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The investor lives from time 0 to time T; their wealth at time T is denoted W T. He starts with a known initial wealth W 0 (which may include the present value of wage income). At time t he must choose what amount of his wealth to consume, c t , and what fraction of wealth to invest in a stock portfolio, π t (the remaining fraction 1 − π t ...
The Brownian motion models for financial markets are based on the work of Robert C. Merton and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William F. Sharpe, and are concerned with defining the concepts of financial assets and markets, portfolios, gains and wealth in terms of continuous-time stochastic processes.
Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a discrete variable. Thus a non-time variable jumps from one value to another as time moves from one time period to the next.
An alternative terminology uses continuous parameter as being more inclusive. [1] A more restricted class of processes are the continuous stochastic processes; here the term often (but not always [2]) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is ...
Poisson process, an example of a jump process; Continuous-time Markov chain (CTMC), an example of a jump process and a generalization of the Poisson process; Counting process, an example of a jump process and a generalization of the Poisson process in a different direction than that of CTMCs; Interacting particle system, an example of a jump ...
If the model is in continuous time, the controller knows the state of the system at each instant of time. The objective is to maximize either an integral of, for example, a concave function of a state variable over a horizon from time zero (the present) to a terminal time T, or a concave function of a state variable at some future date T. As ...
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical ...