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Considering stochastic partial differential equations of parabolic type with random coefficients in vector-valued H\"older spaces, we obtain a sharp Schauder estimate. As an application, the existence and uniqueness of solution to the…
We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of…
We consider an evolution equation with the regularized fractional derivative of an order $\alpha \in (0,1)$ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables.…
We prove the existence of the unique solution of a general Backward Stochastic Differential Equation with quadratic growth driven by martingales. Some kind of comparison theorem is also proved.
We present an elementary Functional Analytic proof of the roughness of Exponential Dichotomy of Ordinary Differential Equations (with exponential growth) on an arbitrary Banach Space.
We introduce a transform on the class of stochastic exponentials for d-dimensional Brownian motions. Each stochastic exponential generates another stochastic exponential under the transform. The new exponential process is often merely a…
We describe the classes of functions $f=(f(x), x\in R)$, for which processes $f(W_t)-Ef(W_t)$ and $f(W_t)/Ef(W_t)$ are martingales. We apply these results to give a martingale characterization of general solutions of the quadratic and the…
For a real Borel measurable function b, which satisfies certain integrability conditions, it is possible to define a stochastic integral of the process b(Y) with respect to a Brownian motion W, where Y is a diffusion driven by W. It is well…
Space-time fractional evolution equations are a powerful tool to model diffusion displaying space-time heterogeneity. We prove existence, uniqueness and stochastic representation of classical solutions for an extension of Caputo evolution…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
We consider stochastic equations for the class of formal mappings. Existence and uniqueness of solution, as well as evolution property are proved.
Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family…
We give a constructive approach for the study of integral representations of classical solutions to Poisson equations under some integrability conditions on data functions.
In this paper, we establish a sharp $C^{2+\alpha}$-theory for stochastic partial differential equations of parabolic type in the whole space.
We introduced a new algebra of stochastic generalized functions which contains to the space of stochastic distributions G, [25]. As an application, we prove existence and uniqueness of the solution of a stochastic Cauchy problem involving…
The paper derives an equation for the Cauchy transform of the solution of a free stochastic differential equation (SDE). This new equation is used to solve several particular examples of free SDEs.
The technique of stochastic solutions, previously used for deterministic equations, is here proposed as a solution method for partial differential equations driven by distribution-valued noises.
We provide a detailed derivation of the Karhunen-Lo\`eve expansion of a stochastic process. We also discuss briefly Gaussian processes, and provide a simple numerical study for the purpose of illustration.
We describe partial differential operators for which we can construct generalised integral means satisfying Pizzetti-type formulas. Using these formulas we give a new characterisation of summability of formal power series solutions to some…
Stochastic exponentials are defined for semimartingales on stochastic intervals, and stochastic logarithms are defined for semimartingales, up to the first time the semimartingale hits zero continuously. In the case of (nonnegative) local…