Portfolio Selection with Jumps under Regime Switching

Remark 2.1. Generally speaking, one uses noncompensated Poisson processes in a jump diffusion model (see Kushner [23]). However, we use compensated Poisson processes in (2.6) instead of using noncompensated Poisson processes, this is because firstly, using relationship (2.1) we can easily transform a jump diffusion model driven by noncompensated Poisson processes into a jump diffusion model driven by compensated Poisson processes; secondly, using compensated Poisson processes we can keep the Riccati Equation (4.2) similar to that of a diffusion model without jump processes, and then H(t, i) in (4.3) has a financial interpretation.

Definition 2.2. A portfolio π(·) is said to be admissible if $$ and the SDE (2.15) has a unique solution x(·) corresponding to π(·). In this case, we refer to (x(·), π(·)) as an admissible (wealth, portfolio) pair.

Remark 2.3. Most works in the literature define a portfolio, say π(·), as the fractions of wealth allocated to different stocks. That is,

Definition 2.4. The mean-variance portfolio selection is a constrained stochastic optimization problem, parameterized by ζ ∈ ℝ¹:

Lemma 2.5. Given a d-dimensional process y(·) satisfying

where f, g, and γ satisfy Lipschitz condition with appropriate dimensions, each column γ^(k) of the matrix γ = [γ_ij] depends on z only through the kth coordinate z_k. Let φ(t, x, i) ∈ C^1,2([0, T] × ℝⁿ × S; ℝ), one then has where where μ is a martingale measure, and $$ is a martingale measure. And γ(dt, dy) is a Poisson random measure with intensity dt × μ(dy), in which μ is the Lebesgue measure on ℝ.

Lemma 3.1. Let ψ(·, i), i = 1,2, …, l, be the solutions to the following system of linear ordinary differential equations (ODEs):

Then the mean-variance problem (2.19) is feasible for every ζ ∈ ℝ¹ if and only if

Proof. To prove the “if" part, construct a family of admissible portfolios π^β(·) = βπ(·) for β ∈ ℝ¹ where

Assume that x^β(t) is the solution of (2.15). Let x^β(t) = x⁰(t) + βy(t), where x⁰(·) satisfies (3.1), and y(·) is the solution to the following equation: Therefore, problem (2.19) is feasible for every ζ ∈ ℝ¹ if there exists β ∈ ℝ such that ζ = 𝔼x^β(T) ≡ 𝔼x⁰(T) + β𝔼y(T). Equivalently, (2.19) is feasible for every ζ ∈ ℝ if 𝔼y(T) ≠ 0. Applying the generalized Itô formula (Lemma 2.5) to φ(t, x, i) = ψ(t, i)x, we have Integrating from 0 to T, taking expectation, and using (3.5), we obtain Consequently, 𝔼y(T) ≠ 0 if (3.4) holds.

Conversely, suppose that problem (2.19) is feasible for every ζ ∈ ℝ¹. Then for each ζ ∈ ℝ, there is an admissible portfolio π(·) so that 𝔼x(T) = ζ. However, we can always decompose x(t) = x⁰(t) + y(t) where y(·) satisfies (3.6). This leads to 𝔼x⁰(T) + 𝔼y(T) = ζ. However, 𝔼x⁰(T) ≡ ζ⁰ is independent of π(·); thus it is necessary that there is a π(·) with 𝔼y(T) ≠ 0. It follows then from (3.8) that (3.4) is valid.

Theorem 3.2. The mean-variance problem (2.19) is feasible for every ζ ∈ ℝ if and only if

Proof. By virtue of Lemma (3.1), it suffices to prove that ψ(t, i) > 0  ∀ t ∈ [0, T],  i = 1,2, …, l. To this end, note that (3.3) can be rewritten as

Treating this as a system of terminal-valued ODEs, a variation-of-constant formula yields Construct a sequence ψ^(k)(·, i) (known as the Picard sequence) as follows: Noting that q_ij ≥ 0 for all j ≠ i, we have On the other hand, it is well known that ψ(t, i) is the limit of the Picard sequence ψ^(k)(t, i) as k → ∞. Thus ψ(t, i) > 0. This proves the desired result.

Corollary 3.3. If (3.9) holds, then for any ζ ∈ ℝ, an admissible portfolio that satisfies 𝔼x(T) = ζ is given by

where x⁰ and ϱ are given by (3.2) and (3.4), respectively.

