scholarly journals Schur–Weyl duality and the product of randomly-rotated symmetries by a unitary Brownian motion

2021 ◽  
pp. 2150002
Author(s):  
Nizar Demni ◽  
Tarek Hamdi
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Hermitian Matrix ◽  
Direct Proof ◽  
Fixed Time ◽  
Stochastic Calculus ◽  
Unitary Matrix ◽  
Weyl Duality ◽  
Jacobi Process ◽  
Tensor Powers

In this paper, we introduce and study a unitary matrix-valued process which is closely related to the Hermitian matrix-Jacobi process. It is precisely defined as the product of a deterministic self-adjoint symmetry and a randomly-rotated one by a unitary Brownian motion. Using stochastic calculus and the action of the symmetric group on tensor powers, we derive an ordinary differential equation for the moments of its fixed-time marginals. Next, we derive an expression of these moments which involves a unitary bridge between our unitary process and another independent unitary Brownian motion. This bridge motivates and allows to write a second direct proof of the obtained moment expression.

scholarly journals ASYMPTOTIC EXPANSION OF THE DENSITY FOR HYPOELLIPTIC ROUGH DIFFERENTIAL EQUATION

2019 ◽  
pp. 1-31
Author(s):  
YUZURU INAHAMA ◽  
NOBUAKI NAGANUMA
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Asymptotic Expansion ◽  
Vector Fields ◽  
Fixed Time ◽  
Hurst Parameter ◽  
Coefficient Vector ◽  
Hörmander’S Condition ◽  
Short Time ◽  
Smooth Density

We study a rough differential equation driven by fractional Brownian motion with Hurst parameter $H$ $(1/4<H\leqslant 1/2)$ . Under Hörmander’s condition on the coefficient vector fields, the solution has a smooth density for each fixed time. Using Watanabe’s distributional Malliavin calculus, we obtain a short time full asymptotic expansion of the density under quite natural assumptions. Our main result can be regarded as a “fractional version” of Ben Arous’ famous work on the off-diagonal asymptotics.

Large deviations and Berry–Esseen inequalities for estimators in nonhomogeneous diffusion driven by fractional Brownian motion

2020 ◽  
Vol 28 (3) ◽  
pp. 183-196
Author(s):  
Kouacou Tanoh ◽  
Modeste N’zi ◽  
Armel Fabrice Yodé
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Maximum Likelihood ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Maximum Likelihood Estimator ◽  
Bayes Estimator ◽  
Likelihood Estimator

AbstractWe are interested in bounds on the large deviations probability and Berry–Esseen type inequalities for maximum likelihood estimator and Bayes estimator of the parameter appearing linearly in the drift of nonhomogeneous stochastic differential equation driven by fractional Brownian motion.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (04) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ E (max0≤t≤τ X t − c τ), where X = (X t ) t≥0 is geometric Brownian motion with drift μ and volatility σ &gt; 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ &lt; 0. The optimal stopping time is given by τ* = inf {t &gt; 0 | X t = g * (max0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 &lt; g(s) &lt; s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g *(s) ∼ ((Δ − 1) / K Δ)1 / Δ s 1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (sup t&gt;0 X t ) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

Stochastic Calculus for Fractional Brownian Motion and Applications

2008 ◽  
Author(s):  
Francesca Biagini ◽  
Yaozhong Hu ◽  
Bernt Øksendal ◽  
Tusheng Zhang
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stochastic Calculus
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