Transition probabilities for vector-valued Brownian motion with boundaries

1978 ◽  
pp. 188-195
Author(s):  
Victor Goodman
Keyword(s):  
Brownian Motion ◽  
Transition Probabilities ◽  
Vector Valued

scholarly journals Diffusion and the Brownian motion

1949 ◽  
Vol 198 (1052) ◽  
pp. 94-116 ◽  
Keyword(s):  
Brownian Motion ◽  
Free Path ◽  
Transition Probabilities ◽  
The Other ◽  
Approximate Theory ◽  
Gaseous Diffusion ◽  
Self Diffusion ◽  
Second Moments ◽  
Diffusion In Liquids ◽  
The One

In the present paper the phenomenon of diffusion is examined in the light of the theory of the Brownian motion. The coefficients of self-diffusion, ordinary diffusion and thermal diffusion are expressed in terms of the first and second moments of certain transition probabilities familiar in the theory of the Brownian motion. It is then found possible in gases of low or moderate density where a fairly well-defined free path exists to follow the future course of a given molecule statistically to as many free flights as required provided the velocity distribution of the molecules in the medium is known. This consideration on the one hand leads to a rigorous expression for the coefficient of self-diffusion directly calculated from a Maxwellian distribution, and on the other serves to clarify the relation between the old free-path theory of gaseous diffusion and the rigorous theory of gaseous diffusion and between self-diffusion and mutual diffusion. Further, an approximate theory of diffusion in liquids corresponding to the old free-path theory in gases is suggested.

Existence of Solutions for Stochastic Differential Equations under G-Brownian Motion with Discontinuous Coefficients

2012 ◽  
Vol 67 (12) ◽  
pp. 692-698 ◽  
Author(s):  
Faiz Faizullah
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence Of Solutions ◽  
Upper And Lower Solutions ◽  
Discontinuous Coefficients ◽  
Existence Theory ◽  
Discontinuous Functions ◽  
Vector Valued

The existence theory for the vector valued stochastic differential equations under G-Brownian motion (G-SDEs) of the type Xt = X0+ ∫to(v;Xv)dv+ ∫t0 g(v;Xv)d(B)v+ ∫t0 h(v;Xv)dBv; t ∊ [0;T]; with first two discontinuous coefficients is established. It is shown that the G-SDEs have more than one solution if the coefficient g or the coefficients f and g simultaneously, are discontinuous functions. The upper and lower solutions method is used and examples are given to explain the theory and its importance.

A Note on the Carathéodory Approximation Scheme for Stochastic Differential Equations under G-Brownian Motion

2012 ◽  
Vol 67 (12) ◽  
pp. 699-704 ◽  
Author(s):  
Faiz Faizullah
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Uniqueness Theorem ◽  
Unique Solution ◽  
Existence And Uniqueness ◽  
Approximation Scheme ◽  
Approximate Solutions ◽  
Existence And Uniqueness Theorem ◽  
Vector Valued

In this note, the Carathéodory approximation scheme for vector valued stochastic differential equations under G-Brownian motion (G-SDEs) is introduced. It is shown that the Carathéodory approximate solutions converge to the unique solution of the G-SDEs. The existence and uniqueness theorem for G-SDEs is established by using the stated method.

scholarly journals Quantum stochastic integrals as belated integrals

1992 ◽  
Vol 34 (2) ◽  
pp. 165-173
Author(s):  
Chris Barnett ◽  
J. M. Lindsay ◽  
Ivan F. Wilde
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Stochastic Integral ◽  
Stochastic Integrals ◽  
Itô Integral ◽  
Ito Integral ◽  
Finite Sum ◽  
Vector Valued ◽  
Quantum Stochastic Integral

Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.

scholarly journals APPROXIMATION METHODS FOR INHOMOGENEOUS GEOMETRIC BROWNIAN MOTION

2019 ◽  
Vol 22 (02) ◽  
pp. 1850055
Author(s):  
LUCA CAPRIOTTI ◽  
YUPENG JIANG ◽  
GAUKHAR SHAIMERDENOVA
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Interest Rates ◽  
High Precision ◽  
Transition Probabilities ◽  
Geometric Brownian Motion ◽  
Approximation Methods ◽  
Time Horizons

We present an accurate and easy-to-compute approximation of the transition probabilities and the associated Arrow-Debreu (AD) prices for the inhomogeneous geometric Brownian motion (IGBM) model for interest rates, default intensities or volatilities. Through this procedure, dubbed exponent expansion, transition probabilities and AD prices are obtained as a power series in time to maturity. This provides remarkably accurate results — for time horizons up to several years — even when truncated after the first few terms. For farther time horizons, the exponent expansion can be combined with a fast numerical convolution to obtain high-precision results.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

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