Damir Kinzebulatov



damir.kinzebulatov@gmail.com







Teaching 

2014 2014 2013 
STA
348F "Introduction to Stochastic Processes" (Fall
2014) MAT 223S "Linear Algebra" (Spring 2014) ACT 460F/STA 2502F "Stochastic Methods for Actuarial Science and Finance" (Fall 2013) ... 

Research 

Complex
Analysis, PDEs and Probability: 

[1] "Feller evolution families and parabolic equations with formbounded vector fields", Preprint, arXiv:1407.4861 (2014), 19 p 1407.4861.pdf We show that the weak solutions of parabolic equation $\partial_t u  \Delta u + b(t,x) \cdot \nabla u=0$ with vector field $b(t,x)$ satisfying formboundedness condition constitute a Feller evolution family and, thus, determine a strong Markov process. Our proof uses a Mosertype 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. 


[2] "Kohn decomposition for forms on coverings of complex manifolds constrained along the fibres" (with A.Brudnyi), Preprint, arXiv:1403.0967 (2014), 18 p 1403.0967v1.pdf The 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 $nq$ 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 (finitedimensional) 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. [3] "Towards OkaCartan 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 [4] "Towards OkaCartan 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 a Complex Function Theory within some Frechet algebras of holomorphic functions on coverings of Stein manifolds. Examples:  Bohr's algebra of holomorphic almost periodic functions on tube domains;  algebra of all fibrewise bounded holomorphic functions (arising e.g. in study of corona problem for H^\infty). In particular, we obtain (within these algebras of holomorphic functions) the results on holomorphic extension from complex submanifolds, properties of divisors, corona type theorems, holomorphic analogues of the PeterWeyl approximation theorem, Hartogs type theorems, characterizations of uniqueness sets, etc. Our approach is based on an extension of the classical OkaCartan theory to coherenttype sheaves on the maximal ideal spaces of these algebras  topological spaces having some features of complex manifolds. [5] "Holomorphic almost periodic functions on coverings of complex manifolds" (with A.Brudnyi) New York J. Math, 17a (2011), p. 267300 nyjm.pdf This paper contains preliminary versions of some results in [3], [4]. We also use some algebrogeometric techniques to give example of a submanifold justifying the need to develop an OkaCartan type theory of [3], [4]. 

[6] "Unique continuation for Schroedinger operators. Towards an optimal result" (with L.Shartser), J. Funct. Anal., 258 (2010), p. 26622681 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 formsum. We apply our result to the problem of absence of positive eigenvalues for selfadjoint Schroedinger operators with formbounded 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). 

[7] "Holomorphic semialmost periodic functions" (with A.Brudnyi), Integr. Equ. Operat. Theory, 66 (2010), p.293325 0911.0954v1.pdf [8] "On algebras of holomorphic functons with semialmost periodic boundary values" (with A.Brudnyi), C. R. Math. Rep. Acad. Sci. Canada, 32 (2010), p.112. [9] "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.495518 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 firstkind discontinuities. However, there are functions \Phi such that \Phi(F)_{\partial D} can be firstkind 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 firstkind boundary discontinuities and Sarason's semialmost periodic functions on \partial D (a priori, there is nothing almost periodic about bounded holomorphic functions and firstkind 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, etc. Our methods admit extension to subalgebras determined by other functions \Phi. 

[10] "A note on GagliardoNirenberg type inequalities on analytic sets", C. R. Math. Rep. Acad. Sci. Canada, 30 (2009), p.97105. 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 Sobolevtype inequalities relating norms of functions on these sets. 

Topological structure of the maximal ideal space 

Algorithmic and HF trading: 

[11] "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. [12] "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 firm to repurchase a signicant 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: 

[13] "Systems with distributions and viability theorem", J. Math. Anal. Appl., 331 (2007), p. 10461067 kinz_viab.pdf [14] "Dynamical generalized functions and the multiplication problem" (with V.Derr), Russian Math., 51 (2007), p.3243 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 firstkind discontinuous functions. [15] "Nicholson blowfiles equation with a distributed delay" (with E. Braverman), Canadian Appl. Math. Q., 14 (2006), p. 107128 brav_kinz.pdf We study properties of some cassical model of Population Dynamics. 
