Damir Kinzebulatov

I am a Postdoctoral Fellow at McGill University and the CRM

mailto: damir.kinzebulatov@utoronto.ca


Earlier, a Postdoctoral Fellow at

The Fields Institute (NSERC),
University of Toronto

Ph.D, University of Toronto, 2012



MAT 223S "Linear Algebra" (Summer 2015, half-term)
MAT 223S "Linear Algebra" (Spring 2015)
MAT 223S "Linear Algebra" (Spring 2014)
ACT 460F/STA 2502F "Stochastic Methods for Actuarial Science and Finance" (Fall 2013)



Analysis and PDEs

[1] "Strong Feller processes with measure-valued drifts", Preprint, arXiv:1508.05983 (2015), 16 p 1508.05983.pdf

We construct a strong Feller process associated with $-\Delta + \sigma \cdot \nabla$, with drift $\sigma$ in a wide class of measures (weakly form-bounded measures, e.g. combining weak $L^d$ and Kato class  measure singularities), by exploiting a quantitative dependence of the smoothness of the domain  of an operator realization of $-\Delta + \sigma \cdot \nabla$ generating a holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$, $p>d-1$, on the form-bound of $\sigma$. Our method admits extension to other types of perturbations of $-\Delta$ or $(-\Delta)^{\frac{\alpha}{2}}$, e.g. to yield new $L^p$-regularity results for Schroedinger operators with form-bounded measure potentials.

"A new approach to the L^p-theory of -\Delta + b\grad, and its applications to Feller processes with general drifts", Preprint, arXiv:1502.07286 (2015), 19 p 1502.07286.pdf

We develop a detailed regularity theory of
$-\Delta + \sigma \cdot \nabla$ in $L^p(R^d)$, for a wide class of vector fields. The L^ptheory allows us to construct associated strong Feller process in $C_\infty(R^d)$. Our starting object is an operator-valued function, which, we prove, determines the resolvent of an operator realization of $-\Delta + b\cdot \nabla$, the generator of a holomorphic C_0-semigroup on $L^p(R^d)$. Then the very form of the operator-valued function yields crucial information about smoothness of the domain of the generator.

[3] "Feller evolution families and parabolic equations with form-bounded vector fields"
, Preprint, arXiv:1407.4861 (2015), 20 p 1407.4861.pdf

We show that the weak solutions of parabolic equation $\partial_t u - \Delta u + b(t,x) \cdot \nabla u=0$, $(t,x) \in (0,\infty) \times \mathbb R^d$, $d \geqslant 3$, for $b(t,x)$ in a wide class of time-dependent vector fields capturing critical singularities both in time and in spatial variables, constitute a Feller evolution family and, thus, determine a Feller process. Our proof uses a Moser-type iterative procedure and an a priori estimate on the $L^p$-norm of the gradient of solution in terms of the $L^q$-norm of the gradient of initial function.

[4] "Kohn decomposition for forms on coverings of complex manifolds constrained along the fibres" (with A.Brudnyi), accepted for publication in
Trans. Amer. Math. Soc. (2014), 19 p 1403.0967v1.pdf

A classical result of J.J.Kohn asserts that over a relatively compact subdomain $D$ with $C^\infty$ boundary of a Hermitian manifold whose Levi form has at least $n-q$ positive eigenvalues or at least $q+1$ negative eigenvalues at each boundary point, there are natural isomorphisms between the $(p,q)$ Dolbeault cohomology groups defined by means of $C^\infty$ up to the boundary differential forms on $D$ and the (finite-dimensional) spaces of harmonic $(p,q)$-forms on $D$ determined by the corresponding complex Laplace operator. In the present paper, using Kohn's technique, we give a similar description of the $(p,q)$ Dolbeault cohomology groups of spaces of differential forms taking values in certain (possibly infinite- dimensional) holomorphic Banach vector bundles on $D$. We apply this result to compute the $(p,q)$ Dolbeault cohomology groups of some regular coverings of $D$ defined by means of $C^\infty$ forms constrained along fibres of the coverings.

[5] "Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds I" (with A.Brudnyi), accepted for publication in Revista Mat. Iberoamericana (2013), 38 p bru_kinz_1.pdf

[6] "Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds II" (with A.Brudnyi), accepted for publication in Revista Mat. Iberoamericana (2013), 56 p bru_kinz_2.pdf

We develop basic elements of Complex Function Theory within some Frechet algebras of holomorphic functions on coverings of Stein manifolds, including
Bohr's holomorphic almost periodic functions, holomorphic functions bounded along the fibres (arising e.g. in study of corona problem for H^\infty), etc. Our method is based on an extension of Oka-Cartan theory to coherent-type sheaves on the maximal ideal spaces of these algebras, topological spaces having some important features of complex analytic manifolds.

[7] "Holomorphic almost periodic functions on coverings of complex manifolds" (with A.Brudnyi)
New York J. Math, 17a (2011), p. 267-300 nyjm.pdf 

We develop basic elements of Complex Analytic Geometry on the maximal ideal spaces of some Frechet algebras of holomorphic functions.

