Researcher:
Soner, Halil Mete

Profile Picture
ORCID

Job Title

Faculty Member

First Name

Halil Mete

Last Name

Soner

Name

Name Variants

Soner, Halil Mete

Email Address

Birth Date

Filters

2002 - 200716

Advanced Search

Filter by

Settings

Researcher Of Publications: search.filters.isAuthorOfPublication.3bf9d2ff-8798-4ff5-8024-516d668c1c7f

Search Results

Now showing 1 - 10 of 16
  • Placeholder
    PublicationMetadata only
    Chapter 6 stochastic representations for nonlinear parabolic pdes
    (Elsevier, 2007) Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We discuss several different representations of nonlinear parabolic partial differential equations in terms of Markov processes. After a brief introduction of the linear case, different representations for nonlinear equations are discussed. One class of representations is in terms of stochastic control and differential games. An extension to geometric equations is also discussed. All of these representations are through the appropriate expected values of the data. Different type of representations are also available through backward stochastic differential equations. A recent extension to second-order backward stochastic differential equations allow us to represent all fully nonlinear scalar parabolic equations.
  • Placeholder
    PublicationMetadata only
    Stochastic representations for nonlinear parabolic pdes
    (North Holland, Elsevier Science Publ Bv, 2007) Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We discuss several different representations of nonlinear parabolic partial differential equations in terms of Markov processes. After a brief introduction of the linear case, different representations for nonlinear equations are discussed. One class of representations is in terms of stochastic control and differential games. An extension to geometric equations is also discussed. All of these representations are through the appropriate expected values of the data. Different type of representations are also available through backward stochastic differential equations. A recent extension to second-order backward stochastic differential equations allow us to represent all fully nonlinear scalar parabolic equations.
  • Placeholder
    PublicationMetadata only
    A stochastic representation for the level set equations
    (Taylor & Francis Inc, 2002) Touzi, Nizar; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    A Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicial choices of controls. The sublevel sets of the unique level set solution of the geometric equation is shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data.
  • Placeholder
    PublicationMetadata only
    Variational and dynamic problems for the Ginzburg-Landau functional
    (Springer-Verlag Berlin, 2003) Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    N/A
  • Placeholder
    PublicationMetadata only
    The multi-dimensional super-replication problem under gamma constraints
    (Elsevier, 2005) Cheridito, Patrick; Touzi, Nizar; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    The classical Black-Scholes hedging strategy of a European contingent claim may require rapid changes in the replicating portfolio. One approach to avoid this is to impose a priori bounds on the variations of the allowed trading strategies, called gamma constraints. Under such a restriction, it is in general no longer possible to replicate a European contingent claim, and super-replication is a commonly used alternative. This paper characterizes the infimum of the initial capitals that allow an investor to super-replicate the contingent claim by carefully choosing an investment strategy obeying a gamma constraint. This infimum is shown to be the unique viscosity solution of a nonstandard partial differential equation. Due to the lower gamma bound, the "intuitive" partial differential equation is not parabolic and the actual equation satisfied by the infimum is the parabolic majorant of this equation. The derivation of the viscosity property is based on new results on the small time behavior of double stochastic integrals.
  • Placeholder
    PublicationMetadata only
    Dynamic programming for stochastic target problems and geometric flows
    (Springer-Verlag Berlin, 2002) Touzi, Nizar; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    Given a controlled stochastic process, the reachability set is the collection of all initial data from which the state process can be driven into a target set at a specified time. Differential properties of these sets are studied by the dynamic programming principle which is proved by the Jankov-von Neumann measurable selection theorem. This principle implies that the reachability sets satisfy a geometric partial differential equation, which is the analogue of the Hamilton-Jacobi-Bellman equation for this problem. By appropriately choosing the controlled process, this connection provides a stochastic representation for mean curvature type geometric flows. Another application is the super-replication problem in financial mathematics. Several applications in this direction are also discussed.