Proof. This is immediate from the proof of the “if" part of Lemma (3.1)

Then one has

Corollary 3.4. If $$, then any admissible portfolio π(·) results in 𝔼x(T) = ζ⁰.

Proof. This is seen from the proof of the “only if" part of Lemma (3.1)

since $$.

Remark 3.5. Condition (3.9) is very mild. For example, (3.9) holds as long as there is one stock whose appreciation-rate process is different from the interest-rate process at any market mode, which is obviously a practically reasonable assumption. On the other hand, if (3.9) fails, then Corollary (3.4) implies that the mean-variance problem (2.19) is feasible only if ζ = ζ⁰. This is pathological and trivial case that does not warrant further consideration. Therefore, from this point on we will assume that (3.9) holds or, equivalently, the mean-variance problem (2.19) is feasible for any ζ.

Proposition 4.1. The solutions of (4.2) and (4.3) must satisfy P(t, i) > 0 and 0 < H(t, i) ≤ 1, ∀t ∈ [0, T], i = 1,2, …, l. Moreover, if for a fixed i, r(t, i) > 0, a.e., t ∈ [0, T], then H(t, i) < 1, ∀t ∈ [0, T).

Proof. The assertion P(t, i) > 0 can be proved in exactly the same way as that of ψ(t, i) > 0; see the proof of Theorem 3.2. Having proved the positivity of P(t, i), one can then show that H(t, i) > 0 using the same argument because now P(t, j)/P(t, i) > 0.

To prove that H(t, i) ≤ 1, first note that the following system of ODEs:

has the only solutions $$,  i = 1,2, …, l, due to the uniqueness of solutions. Set which solves the following equations: A variation-of-constant formula leads to A similar trick using the construction of Picard′s sequence yields that $$. In addition, $$, ∀t ∈ [0, T), if r(t, i) > 0, a.e., t ∈ [0, T]. The desired result then follows from the fact that $$.

Remark 4.2. Equation (4.2) is a Riccati type equation that arises naturally in studying the stochastic LQ control problem (3.19) whereas (4.3) is used to handle the nonhomogeneous terms involved in (3.19); see the proof of Theorem 4.3. On the other hand, H(t, i) has a financial interpretation: for fixed (t, i), H(t, i) is a deterministic quantity representing the risk-adjusted discount factor at time t when the market mode is i (note that the interest rate itself is random).

Theorem 4.3. Problem (3.19) has an optimal feedback control

Moreover, the corresponding optimal value is where with the transition probabilities $$ given by (2.4).

Proof. Let π(·) be any admissible control and x(·)  the corresponding state trajectory of (2.15). Applying the generalized Itô formula (Lemma 2.5) to

we obtain where π^*(t, x, i) is defined as the right-hand side of (4.8). Integrating the above from 0 to T and taking expectations, we obtain Consequently, Since P(t, α(t)) > 0 by Proposition (4.1), it follows immediately that the optimal feedback control is given by (4.8) and the optimal value is given by (4.9), provided that the corresponding equation (2.15) under the feedback control (4.8) has a solution. But under (4.8), the system (2.15) is a nonhomogeneous linear SDE with coefficients modulated by α(t). Since all the coefficients of this linear equation are uniformly bounded and α(t) is independent of W(t), the existence and uniqueness of the solution to the equation are straightforward based on a standard successive approximation scheme.

Finally, since

and q_ij ≥ 0 for all  i ≠ j, we must have θ ≥ 0. This completes the proof.

Theorem 5.1 (efficient portfolios and efficient frontier). Assume that (3.9) holds. Then one has

Moreover, the efficient portfolio corresponding to z, as a function of the time t, the wealth level x, and the market mode i, is where Furthermore, the optimal value of   Var x(T), among all the wealth processes x(·) satisfying 𝔼x(T) = ζ, is