[8] "Unique continuation for Schroedinger operators. Towards an optimal result" (with L.Shartser),
J.Funct. Anal., 258 (2010), p. 2662-2681 UC_jfa.pdf  

We prove the property of unique continuation for solutions of differential inequality |\Delta u| \leq |Vu|, with potential V belonging to a local analogue of a class for which
Schroedinger operator -\Delta+V is well defined in the sense of form-sum. We apply our result to the problem of absence of positive eigenvalues for self-adjoint Schroedinger operators with form-bounded potentials V, i.e. potentials that admit critical singularities. Our result stengthens a number of classical results (by E. Stein, D. Jerison and C. Kenig, C. Chanillo and E. Sawyer).

[9] "Holomorphic semi-almost periodic functions" (with A.Brudnyi), Integr. Equ. Operat. Theory, 66 (2010), p.293-325 0911.0954v1.pdf

[10] "On algebras of holomorphic functons with semi-almost periodic boundary values" (with A.Brudnyi), C. R. Math. Rep. Acad. Sci. Canada, 32 (2010), p.1-12.

[11] "On uniform subalgebras of L\infty on the unit circle generated by almost periodic functions" (with A.Brudnyi), St. Petersburg Math. J. 19 (2008), p.495-518 BK - SPbJMath.pdf

How does fixing the type of discontinuity of boundary values of a bounded holomorphic function F on the unit disk D affects the properties of F inside of D? Lindelof theorem states that the boundary values f=F|_{\partial D} of F can not have first-kind discontinuities. However, there are functions \Phi such that \Phi(F)|_{\partial D} can be first-kind disconinuous. It turns out that in the case \Phi(z)=|z| there is a connection between bounded holomorphic functions on D whose moduli can have only first-kind boundary discontinuities and Sarason's semi-almost periodic functions on \partial D (a priori, there is nothing almost periodic about bounded holomorphic functions and first-kind discontinuities!).
This link allows us to apply, in our study, the results on almost periodic functions. In particular, we establish
for this algebra of bounded of holomorphic functions:
 - Grothendieck's approximation property (still a conjecture for H^\infty(D)).
 - Corona theorem.
 - Results on completion of matrices with entries in this algebra,
Our methods admit extension to subalgebras determined by other functions \Phi.

"A note on Gagliardo-Nirenberg type inequalities on analytic sets", C. R. Math. Rep. Acad. Sci. Canada, 30 (2009), p.97-105. kinzebulatovC351.pdf

We study a new local invariant of
singularities of complex analytic sets. This invariant arises as a 'correcting exponent' in a family of Sobolev-type inequalities relating norms of functions on these sets.

Topological structure of the maximal ideal space

General classes of singular vector fields studied in connection with operator -\Delta + b\grad

Algorithmic and HF trading:

[13] "Algorithmic trading with learning" (with A.Cartea, S.Jaimungal)
, Preprint, SSRN eLibrary, http://ssrn.com/abstract=2373196 (2013), 20 p. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2373196
We propose a model where an algorithmic trader takes a view on the distribution of prices at a future date and then decides how to trade in the direction of her predictions using the optimal mix of market and limit orders. Namely, we model the asset midquote price as a randomized Brownian bridge: 

S_t=S_0+\sigma\beta_{tT}+ tD/T 

where D is a random variable that encodes the trader's prior belief on the asset's future price distribution, the `noise' term \beta_{tT} is a standard Brownian bridge over time interval independent of D. As time flows, the trader learns the realized value of D.

[14] "Optimal accelerated share repurchase" (with S.Jaimungal, D.Rubisov), Preprint, SSRN eLibrary, http://ssrn.com/abstract=2360394 (2013), 28 p. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2360394

An accelerated share repurchase (ASR) allows a fi rm to repurchase a signi cant portion of its shares immediately, while shifting the burden of reducing the impact and uncertainty in the trade to an intermediary. The intermediary must then purchase the shares from the market over several days, weeks, or as much as several months. In this work, we address the intermediary's optimal execution and exit strategy taking into account the impact that trading has on the market. We demonstrate that it is optimal to exercise when the TWAP exceeds \zeta(t)S_t, where S_t is the fundamental price of the asset and \zeta(t) is deterministic. Moreover, we develop a dimensional reduction of the Stochastic Control and Stopping problem and implement an efficient numerical scheme to compute the optimal trading and exit strategies.

Earlier research while stuyding for B.Sc and M.Sc:

"Systems with distributions and viability theorem", J. Math. Anal. Appl., 331 (2007), p. 1046-1067 kinz_viab.pdf

[16] "Dynamical generalized functions and the multiplication problem" (with V.Derr), Russian Math., 51 (2007), p.32-43 0603351.pdf

We study some qualitative properties of ordinary differential equations arising in singular Optimal Control problems. To carry out this study, we had to introduce a new space of measures (distributions) with a continuous operation of multiplication by first-kind discontinuous functions.

[17] "Nicholson blowfiles equation with a distributed delay" (with E. Braverman), Canadian Appl. Math. Q., 14  (2006), p. 107-128 brav_kinz.pdf

We study properties of some cassical model of Population Dynamics.