რევაზ თევზაძე

We consider an incomplete financial market model, where the dynamics of asset prices is determined by an i?d-valued continuous semimartingale. Using the dynamic programming approach we give a description of the p-optimal martingale measure in terms of the value process for a suitable problem of an optimal equivalent change of measure and show that this value process uniquely solves the corresponding semimartingale backward equation. This result is applied to approximate the lower price and the corresponding hedging strategy of a contingent claim. Key Words Semimartingale backward equation, contingent claim pricing, p-optimal martingale measure, incomplete markets, lower and upper prices.

 We consider a problem of minimization of a hedging error in an incomplete financial market model. The hedging error is measured by a positive strictly convex random function and the dynamics of asset price is given by a continuous one dimensional semimartingale defined on a complete probability space with continuous filtration. Under some regularity assumptions we derive a backward stochastic PDE for the value function of the problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As an example the case of mean-variance hedging is considered.

An incomplete financial market model is considered, where the dynamics of the assets price is described by an Rd-valued continuous semimartingale. We express the density of the minimal entropy martingale measure in terms of the value process of the related optimization problem and show that this value process is determined as the unique solution of a semimartingale backward equation. We consider some extreme cases when this equation admits an explicit solution.

 We give a unified characterization of q-optimal martingale measures for q ∈ [0, ∞) in an incomplete market model, where the dynamics of

asset prices are described by a continuous semimartingale. According to this characterization the variance-optimal, the minimal entropy and the minimal martingale measures appear as the special cases q = 2, q = 1 and q = 0 respectively. Under assumption that the Reverse H¨older condition is satisfied, the continuity (in L1 and in entropy) of densities of q-optimal martingale measures with respect to q is proved.

We consider a problem of minimization of a hedging error, measured by a positive convex random function, in an incomplete financial market model, where the dynamics of asset prices is given by an Rd-valued continuous semimartingale. Under some regularity assumptions we derive a backward stochastic PDE for the value function of the problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As an example the case of mean-variance hedging is considered.

Keywords:

Backward stochasic partial differential equationsemimartinggale market modelincomplete marketsmean-variance hedging

We consider a financial market model, where the dynamics of asset prices are given by an Rd-valued continuous semimartingale and the information flow is right-continuous. Using the dynamic programming approach we express the variance-optimal martingale measure in terms of the value process of a suitable optimization problem and show that this value process uniquely solves the corresponding semimartingale backward equation. We consider two extreme cases when this equation admits an explicit solution. In particular, we give necessary and sufficient conditions in order that the variance-optimal martingale measure coincides with the minimal martingale measure as well as with the martingale measure appearing in the second extreme case.

We study distribution laws of diffusion type processes and corresponding generalized functions. The stochastic integral equation satisfed by these generalized functions is derived.

We derive a backward stochastic differential equation and a Bellman equation characterizing the minimal entropy martingale measure for market models, where asset prices are driven by Markov diffusion processes. A relation between these equations is established.

Keywords: Minimal entropy martingale measure, backward stochastic differential equation, Bellman equation, incomplete market, stochastic volatility model.

We prove an existence of a unique solution of an exponential martingale equation in the class of BMO martingales. The solution is used to characterize optimal martingale measures

We establish the existence of unique solution of an exponential martingale equation in the class of BM O martingales. The solution is used to characterize variance-optimal martingale measures.

We prove the existence of the unique solution of a general backward stochastic differential equation with quadratic growth driven by martingales. A kind of comparison theorem is also proved.

We study the utility maximization problem, the problem of minimization of the hedging error and the corresponding dual problems using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an ℝd-valued continuous semimartingale. Under some regularity assumptions, we derive the backward stochastic PDEs for the value functions related to these problems, and for the primal problem, we show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward SDE. As examples we consider the mean-variance hedging problem and the cases of power, exponential, logarithmic utilities, and corresponding dual problems.

 We discuss possible applications of the 1D direct and inverse scattering problem to the design of universal quantum gates for quantum computation. The potentials generating some universal gates are described.

We consider the mean-variance hedging problem under partial information. The underlying asset price process follows a continuous semimartingale, and strategies have to be constructed when only part of the information in the market is available. We show that the initial mean-variance hedging problem is equivalent to a new mean-variance hedging problem with an additional correction term, which is formulated in terms of observable processes. We prove that the value process of the reduced problem is a square trinomial with coefficients satisfying a triangle system of backward stochastic differential equations and the filtered wealth process of the optimal hedging strategy is characterized as a solution of a linear forward equation.

