Dmitry Kramkov

Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets

SummaryLetM(X) be the family of all equivalent local martingale measuresQ for some locally boundedd-dimensional processX, andV be a positive process. The main result of the paper (Theorem 2.1) states that the processV is a supermartingale whateverQ∈M(X), if and only if this process admits the following decomposition:

Asymptotic arbitrage in large financial markets

V_t

Existence of an endogenously complete equilibrium driven by a diffusion

whereH is an integrand forX, andC is an adapted increasing process. We call such a representationoptional because, in contrast to the Doob-Meyer decomposition, it generally exists only with an adapted (optional) processC. We apply this decomposition to the problem of hedging European and American style contingent claims in the setting ofincomplete security markets.

A system of quadratic BSDEs arising in a price impact model

In this paper, we study the problem of expected utility maximization of an agent who, in addition to an initial capital, receives random endowments at maturity. Contrary to previous studies, we treat as the variables of the optimization problem not only the initial capital but also the number of units of the random endowments. We show that this approach leads to a dual problem, whose solution is always attained in the space of random variables. In particular, this technique does not require the use of finitely additive measures and the related assumption that the endowments are bounded.

Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model

Abstract. A large financial market is described by a sequence of standard general models of continuous trading. It turns out that the absence of asymptotic arbitrage of the first kind is equivalent to the contiguity of sequence of objective probabilities with respect to the sequence of upper envelopes of equivalent martingale measures, while absence of asymptotic arbitrage of the second kind is equivalent to the contiguity of the sequence of lower envelopes of equivalent martingale measures with respect to the sequence of objective probabilities. We express criteria of contiguity in terms of the Hellinger processes. As examples, we study a large market with asset prices given by linear stochastic equations which may have random volatilities, the Ross Arbitrage Pricing Model, and a discrete-time model with two assets and infinite horizon. The suggested theory can be considered as a natural extension of Arbirage Pricing Theory covering the continuous as well as the discrete time case.

On a stochastic differential equation arising in a price impact model

We develop a single-period model for a large economic agent who trades with market makers at their utility indifference prices. We compute the sensitivities of these market indifference prices with respect to the size of the investor’s order. It turns out that the price impact of an order is determined both by the market makers’ joint risk tolerance and by the variation of individual risk tolerances. On a technical level, a key role in our analysis is played by a pair of conjugate saddle functions associated with the description of Pareto optimal allocations in terms of the aggregate utility function.

Muckenhoupt’s (Ap) condition and the existence of the optimal martingale measure

The existence of complete Radner equilibria is established in an economy whose parameters are driven by a diffusion process. Our results complement those in the literature. In particular, we work under essentially minimal regularity conditions and treat the time-inhomogeneous case.

Optional decompositions under constraints

We consider a financial model where the prices of risky assets are quoted by a representative market maker who takes into account an exogenous demand. We characterize these prices in terms of a system of BSDEs with quadratic growth. We show that this system admits a unique solution for every bounded demand if and only if the market makers risk-aversion is sufficiently small. The uniqueness is established in the natural class of solutions, without any additional norm restrictions. To the best of our knowledge, this is the first study that proves such (global) uniqueness result for a system of fully coupled quadratic BSDEs.