[Paper Review] Optimal Investment in a Dual Risk Model

Arash Fahim, Lingjiong Zhu|arXiv (Cornell University)|Oct 16, 2015

Probability and Risk Models23 references3 citations

TL;DR

This paper proposes an optimal investment strategy in research and development (R&D) for a dual risk model to minimize the ruin probability of a high-tech or venture capital firm. By modeling R&D investment as a control variable that affects both the drift and jump intensity of the surplus process, the authors derive closed-form expressions for the minimal ruin probability under state-dependent dynamics, showing that optimal R&D investment can lead to exponential decay in ruin probability when wealth exceeds a critical threshold.

ABSTRACT

Dual risk models are popular for modeling a venture capital or high tech company, for which the running cost is deterministic and the profits arrive stochastically over time. Most of the existing literature on dual risk models concentrated on the optimal dividend strategies. In this paper, we propose to study the optimal investment strategy on research and development for the dual risk models to minimize the ruin probability of the underlying company. We will also study the optimization problem when in addition the investment in a risky asset is allowed.

Motivation & Objective

• To address the lack of optimization studies on R&D investment in dual risk models, which traditionally focus only on dividend strategies.
• To model how increasing R&D investment affects the company's running cost and profit arrival intensity, thereby influencing ruin risk.
• To derive the optimal R&D investment strategy that minimizes the ultimate ruin probability of a high-tech or venture capital firm.
• To extend the dual risk model to include state-dependent dynamics where R&D investment alters both the drift and jump intensity of the surplus process.
• To provide explicit analytical expressions for the minimal ruin probability under optimal R&D investment, enabling quantitative assessment of strategic decisions.

Proposed method

• The paper formulates a state-dependent dual risk model where the running cost and profit arrival rate depend on the company's current wealth and R&D investment level.
• R&D investment is modeled as a control variable that increases the profit arrival intensity while raising the running cost, leading to a trade-off in ruin probability.
• The optimal investment strategy is derived using stochastic control theory, solving a Hamilton-Jacobi-Bellman (HJB) equation for the value function of the ruin probability.
• The solution involves constructing a candidate optimal control and verifying it via verification theorems, leading to explicit expressions for the minimal ruin probability.
• For the case with γ = 1, the optimal strategy is bang-bang: invest nothing if wealth is below a critical threshold x*, and invest infinitely if above it.
• The paper derives closed-form expressions for the minimal ruin probability using integrals involving the modified Bessel function and the complementary error function (erfc), particularly for exponential and power-law jump size distributions.

Experimental results

Research questions

• RQ1 What is the optimal level of R&D investment that minimizes the ruin probability in a dual risk model with state-dependent dynamics?
• RQ2 How does the optimal R&D investment strategy depend on the initial wealth and model parameters such as the profit arrival intensity and cost rate?
• RQ3 What is the analytical form of the minimal ruin probability when R&D investment is optimally chosen?
• RQ4 Does optimal R&D investment lead to faster decay (exponential vs. polynomial) of the ruin probability as initial wealth increases?
• RQ5 Under what conditions does it become optimal to invest heavily in R&D, and when is it better to refrain from investment?

Key findings

• The minimal ruin probability under optimal R&D investment decays exponentially in the initial wealth when the company’s wealth exceeds a critical threshold x*, while it decays polynomially when wealth is below x*.
• For the case γ = 1, the optimal strategy is bang-bang: no investment when wealth x ≤ x*, and infinite investment when x > x*, with x* = (λ₀ − δ₀ + ν)/(δ₀ − ν).
• The closed-form expression for the minimal ruin probability is given by a ratio of integrals involving the modified Bessel function and the complementary error function, with explicit dependence on model parameters.
• When no R&D investment is made (C₀ = 0), the ruin probability decays polynomially and is strictly between 0 and 1, as shown in Equation (5.8).
• Numerical illustrations confirm that optimal R&D investment significantly reduces ruin probability, especially for higher initial wealth, with the red dashed curve (optimal) lying well below the blue curve (no investment).
• The model exhibits subexponential, exponential, or superexponential decay of ruin probability depending on the state-dependent parameters, challenging the assumption of convexity in ruin probability as a function of initial wealth.