By Dauxois J.-Y., Druihlet P., Pommeret D.

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**Extra resources for A Bayesian Choice Between Poisson, Binomial and Negative Binomial Models**

**Example text**

The investor is to seek an optimal consumption-trading strategy in order to maximize the expected utility from the total discounted consumption. The portfolio optimization problem is formulated as an infinite-dimensional stochastic classical-impulse control problem due to the hereditary nature of the stock price dynamics and inventories. The HJBQVI together with its boundary conditions for the value function is derived under some smoothness conditions. The verification theorem for the optimal strategy is given.

Y(0)) ∈ (S)N +1 , P {ζ(k+1)h = y | ζ(ih), u(ih), i ≤ k} = p(N ) (ζkh , u(kh); y) if y(−ih) = ζ((−i + 1)h) for 1 ≤ i ≤ N 0 otherwise Introduction and Summary 15 (iii) Local Consistency with the Drift Coefficient: (N,u) b(kh) ≡ Eψ = hf (Π ≡ hf [ζ((k + 1)h) − ζ(kh)] (N ) (N ) (ζkh ), u(kh)) + o(h) (ζkh , u(kh)), where f (N ) : ℜN +1 × U → ℜ is defined by f (N ) (x, u) = f (Π (N ) (x), u), ∀(x, u) ∈ ℜN +1 × U, N,u(·) is the conditional expectation given the discrete admissible control Eψ (N ) u (·) = {u(kh), k = 0, 1, 2, .

5). At the time of a transaction (say, t ≥ 0), the investor is required to pay a capital gains tax (respectively, be paid a capital loss credit) in the amount that is proportional to the amount of profit (respectively, loss). A sale of stock shares is said to result in a profit if the current stock price S(t) is higher than the base price B(t) of the stock and it is a loss otherwise. The base price B(t) is defined to be the price at which the stock shares were previously bought or short sold; that is, B(t) = S(t − τ (t)), where τ (t) > 0 is the time duration for which those shares (long or short) have been held at time t.

### A Bayesian Choice Between Poisson, Binomial and Negative Binomial Models by Dauxois J.-Y., Druihlet P., Pommeret D.

by Brian

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