Non-smooth analysis method in optimal investment- a BSDE approach

In this paper, our aim is to investigate necessary conditions for optimal investment. We model the wealth process by Backward differential stochastic equations (shortly for BSDE) with or without constraints on wealth and portfolio process. The constraints can be very general thanks the non-smooth analysis method we adopted.


Introduction
In the sequel, let (Ω, F , P ) be a probability space equipped with a standard Brownian motion W . For a fixed real number T > 0, we consider the filtration F := (F t ) 0≤t≤T which is generated by W and augmented by all P -null sets. The filtered probability space (Ω, F , F, P ) satisfies the usual conditions.
The BSDE approach is a backward view for investment. For comparison, if we take a forward view for the equation (1), then the complicated process z t acts as a control. However, by the theory of BSDE, z t is determined by the terminal value ξ via a one-one correspondence. Thus the BSDE approach has the virtue to handle similar control problem taking ξ as a control instead. Moreover, if we consider the function y 0 := E g t,T (ξ) induced by BSDE with terminal value ξ, a terminal perturbation method, which was first used in Bielecki et al. [1], can be used to analyze the optimal investment problem. Along with this line, later in many years, Ji and Peng [8] used it to obtain a necessary condition via Ekeland variation principle. In this paper, as a generalization, we study optimal investment problems by non-smooth analysis method, which makes more general optimal problems inside our consideration.
Supposing that the investor has initial wealth x, he invest it in the financial market according to the equation (1). By the above analysis, his investment strategy is determined by all available terminal value for him. Let where ρ(·) is a function defined on L 2 T (R) which usually represents a risk measure. However in our paper, it can be a general Lipschitz function. 1 The work of Helin Wu is Supported by the Scientific and Technological Research Program of Chongqing Municipal Education Sometimes, one ask (y t , z t ) satisfy some constraint condition (y(t), z(t)) ∈ Γ t , a.e., a.s. on [0, T ] × Ω, (C) where Γ t := {(y, z)|φ(t, y, z) = 0} ⊂ R × R d and φ satisfies conditions (A1) and (A2). In such constrained case, the investment model should be changed to a Constrained Backward Differential Equation (shortly for CBSDE) as follows, where C t is an increasing RCLL (right continuous an left limit exists) process with C 0 = 0, y t is often called a super-solution of BSDE in the literature. The idea of constrained investment comes from incompleteness or other constraints on investment in financial market. In such case, super-hedging strategies are often adopted. Corresponding to such strategies, the minimal super-solution defined as follow is meaningful.
is said to be the minimal solution, given y T = ξ, subjected to the constraint (C) if for any other g-super-solution (y ′ t , z ′ t , C ′ t ) satisfying (C) with y ′ T = ξ, we have y t ≤ y ′ t a.e., a.s.. We call the minimal solution g Γ -solution and denote it as y t := E g,φ t,T (ξ). In no constrained case, i.e. when φ(t, y, z) ≡ 0 P -a.s. for any t ∈ [0, T ], we denote it as E g t,T (ξ) for convenience.
Our problem in the constrained case is similar to (2) Our paper is organized as follows. In section 2, we first study optimal investment problem without constraints on wealth and portfolio process. With the help of non-smooth analysis, we obtain a necessary condition for an optimal solution, which generalize those obtained in Ji and Peng [8]. Secondly, we continue to consider constrained case. We point out serious difficulties we met in this case and discussed such problems briefly, more details and fully discussion about such constrained problem will be included in our future papers. In sections 3, we give some examples to verify our analysis. At last section, some necessary backgrounds about non-smooth analysis are gathered.

