  
Fourier Transform ◽  
Brownian Motion ◽  
Hitting Time ◽  
First Hitting Time ◽  
Brownian Motion Process ◽  
Classical Technique ◽  
Motion Process ◽  
Mouvement Brownien ◽  
The Law ◽  
The Right

Let be the Brownian motion process starting at the origin, its primitive and Ut = (Xt+x + ty, Bt + y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab), where σ α and σ ab denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process .Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb).

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