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We consider the problem of optimal consumption and portfolio in a jump diffusion market in the presence of proportional transaction costs for an agent with constant relative risk aversion utility. We show that the solution in the jump diffusion case has the same form as in the pure diffusion case first solved by Davis and Norman [Mathematics of Operations Research 15 (1990) 676–713]. In particular, we show that (under some assumptions) there is a no transaction coneD in the (x, y)-plane such that it is optimal to make no transactions as long as the wealth position remains in D and to sell/buy stocks according to local time on the boundary of D.

In this paper we study the problem of optimal consumption and investment policy in a jump diffusion market consisting of a bank account and a stock. A well established model for the stock price is the log-normal diffusion or geometric Brownian motion, which has several computational advantages. The rate of return is independent of the past, stationary (i.e. time-homogenous in law) and follows the normal distribution. In this paper we will drop the latter assumption; returns will not be assumed to be Gaussian. A stochastic process with stationary increments independent of the past, and in addition satisfying the mild technical condition of continuity in probability (implying that it has no fixed jump times) is called a Lévy process, and is essentially a piecewise Brownian motion with both drift and Poisson jumps with uniform intensity. We briefly note that we pick the unique right continuous (with left limits) version of all processes, which is the natural choice from a stochastic integration point of view; this also applies to the control processes (View the MathML source) below, ever though that choice may seem less natural from an impulse control point of view. In view of the above, we shall assume that the bank gives a fixed interest rate r, and the bank deposit then follows the equation equation(1.1) View the MathML source The price P2(t) at time t of the stock, is then assumed to be a geometric Lévy process, following the stochastic differential equation equation(1.2) View the MathML source Here α≥r and σ>0 are constants, and W(t) is a Wiener process (Brownian motion) on a filtered probability space View the MathML source. The Ñ entity is View the MathML source-centered Poisson random measure, View the MathML source where N(t, A) is a Poisson random measure measuring the number of jumps with amplitude in (a Borel set) A⊆(−1,∞) up to and including time t. N has a time-homogenous intensity measure View the MathML source; q is then called the Lévy measure associated to N. See e.g. Bensoussan and Lions (1984), Jacod and Shiryaev (1987) and Protter (1990) for more information about such stochastic differential equations. Note that since we only allow jump sizes η which are bigger than −1, the process P2(t) will remain positive for all t≥0, a.s., and will not violate limited liability. On the technical side, we shall assume that equation(1.3) View the MathML source We assume that at any time t the investor can choose a rate c(t)≥0 of consumption taken from the bank account. We also assume that he can transfer money at any time from one asset to the other with a transaction cost which is proportional to the size of the transaction. Let X(t), Y(t) denote the amount of money invested in asset numbers 1 and 2, respectively. Then the evolution equations for X(t), Y(t) are equation(1.4) View the MathML source Here View the MathML source, View the MathML source represent cumulative purchase and sale, respectively, of stocks up to time t. The coefficients λ≥0,μ∈[0,1] represent the constants of proportionality of the transaction costs. Remark 1. By multiplying all processes by e−rt and differentiating using the Itô formula, one will see that the problem only depends on α and r through their difference, just like the Merton problem. It would in fact suffice to consider the case r=0. In this case, X(t) would be non-increasing except at the times we sell stocks. Our controls will have to meet certain conditions. The solvency regionView the MathML source is defined to be the set of states where the net wealth is non-negative: equation(1.5) View the MathML source with boundaries View the MathML source as in Fig. 1. It is natural to require that equation(1.6a) View the MathML source Note that in the presence of a jump term, we need to make sure that we can cover any position we could happen to jump to. Hence, if we define View the MathML source as equation(1.6b) View the MathML source then it is necessary and sufficient for (1.6a) to hold that equation(1.6c) View the MathML source Since we already have to deal with a cone contained in View the MathML source (with equality iff q=0), we get the following generalization more or less for free: let U⊆U′ be a given open convex cone with vertex at the origin. It will later be convenient to characterize U in terms of polar coordinates; let ∂U be given by angles θ1∈[−(π/4),3π/4) and θ2∈(θ1,3π/4] (and such that U⊆U′). Thus, View the MathML source So what we will require, is the following: equation(1.7) View the MathML source The restriction to a (possibly) smaller cone U may be given an economic interpretation as a (say, law enforced) limitations on short sale or leverage. Of particular interest is the case where U is the first quadrant. This serves as the authors’ “moral justification” for the restrictive assumption of Theorem 5, that the no transaction region is contained in the first quadrant (Eq. (4.3)) — an assumption we conjecture not to hold if the Merton line lies outside the first quadrant. Definition 1. The set View the MathML source of admissible controls is the set of predictable consumption–investment policies (View the MathML source) with c(t,ω)≥0 (a.a. (t, ω)) and View the MathML source right-continuous, non-decreasing and View the MathML source, and such that (1.7) holds.The intuition behind requiring (1.7), is that if a jump should bring us out of Ū, then an admissible control will bring us back into Ū immediately. Now since we have chosen to work with the