**2. Formulation**

We focus exclusively on a simple context with just two assets available in the financial market, namely a risky asset and a risk-free asset. In practice, the risky asset would be a broad market index fund. An investor saves for retirement at time *T*. The amounts that this investor's portfolio contains of the risky and risk-free assets at time *t* are denoted by *St* and *Bt*, respectively. The investor's total wealth from the portfolio (i.e., the total value of the portfolio) at *t* is *Wt* = *St* + *Bt*. The fraction of total wealth invested in the risky asset is *pt* = *St*/*Wt*. The investment period runs from the inception time *t* = 0 to the horizon date *t* = *T*. There is a set of *M* + 1 pre-determined *action times* denoted by T ,

$$\mathcal{T} \equiv \{ t\_0 = 0 \le t\_1 \le \cdots \le t\_M = T \}. \tag{1}$$

At the horizon date *tM* = *T*, the portfolio is liquidated. At each action time *ti* ∈ T1 = T \{*tM*} (i.e., each action time prior to *T*), (i) an amount of cash *qi* is contributed to the portfolio and then (ii) the portfolio is rebalanced.<sup>7</sup>

Let the instant before action time *ti* be *t* − *i* = *ti* − , where → 0+. Similarly, the instant after *ti* is denoted by *t* + *i* = *ti* + . To simplify notation, let *S*<sup>+</sup> *i* = *St* + *i* , *S*− *i* = *St* − *i* , *B*<sup>+</sup> *i* = *Bt* + *i* , *B*− *i* = *Bt* − *i* , *W*<sup>+</sup> *i* = *Wt* + *i* , and *W*− *i* = *Wt* − *i*. Similarly, let *pt* + *i*= *pi*.

 Between action times (i.e., *t* ∈ T / ), the value of the investor's portfolio will fluctuate in accordance with changes in the unit prices of the two assets. We assume a constant risk-free rate *r*, so that the evolution of the amount invested in the risk-free asset is

$$dB\_t = rB\_t \, dt,\qquad t \notin \mathcal{T}.\tag{2}$$

The dynamics of the changes in the amount invested in the risky asset between action times are given by the jump diffusion process

$$\frac{dS\_t}{S\_{t^-}} = (\mu - \lambda \kappa) \, dt + \sigma \, dZ + d\left(\sum\_{i=1}^{\pi\_t} (\xi\_i - 1)\right), \qquad t \notin \mathcal{T},\tag{3}$$

where *μ* is the (uncompensated) drift rate, *σ* is the volatility, *dZ* is the increment of a Wiener process, *πt* is a Poisson process with intensity *λ*, and *ξ* denotes the random jump multiplier. When a jump

<sup>7</sup> As discussed below, in the case of an optimal QS strategy, the investor may also withdraw cash from the portfolio at an action time.

occurs, *St* = *ξSt*− , and *κ* = *E*[*ξ* − 1] where *<sup>E</sup>*[·] is the expectation operator. We assume that *ξi* are i.i.d. positive random variables characterized by a double exponential distribution (Kou and Wang, 2004). Given that a jump occurs, *pup* is the probability of an upward jump and 1 − *pup* is the probability of a downward jump. The density function *f*(*y* = log *ξ*) is then

$$f(y) = p\_{up} \eta\_1 e^{-\eta\_1 y} \mathbf{1}\_{y \ge 0} + (1 - p\_{up}) \eta\_2 e^{\eta\_2 y} \mathbf{1}\_{y < 0}.\tag{4}$$

We do not permit short sales of the risky asset, and we impose an upper bound on the use of leverage, i.e., borrowed funds obtained through short sales of the risk-free asset. This means that there is an upper bound on the weight that the investor can place on the risky asset, which we denote by *L*max. In other words, 0 ≤ *pi* ≤ *L*max for all action times *ti* ∈ T1. Generally, with a DC account, it is reasonable to specify *L*max = 1, ruling out the use of any leverage. Since the value of the risky asset follows the jump diffusion (3), if we allow leverage by setting *L*max > 1, the investor can become insolvent.<sup>8</sup> We add the further constraint that, if the investor becomes insolvent at any time, then trading stops and all positions in the risky asset are liquidated.<sup>9</sup> In insolvency, debt accumulates until it is (possibly) eliminated by cash contributions. We emphasize that insolvency can only occur if leverage is allowed, i.e., *L*max > 1.