  • Placeholder
    PublicationMetadata only
    The jacobian and the ginzburg-landau energy
    (Springer Nature, 2002) Jerrard, Robert L.; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We study the Ginzburg-Landau functional Iε(u):= 1/ln(1/ε) ∫U 1/2|∇u|2 + 1/4∈2 (1- |u|2)2 dx, for u ∈ H1 (U; ℝ2), where U is a bounded, open subset of R2. We show that if a sequence of functions uε satisfies sup Iε(uε) andlt; ∞, then their Jacobians Juε are precompact in the dual of Cc0,α for every α ∈ (0, 1]. Moreover, any limiting measure is a sum of point masses. We also characterize the Γ-limit I(·) of the functionals Iε (·), in terms of the function space B2V introduced by the authors in [16, 17]: we show that I(u) is finite if and only if u ∈ B2V(U; S1), and for u ∈ B2V(U; S1), I(u) is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if Iε (uε) ≤ C then the Jacobians Juε are again precompact in (Cc0,α)* for all α ∈ (0, 1], and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation of the Jacobian measure is a lower bound for the Γ limit of the Ginzburg-Landau functional.
  • Placeholder
    PublicationMetadata only
    Limiting behavior of the ginzburg-landau functional
    (Elsevier, 2002) Jerrard, Robert L.; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We continue our study of the functional E-0(u) := integral(0)1/2\delu\(2) + 1/4epsilon(2) (1 - \u\(2))(2) dx, for u is an element of H-1(U;R-2), where U is a bounded, open subset of R-2. Compactness results for the scaled Jacobian of u(E) are proved under the assumption that E, (u(r)) is bounded uniformly by a function of epsilon. In addition, the Gamma limit of E(u(r))/(ln epsilon)(2) is shown to be E(upsilon) := 1/2parallel toupsilonparallel to(2)(2) + parallel todel x upsilonparallel to(.H), where upsilon is the limit of j(u(0))/\ln epsilon\, j(u(0)) = u(0) x Du(0), and parallel to(.)parallel to(.H) is the total variation of a Radon measure. These results are applied to the Ginzburg-Landau functional F-0(u,A;h(ext)) := integral(upsilon)1/2 \del(A)u\(2) + 1/4epsilon(2) (1 - \u\(2))(2) + 1/2\del x A - h(ext)\ dx, with external magnetic field h(ext) approximate to H\ln epsilon\. The Gamma limit of F-0/(ln epsilon)(2) is calculated to be F(upsilon, a; H) := 1/2[parallel toupsilon - aparallel to(2)(2) +parallel todel x upsilonparallel to(.H) + parallel todel x a - Hparallel to(2)(2)], where upsilon is as before, and a is the limit of A(p)/\ln epsilon\. (C) Elsevier Science (USA).
  • Placeholder
    PublicationMetadata only
    Functions of bounded higher variation
    (Indiana Univ Math Journal, 2002) Jerrard, Robert L.; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We say that a function u : R-m --> R-n, with m greater than or equal to n, has bounded n-variation if Der (u(xalpha1), . . . , u(xalphan)) is a measure for every 1 less than or equal to alpha(1) < (. . .) < alpha(n) less than or equal to m. Here Det(v1, . . . , v(n)) denotes the distributional determinant of the matrix whose columns are the given vectors, arranged in the given order. In this paper we establish a number of properties of BnV functions and related functions. We establish general (and rather weak) versions of the chain rule and the coarea formula; we show that stronger forms of the chain rule can fail, and we also demonstrate that BnV functions cannot, in general, be strongly approximated by smooth functions; and we prove that if u e BnV(R-m, R-n) and \u\ = 1 a.e., then the jacobian of u is an m - n-dimensional rectifiable current.
  • Placeholder
    PublicationMetadata only
    Stochastic control for a class of random evolution models
    (Springer-Verlag, 2004) Hongler, Max-Olivier; Streit, Ludwig; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We construct the explicit connection existing between a solvable model of the discrete velocities non-linear Boltzmann equation and the Hamilton-Bellman-Jacobi equation associated with a simple optimal control of a piecewise deterministic process. This study extends the known relation that exists between the Burgers equation and a simple controlled diffusion problem. In both cases the resulting partial differential equations can be linearized via a logarithmic transformation and hence offer the possibility to solve physically relevant non-linear field models in full generality.