right-continuous version, then “out of Ū” should be interpreted as View the MathML source where equation(1.8) View the MathML source and N({t}, · ) denotes the jump in the Poisson random measure occurring exactly at time t. Define the performance criterion by equation(1.9) View the MathML source where δ>0, γ∈(0,1) are constants and Ex,y is the expectation with respect to the probability law Px,y of (X(t), Y(t)) when View the MathML source. The problem is to find V and (if exists) View the MathML source such that equation(1.10) View the MathML source Due to the choice of utility function, the solvency restriction is necessary for the problem to be well defined; obviously, the only concave extension of CRRA utility, is to put utility equal to minus infinity for negative consumption. In the special case when the stock price is a geometric Brownian motion (i.e. q=0) and there are no transaction costs (i.e. λ=μ=0) this problem was first studied by Merton (1971). He proved that if equation(1.11) View the MathML source then the value function V0(x,y) is given by equation(1.12) View the MathML source where equation(1.13) View the MathML source Moreover, the corresponding optimal consumption View the MathML source is given (in feedback form) by equation(1.14) View the MathML source and the corresponding optimal portfolio is to keep the fraction Y(t)/(X(t)+Y(t)) of wealth invested in the stocks constantly equal to the value equation(1.15) View the MathML source at all times. In other words, it is optimal to perform transactions in such a way that the state (X(t), Y(t)) is always situated on the line View the MathML source in the (x, y)-plane (the Merton line). In Aase (1984) and later in Framstad et al. (1998) (see also Benth et al. (1999)) the results of Merton (1971) are extended to the case when the stock price is a geometric Lévy process, i.e. as (1.2), still assuming that there are no transaction costs, i.e. λ=μ=0. It is proved that the value function V(x, y) still has the same form, namely equation(1.16) V(x,y)=K(x+y)γ but with a different constant K (under an assumption similar to (1.11)). The corresponding optimal consumption c∗ is given by equation(1.17) c∗(x,y)=(Kγ)1/(γ−1)(x+y) and it is still optimal to keep the fraction Y(t)/(X(t)+Y(t)) constantly equal to a value u∗ (see Framstad et al. (1998, Theorem 2.3)). In Framstad et al. (1998, Corollary 2.4) it is proved that if q≠0 then View the MathML source so the introduction of the jump term involving the integral with respect to Ñ has the same effect on the solution as increasing the volatility σ. The purpose of this paper is to study the general case with the stock price given by a geometric Lévy process (1.2) and with proportional transaction costs. As the constant ratio portfolio in the no transaction cost case implies that one must rebalance continuously, it would lead to instant bankruptcy when transaction costs are non-zero, and so it cannot be optimal to keep the fraction of wealth invested in the risky asset constant. We shall, however, prove that it will be optimal to keep that fraction in a fixed interval, i.e. there exists a no transaction region D in the (x, y)-plane with the shape of a cone with vertex at the origin, such that it is optimal to make no transactions as long as (X(t),Y(t))∈D and to sell stocks at the rate of local time (of the reflected process) at the upper/left boundary of D and purchase stocks at the rate of local time at the lower/right boundary. These results generalize the results of Davis and Norman (1990) who obtained similar results in the no jump case (q=0). Our paper is also inspired by the paper of Shreve and Soner (1994), who also considered the case q=0. They used, as we do, a viscosity solution approach and were able to remove some of the assumptions in Davis and Norman (1990). Viscosity solutions of combined stochastic control and optimal stopping problems for jump diffusion processes are studied by Pham (1998). However, his conditions are not satisfied in the case we consider because our utility rate cγ/γ is not bounded as a function of c≥0.

We have obtained, in some sense, the simplest possible generalization to the Merton problem. As noted in Section 1, the optimal portfolio for the CRRA investor in the log-normal case with transaction costs, is obtained by keeping the fraction of wealth invested in the risky asset in a fixed interval, appearing as a “no transaction cone” in the (x, y)-plane, and rebalancing according to the local time at its boundary. We have shown that this result does not depend on log-normality; since the process may jump out of the no transaction cone, we must allow for the generalization of the local time concept in order to have the connection to the Skorohod problem as in Theorem 4. For future research, economic intuition suggests the following properties: • The continuation region tends to the first quadrant as λ→∞ and μ→1. It is tempting to guess that the boundaries of the no transaction region tend monotonically to the axes. This agrees with the conjecture in Shreve and Soner (1994, Remark 11.3) in that when leverage is optimal in the Merton problem, then the presence of transaction cost will reduce the leverage. We expect to see the similar for short-selling as well. Furthermore, if these properties hold then the θi boundary coincides with the ith axis iff the Merton line does, again in accordance with the remark in Shreve and Soner (1994). • Let us note that if θ2=π/2, then we face the following interesting situation. Once on the y-axis, we have dX=0 so that View the MathML source is absolutely continuous for t>0 and we face a pure consumption optimization problem. (A similar thing happens on the x-axis if θ1=0.) It is fairly obvious that if the no transaction region has no boundaries coinciding with axes, then and View the MathML source and View the MathML source are dt-singular, while they are absolutely continuous (for t>0) on the axes. This may be the explanation why it has turned out to be difficult to prove the value function t o be C2 on the axes. In all cases, we conjecture that the assumptions made to ensure that the Merton line lie in the first quadrant, are not needed.