اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the location of the maximum of a continuous stochastic process
113
0
0.0
(
0
)
تأليف
Leandro P. R. Pimentel
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this short note we will provide a sufficient and necessary condition to have uniqueness of the location of the maximum of a stochastic process over an interval. The result will also express the mean value of the location in terms of the derivative of the expectation of the maximum of a linear perturbation of the underlying process. As an application, we will consider a Brownian motion with variable drift. The ideas behind the method of proof will also be useful to study the location of the maximum, over the real line, of a two-sided Brownian motion minus a parabola and of a stationary process minus a parabola.
قيم البحث
اقرأ أيضاً
We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer
Lipschitz. The first type is the equation label{eq1} X_{t}=int_{0}^{t}sigma (s,X_{s})dW_{s}+int_{0}^{t}b(s,X_{s})ds+alpha max_{0leq sleq t}X_{s}. The second type is the equation label{eq2} {l} X_{t} =ig{0}{t}sigma (s,X_{s})dW_{s}+ig{0}{t}b(s,X_{s})ds+alpha max_{0leq sleq t}X_{s},,+L_{t}^{0}, X_{t} geq 0, forall tgeq 0. The third type is the equation label{eq3} X_{t}=x+W_{t}+int_{0}^{t}b(X_{s},max_{0leq uleq s}X_{u})ds. We end the paper by establishing the existence of strong solution and pathwise uniqueness, under Lipschitz condition, for the SDE label{e2} X_t=xi+int_0^t si(s,X_s)dW_s +int_0^t b(s,X_s)ds +almax_{0leq sleq t}X_s +be min_{0leq s leq t}X_s.
In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problems can be interpreted as a stochastic cont
rol problem for an evolution system in an Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays a crucial role in the formulation of the maximum principle.
We study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows
exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $lambda_1$ for weak survival, and the survival probability $p(lambda)$ is continuous with respect to the infection rate $lambda$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $lambda_1<lambda_2$, which confirms a conjecture of Staceys cite{Stacey}. We also prove that if the contact process survives strongly at $lambda$ then it survives strongly at a $lambda<lambda$, which implies that the process does not survive strongly at the critical value $lambda_2$ for strong survival.
We study the stochastic growth process in discrete time $x_{i+1} = (1 + mu_i) x_i$ with growth rate $mu_i = rho e^{Z_i - frac12 var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $dZ_t = - gamma Z_t dt + sigma dW_t$ sam
pled on a grid of uniformly spaced times ${t_i}_{i=0}^n$ with time step $tau$. Using large deviation theory methods we compute the asymptotic growth rate (Lyapunov exponent) $lambda = lim_{nto infty} frac{1}{n} log mathbb{E}[x_n]$. We show that this limit exists, under appropriate scaling of the O-U parameters, and can be expressed as the solution of a variational problem. The asymptotic growth rate is related to the thermodynamical pressure of a one-dimensional lattice gas with attractive exponential potentials. For $Z_t$ a stationary O-U process the lattice gas coincides with a system considered previously by Kac and Helfand. We derive upper and lower bounds on $lambda$. In the large mean-reversion limit $gamma n tau gg 1$ the two bounds converge and the growth rate is given by a lattice version of the van der Waals equation of state. The predictions are tested against numerical simulations of the stochastic growth model.
We consider the Brownian ``spider process, also known as Walsh Brownian motion, first introduced in the epilogue of Walsh 1978. The paper provides the best constant $C_n$ for the inequality $$ E D_tauleq C_n sqrt{E tau},$$ where $tau$ is the class of
all adapted and integrable stopping times and $D$ denotes the diameter of the spider process measured in terms of the British rail metric. The proof relies on the explicit identification of the value function for the associated optimal stopping problem.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