 An incomplete financial market model is considered, where the dynamics of the assets price is described by an Rd-valued continuous semimartingale. We express the density of the minimal entropy martingale measure in terms of the value process of the related optimization problem and show that this value process is determined as the unique solution of a semimartingale backward equation. We consider some extreme cases when this equation admits an explicit solution.

Key words: Semimartingale backward equation, contingent claim pricing, minimal entropy martingale measure, incomplete markets

We consider the mean-variance hedging problem under partial information in the case where the flow of observable events does not contain the full information on the underlying asset price process. We introduce a certain type martingale equation and characterize the optimal strategy in terms of the solution of this equation. We give relations between this equation and backward stochastic differential equations for the value process of the problem.

We study utility maximization problem for general utility functions using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an $R^d$-valued continuous semimartingale. Under some regularity assumptions we derive backward stochastic partial differential equation (BSPDE) related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As examples the cases of power, exponential and logarithmic utilities are considered.

We consider the mean-variance hedging problem under partial information. The underlying asset price process follows a continuous semimartingale, and strategies have to be constructed when only part of the information in the market is available. We show that the initial mean-variance hedging problem is equivalent to a new mean-variance hedging problem with an additional correction term, which is formulated in terms of observable processes. We prove that the value process of the reduced problem is a square trinomial with coefficients satisfying a triangle system of backward stochastic differential equations and the filtered wealth process of the optimal hedging strategy is characterized as a solution of a linear forward equation.

In this paper, we consider the mean-variance hedging problem of contingent claims in a financial market model composed of assets with uncertain price parameters. We consider the worst case of model parameters required to solve the minimax problem. In general, such minimax problems cannot be changed to maximin problems. The main approach we develop is the randomization of the parameters, which allows us to change minimax to maximin problems, which are easier to solve. We provide an explicit solution for the robust mean-variance hedging problem in the single-period model for some types of contingent claims.

The paper studies the robust maximization of utility from terminal wealth in a diffusion financial market model. The underlying model consists of a tradable risky asset whose price is described by a diffusion process with misspecified trend and volatility coefficients, and a non-tradable asset with a known parameter. The robust functional is defined in terms of a utility function. An explicit characterization of the solution is given via the solution of the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation.

We study the analytical properties of a dynamic value function and of an optimal solution to the utility maximization problem in incomplete markets for utility functions defined on the whole real line. It was shown by that if the relative risk-aversion coefficient of the utility function defined on the half real line is uniformly bounded away from zero and infinity, then the value function at time t = 0 of utility maximization problem is two-times differentiable and the optimal wealth is differentiable in probability with respect to the initial capital. Similar results are true for utility functions defined on the whole real line if instead of relative risk-aversion the same condition on the risk-aversion coefficient is imposed. Besides, assuming the continuity of the filtration F we prove that the derivative of the optimal wealth is strictly positive and that the derivative exists also in the sense of L1-convergence. This enables us to show the existence of a strictly increasing (with respect to the initial capital) and absolutely continuous modification of the optimal wealth. We also study the differentiability properties of the value function V(t,x) and of the optimal wealth process Xt(x) for any t ∈ [0,T] and give the conditions under which the second derivative of the value function and the derivative of the optimal wealth process are continuous. We need these properties to show that the value function satisfies a certain backward stochastic partial differential equation used to characterize the optimal wealth process.

 We study the regularity properties of both the dynamic value function and the optimal solution to the utility maximization problem for utility functions defined on the whole real line. These properties are needed to show that the value function satisfies the corresponding backward stochastic partial differential equation. In particular, in the case of complete markets we give conditions on the utility function when this equation admits a solution.

 We study regularity properties of the dynamic value functions of primal and dual problems of optimal investing for utility functions defined on the whole real line. Relations between decomposition terms of value processes of primal and dual problems and between optimal solutions of basic and conditional utility maximization problems are established. These properties are used to show that the value

function satisfies a corresponding backward stochastic partial differential equation. In the case of complete markets we give conditions

on the utility function when this equation admits a solution.

Keywords: Utility maximization, complete and incomplete markets, duality, Backward stochastic partial differential equation, value function.

Connections between a system of Forward-Backward SDEs and Backward Stochastic PDEs related to the utility maximiza-tion problem is established. Besides, we derive another version of FBSDE of the same problem and prove an existence of a solution

For non-anticipative functionals, differentiable in Chitashvili’s sense, the Itô formula for cadlag semimartingales is proved. Relations between different notions of functional derivatives are established.

We describe the class of functions f:Rn→Rm which transform a vector Brownian Motion into a martingale and use this description to give martingale characterization of the general measurable solution of the multidimensional Cauchy functional equation.