Market Forces and Dynamic Asset Pricing

Risk minimizing portfolios and HJBI equations for stochastic differential games

“…Example 4.1 Let us return to the market (2.3), (2.4) in Section 2. However, as suggested by [27], let us assume that the mean rate of return of the stock, α(. ), is not given a priori, but is the consequence of the portfolio choice π(.)…”

“…This corresponds to changing the original probability measure P to some equivalent martingale measure Q, which is in agreement with the principle of arbitrage free option pricing. See [27] for more discussion on this. The result above is an extension of the result in [27] to (Markovian) stochastic processes r(s), α(s), β(s) and to the jump diffusion case.…”

“…See [27] for more discussion on this. The result above is an extension of the result in [27] to (Markovian) stochastic processes r(s), α(s), β(s) and to the jump diffusion case.…”

Relationship between Maximum Principle and Dynamic Programming in Stochastic Differential Games and Applications

“…Proof. (13), ( 15) can be obtained from the HJBI Equation (10), by the definitions of the generator ,π A  in (9) and the generalized Hamiltonian function G in (12).…”

“…As applications, we discuss a portfolio optimization problem under model uncertainty in the financial market. In this problem, the optimal portfolio strategies for the trader (representative agent) and the "worst case scenarios" (see Peskir and Shorish [13], Korn and Menkens [14]) for the market, derived from both maximum principle and dynamic programming approaches independently, coincide. The relation that we obtained in our main result is illustrated.…”