Proof. By assumption (3.9) and Theorem 3.2, the mean-variance problem (2.19) is feasible for any ζ ∈ ℝ¹. Moreover, using exactly the same approach in the proof of Theorem 4.3, one can show that problem (2.19) without the constraint 𝔼x(T) = ζ must have a finite optimal value, hence so does the problem (2.19). Therefore, (2.19) is finite for any ζ ∈ ℝ¹. Now we need to prove that J_MV(x₀, i₀, π(·)) is strictly convex in π(·). We can easily get

where κ ∈ [0,1]. So, we obtain which proves J_MV(x₀, i₀, π(·)) is strictly convex in π(·). that Affine space means the complement of points at infinity. It can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one. Since J_MV(x₀, i₀, π(·)) is strictly convex in π(·) and the constraint function 𝔼x(T) − ζ is affine in π(·), we can apply the well-known duality theorem (see [26, page 224, Theorem 1]) to conclude that for any ζ ∈ ℝ¹, the optimal value of (2.19) is By Theorem 4.3, inf _{π(·)admissible}J(x₀, i₀, π(·), λ) is a quadratic function (4.9) in λ − ζ. It follows from the finiteness of the supremum value of this quadratic function that Now if then again by Theorem 4.3 and (5.7) we must have for every ζ ∈ ℝ¹, which is a contradiction. This proves (5.1). On the other hand, in view of (5.7), we maximize the quadratic function (4.9) over λ − ζ and conclude that the maximizer is given by (5.3) whereas the maximum value is given by the right-hand side of (5.4). Finally, the optimal control (5.2) is obtained by (4.8) with λ = λ^*.The efficient frontier (5.4) reveals explicitly the tradeoff between the mean (return) and variance (risk) at the terminal. Quite contrary to the case without Markovian jumps [17], the efficient frontier in the present case is no longer a perfect square (or, equivalently, the efficient frontier in the mean-standard deviation diagram is no more a straight line). As a consequence, one is not able to achieve a risk-free investment. This, certainly, is expected since now the interest rate process is modulated by the Markov chain, and the interest rate risk cannot be perfectly hedged by any portfolio consisting of the bank account and stocks [27], because the Markov chain is independent of the Brownian motion. Nevertheless, expression (5.4) does disclose the minimum variance, namely, the minimum possible terminal variance achievable by an admissible portfolio, along with the portfolio that attains this minimum variance.

Theorem 5.2 (minimum variance). The minimum terminal variance is

with the corresponding expected terminal wealth and the corresponding Lagrange multiplier $$. Moreover, the portfolio that achieves the above minimum variance, as a function of the time t, the wealth level x, and the market mode i, is

Proof. The conclusions regarding (5.11) and (5.12) are evident in view of the efficient frontier (5.4). The assertion $$ can be verified via (5.3) and (5.12). Finally, (5.13) follows from (5.2).

Remark 5.3. As a consequence of the above theorem, the parameter s can be restricted to ζ ≥ ζ_min when one defines the efficient frontier for the mean-variance problem (2.19).

Theorem 5.4 (mutual fund theorem). Suppose that an efficient portfolio $$ is given by (5.2) corresponding to ζ = ζ₁ > ζ_min. Then a portfolio π^*(·) is efficient if and only if there is a μ ≥ 0 such that

where $$ is the minimum variance portfolio defined in Theorem 5.2.

Proof. We first prove the “if" part. Since both $$ and $$ are efficient, by the explicit expression of any efficient portfolio given by (5.2), $$ must be in the form of (5.2) corresponding to ζ = (1 − μ)ζ_min + μζ₁ (also noting that x^*(·) is linear in π^*(·)). Hence π^*(t) must be efficient.

Conversely, suppose that π^*(·) is efficient corresponding to a certain ζ ≥ ζ_min. Write ζ = (1 − μ)ζ_min + μζ₁ with some μ ≥ 0. Multiplying

by (1 − μ), multiplying by μ, and summing them up, we obtain that $$ is represented by (5.2) with $$ and ζ = (1 − μ)ζ_min + μζ₁. This leads to (5.14).

Remark 5.5. The above mutual fund theorem implies that any investor needs only to invest in the minimum variance portfolio and another prespecified efficient portfolio in order to achieve the efficiency. Note that in the case where all the market parameters are deterministic [17], the corresponding mutual fund theorem becomes the one-fund theorem, which yields that any efficient portfolio is a combination of the bank account and a given efficient risky portfolio (known as the tangent fund). This is equivalent to the fact that the fractions of wealth among the stocks are the same among all efficient portfolios. However, in the present Markov-modulated case this feature is no longer available.