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Evelina Shamarova

Publications and source records attributed to Evelina Shamarova.

26 records · Page 2Linked to original sources
arXiv2013

Forward-Backward SDEs driven by Lévy Processes and Application to Option Pricing

Recent developments on financial markets have revealed the limits of Brownian motion pricing models when they are applied to actual markets. Lévy processes, that admit jumps over time, have been found more useful for applications. Thus, we suggest a Lévy model based on Forward-Backward Stochastic Differential Equations (FBSDEs) for option pricing in a Lévy-type market. We show the existence and uniqueness of a solution to FBSDEs driven by a Lévy process. This result is important from the mathematical point of view, and also, provides a much more realistic approach to option pricing.

math.PR↗
arXiv2013

A version of Hörmander's theorem in 2-smooth Banach spaces

We consider a stochastic evolution equation in a 2-smooth Banach space with a densely and continuously embedded Hilbert subspace. We prove that under Hörmander's bracket condition, the image measure of the solution law under any finite-rank bounded linear operator is absolutely continuous with respect to the Lebesgue measure. To obtain this result, we apply methods of the Malliavin calculus.

math.PR↗
arXiv2011

Chernoff's theorem for backward propagators and applications to diffusions on manifolds

The classical Chernoff's theorem is a statement about discrete-time approximations of semigroups, where the approximations are consturcted as products of time-dependent contraction operators strongly differentiable at zero. We generalize the version of Chernoff's theorem for semigroups proved in a paper by Smolyanov et al., and obtain a theorem about descrete-time approximations of backward propagators.

math.FA↗
arXiv2010

Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus

We establish a connection between the strong solution to the spatially periodic Navier-Stokes equations and a solution to a system of forward-backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier-Stokes equations in terms of diffusion processes.

math.PR↗
arXiv2008

Differential forms on locally convex spaces and the Stokes formula

We prove a version of the Stokes formula for differential forms on locally convex spaces. The main tool used for proving this formula is the surface layer theorem proved in another paper by the author. Moreover, for differential forms of a Sobolev-type class relative to a differentiable measure, we compute the operator adjoint to the exterior differential in terms of standard operations of calculus of differential forms and the logarithmic derivative. Previously, this connection was established under essentially stronger assumptions.

math.FA↗
arXiv2008

A mathematical approach to the nonequilibrium work theorem

We develop a mathematical approach to the nonequilibrium work theorem which is traditionally referred to in statistical mechanics as Jarzynski's identity. We suggest a mathematically rigorous formulation and proof of the identity.

math.PR↗
arXiv2007

Chernoff's theorem for evolution families

A generalized version of Chernoff's theorem has been obtained. Namely, the version of Chernoff's theorem for semigroups obtained in a paper by Smolyanov, Weizsaecker, and Wittich is generalized for a time-inhomogeneous case. The main theorem obtained in the current paper, Chernoff's theorem for evolution families, deals with a family of time-dependent generators of semigroups $A_t$ on a Banach space, a two-parameter family of operators $Q_{t,t+Δt}$ satisfying the relation: $\frac{\partial}{\partial Δt}Q_{t,t+Δt}|_{Δt = 0}=A_t$, whose products $Q_{t_i,t_{i+1}}... Q_{t_{k-1},t_k}$ are uniformly bounded for all subpartitions $s = t_0 < t_1 < >... < t_n = t$. The theorem states that $Q_{t_0,t_1}... Q_{t_{n-1},t_n}$ converges to an evolution family $U(s,t)$ solving a non-autonomous Cauchy problem. Furthermore, the theorem is formulated for a particular case when the generators $A_t$ are time dependent second order differential operators. Finally, an example of application of this theorem to a construction of time-inhomogeneous diffusions on a compact Riemannian manifold is given. Keywords: Chernoff's theorem, evolution family, strongly continuous semigroup, evolution families generated by manifold valued stochastic processes.

math.FA↗