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Merton's portfolio problem is a problem in continuous-time finance and in particular intertemporal portfolio choice.An investor must choose how much to consume and must allocate their wealth between stocks and a risk-free asset so as to maximize expected utility.
Now consider the calculation of , the expectation in the worst 20 out of 100 cases. These cases are as follows: 10 cases from row one, and 10 cases from row two (note that 10+10 equals the desired 20 cases). For row 1 there is a profit of −100, while for row 2 a profit of −20.
The process continues forever, indexed by the natural numbers. An example of a stochastic process which is not a Markov chain is the model of a machine which has states A and E and moves to A from either state with 50% chance if it has ever visited A before, and 20% chance if it has never visited A before (leaving a 50% or 80% chance that the ...
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values.
The ingredients of a stochastic game are: a finite set of players ; a state space (either a finite set or a measurable space (,)); for each player , an action set (either a finite set or a measurable space (,)); a transition probability from , where = is the action profiles, to , where (,) is the probability that the next state is in given the current state and the current action profile ; and ...
Starting from a constant volatility approach, assume that the derivative's underlying asset price follows a standard model for geometric Brownian motion: = + where is the constant drift (i.e. expected return) of the security price , is the constant volatility, and is a standard Wiener process with zero mean and unit rate of variance.
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space.
His PhD thesis, obtained in 1971, initiated the martingale theory of stochastic control. [3] Returning to the UK in 1972, Davis joined the Control Group at Imperial College London . From 1995 to 1999 he was Head of Research and Product Development at Tokyo-Mitsubishi International, leading a quantitative research team providing pricing models ...