On The Existence and Uniqueness of Wealth Dynamics and Derivation of Optimal Portfolio-Consumption Strategies For an Investor

Abstract

We consider the existence and uniqueness of investor’s wealth dynamics and optimization of investment portfolio and consumption processes. We described the existence and uniqueness of our dynamics using already existing approaches. Using the method of successive approximation, we found that
P{∫₀μ(s,X^(K)(s))ds→∫₀μ(s,X(s))ds}=1 and

P{∫₀σ(s,X^(K)(s))dW(s)→∫₀μ(s,X(s))dW(s)}=1
in probability as k→∞ for each t[0,T]. This shows that the limit process X(t) satisfies our Stochastic wealth equation (1.9). We assume that the investor invested his short positions into a riskless asset and N risky assets. We also assume that the market is complete, arbitrage-free and continuously open. We derived the optimal portfolio as well as the optimal consumption strategies for an investor.