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Brownian Motion and Stochastic Calculus
- I. Karatzas
- Mathematics, Computer Science
This chapter discusses Brownian motion, which is concerned with continuous, Square-Integrable Martingales, and the Stochastic Integration, which deals with the integration of continuous, local martingales into Markov processes.
Stochastic Differential Equations
We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion…
Methods of Mathematical Finance
A Brownian Motion of Financial Markets.- Contingent Claim Valuation in a Complete Market.- Single-Agent Consumption and Investment.- Equilibrium in a Complete Market.- Contingent Claims in Incomplete…
Convex Duality in Constrained Portfolio Optimization
We study the stochastic control problem of maximizing expected utility from terminal wealth and/or consumption, when the portfolio is constrained to take values in a given closed, convex subset of…
Optimal portfolio and consumption decisions for a “small investor” on a finite horizon
A general consumption/investment problem is considered for an agent whose actions cannot affect the market prices, and who strives to maximize total expected discounted utility of both consumption ...
The numéraire portfolio in semimartingale financial models
It is shown that the notion of a no-free-lunch-type notion is the minimal a-priori assumption required in order to proceed with utility optimization, something that the stronger NFLVR condition lacks.
On the optimal stopping problem for one-dimensional diffusions
A new characterization of excessive functions for arbitrary one-dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to…
Martingale and duality methods for utility maximization in a incomplete market
The problem of maximizing the expected utility from terminal wealth is well understood in the context of a complete financial market. This paper studies the same problem in an incomplete market…
Backward stochastic differential equations with reflection and Dynkin games
We establish existence and uniqueness results for adapted solutions of backward stochastic differential equations (BSDE's) with two reflecting barriers, generalizing the work of El Karoui,…