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Register a new userTime inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators
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Authors
Pierre Etore
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In this paper we study time-inhomogeneo
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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.
This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates.
In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not Levy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bernv{s}tein functions for each time $t$. By means of these processes, a generalization of subordinate semigroups in the sense of Bochner is proposed. Because of time-inhomogeneity, two-parameter semigroups (propagators) arise and we provide a Phillips formula which leads to time dependent generators. The inverse processes are also investigated and the corresponding governing equations obtained in the form of generalized variable order fractional equations. An application to a generalized subordinate Brownian motion is also examined.
We investigate the space-time regularity of the local time associated to Volterra-Levy processes, including Volterra processes driven by $alpha$-stable processes for $alphain(0,2]$. We show that the spatial regularity of the local time for Volterra-Levy process is $P$-a.s. inverse proportionally to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbaPtion of ODEs by a Volterra-Levy process which has sufficiently regular local time. Following along the lines of [15], we show existence, uniqueness and differentiablility of the flow associated to such equations.
We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ partial_t^alpha u - Lu= f quad mathrm{in} quad (0,T) times mathbb{R}^d,$$ where $partial_t^alpha u$ is the Caputo fractional derivative of order $alpha in (0,1]$, $Tin (0,infty)$, and $$Lu(t,x) := int_{ mathbb{R}^d} bigg( u(t,x+y)-u(t,x) - ycdot abla_xu(t,x)chi^{(sigma)}(y)bigg)K(t,x,y),dy $$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel $K$ with respect to $t$ and $y$. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., $alpha = 1$, which extend the previous results by using a quite different method.
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