Damir Kinzebulatov



damir.kinzebulatov@gmail.com


Ph.D 
University
of Toronto 



Teaching 

2014 2013 2011 2010 
MAT
223S  "Linear Algebra" (Spring 2014) ACT 460F/STA 2502F  "Stochastic Methods for Actuarial Science and Finance" (Fall 2013) MAT 135Y "Single Variable Calculus" (Summer 2011) MAT 235Y "Multivariable Calculus" (Spring 2010) 

Complex
Analysis and PDEs: 


[1] "On parabolic equations
in Feller space C_0, and some a priori estimates",
Preprint
(available upon request), 15 p. We obtain new regularity results for
parabolic differential equation \partial_t u =
\Delta u + b(t,x) \nabla u with drift b(t,x) having
critical singularities (both in time and in spacial
variables), in space C_\infty of continuous
functions on R^d vanishing at infinity, and also in
L^p. Our approach uses the iterative procedure of
KovalenkoSemenov and an apriori estimate on L^q
norm of the gradient of solution. [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 We extend the classical result of J.J. Kohn on Hodgetype decomposition for forms on compact subdomains of Hermitian manifolds to an infinitedimensional setting: forms taking values in holomorphic Banach vector bundles. Motivation: applications to study of certain spaces of forms on coverings of complex manifolds. [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. 

Topological structure of the maximal ideal space 

[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. 

[11] "Systems with
distributions and viability theorem", J. Math. Anal. Appl., 331 (2007), p.
10461067 kinz_viab.pdf [12] "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. 

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. The trader has access to filtration F_t generated by process S_t, but not to the filtration of \beta_{tT} (the latter would mean that the trader has perfect information, i.e. she would be able to find the realized value of D prior to terminal time T). In particular, we establish that compared to a trader that cannot learn from market dynamics or form a view of the market (i.e. D is normally distributed with variance s\igma^2T, in which case term (E[DF_t]S_t+S_0)/(Tt) in the SDE is zero, and S_t becomes an arithmetic Brownian motion), the algorithmic trader's profits are higher and more certain. Even though the trader executes a strategy based on a directional view, the sources of profits are both from making the spread as well as capital appreciation of inventories. Higher volatility of prices considerably impairs the trader's ability to learn from price innovations, but this adverse effect can be circumvented by learning from a collection of assets that comove. [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 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. Some contracts allow the intermediary to specify when the repurchase ends, at which point the corporation and the intermediary exchange the dierence between the arrival price and the TWAP over the trading period plus a spread. Hence, the intermediary effectively has an American option embedded within an optimal execution problem. As a result, the firm receives a discounted spread relative to the no early exercise case. 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. 

Industry
papers 

[15] "Optimal acquisition with a price limiter" (with S.Jaimungal), to appear in RISK J.  
Earlier research (B.Sc and M.Sc) 

[16] "On the extension of
Schwartz distributions to the space of
discontinuous test functions of several variables"
(with V. Derr), Rocky Mountain J. Math, 39
(2009), p. 3243 (applications to Game Theory,
accepted for publication in 2006). 

[17] "On linear
perturbations of the Ricker model" (with E. Braverman), Math Biosci., 202 (2006), p.
323339 brav_kinz0.pdf [18] "Nicholson's blowflies 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. 

Last update: March 2014 