Maximum principle for the optimal investment problem
In this section, we aim to derive some necessary conditions for the optimality of our problem. Suppose the wealth process of an investor evolving according to (1) with limited initial capital x. The optimal problem (2) is a constrained problem. Just as usual, Suppose no constraints on wealth and portfolio process, by an exact penalization method used in non-smooth analysis, we need to assume that i) The risk measure ρ(·) is Lipschitz ii) If we write y 0 E g 0,T (·) as a function of terminal value, it is Lipschitz. If no constraints on wealth and portfolio process, by the theory of BSDE, E g 0,T (·) is obviously Lipschitz, see E. Pardoux, S.G. Peng [7] for example. Proposition 2.1. Suppose g satisfies conditions (A1) and (A2), ξ i ∈ L 2 T (R), (y i t , z i t ), i = 1, 2 are solutions of (1) with terminal values ξ i , then there exists a constant C > 0 such that In order to use results (4.2) in appendix, we need to show Lemma 2.1. Supposing that g satisfies conditions (A1) and (A2), then for any The proof is very similar to the proof of the strict comparison theorem of BSDE.
holds for any η ∈ L 2 T (R). But by the definition In the above equation, set η = −1, then for any ξ, we get a contradiction with f o (ξ * , η) ≥ 0.
Since functions E g 0,T (·) y 0 generated by BSDE via (1) is Lipschitz, then according to in appendix, ∂ o E g 0,T (ξ) is not empty for any ξ, in fact, we have following results.
Theorem 2.1. Suppose that g and ξ are the standard parameters for BSDE, The meaning of the equation above is that, for any where (ỹ t ,z t ) is the solution of BSDE generated by h(t, y, z) = ϕ t y + ψ t z with terminal value ζ.
Proof. By Lemma 2.1, we have 0 / ∈ ∂ o E g 0,T (ξ * ) and thus Proposition 4.2 can be used to deduce that If ξ * is an optimal solution of (2) satisfying E g 0,T (ξ * ) = x, then by the Fermat condition (23), there exists a nonnegative number λ ≥ 0 and some ζ ∈ ∂ o ρ(ξ * ), η ∈ ∂ o E g 0,T (ξ * ) such that ζ + λη = 0. Now supposingx = E g 0,T (ξ * ) < x, then we can set and solve optimal problem on C. It is easy to see that ξ * is optimal on C if it is optimal on C for ρ(·).
In Ji and Peng [8], they assume that the generator is continuously differentiable with variables, in this special case, we can get a explicit form of E g 0,T (·). To do so, we need a notation named strict differentiable for a function in Banach Space and a related theorem.
Theorem 2.3. (Clark [2]) A function f (·) defined on Banach space X is strict differentiable at x ∈ X as in the above definition, then ∂f (x) = {x * }.
Based on the above notations and results, we have Lemma 2.2. If g is continuously differentiable in (y, z) ∈ R × R d with bounded derivatives, (y t = E g 0,t (ξ), z t ) is the solution of BSDE with terminal value ξ, then where q T ∈ L 2 T (R), ∀η ∈ L 2 T (R), q T , η =ỹ 0 = Eg 0,T (η),g(t, y, z) = g y (y t , z t )y + g z (y t , z t )z, i.e.,ỹ t is the solution of BSDE with generatorg(t, y, z).
Proof. By Ji and Peng [8], if g is continuously differentiable in (y, z) ∈ ×R × R d with bounded derivatives, then E g 0,T (·) is strict differentiable and for any η ∈ L 2 T (R), It is obviously that the functionỹ 0 = Eg 0,T (η) deduced by (10) is linear continuous on L 2 T (R), hence by the Riesz representation theorem, there exists q T ∈ L 2 T (R) such that q T , η = Eg 0,T (η) holds for any η ∈ L 2 T (R) and the corresponding sub-differential set contains only one element q T .
Corollary 2.1. Suppose g(t, y, z) has bounded continuous derivatives in (y, z) and (y t = E g 0,t (ξ), z t ) is a solution of BSDE with terminal value ξ. If ξ * is an optimal solution of (2), then there exists ζ ∈ ∂ o ρ(ξ * ) and some positive number λ such that ζ + λq T = 0, where q T is obtained by the Riesz representation theorem through (10).
The key points for our successes to use non-smooth results are Proposition 2.1 and Lemma 2.1 and we can take h(·) as E g 0,T (·) in Proposition 4.2. But in constrained case, the function E g,φ 0,T (·) fails in both Proposition (2.1) and Lemma (2.1). In such case, we try to describe ∂d o Thanks to the lower-semi continuity of E g,φ t,T (·), the constrained set C in

Examples
In this section, some examples are proposed to illustrate the obtained result. In Ji and Peng [8], they considered the following optimal problem to find ξ * such that is minimized under the following constraints s.} and functions u, ϕ are both continuous differentiable with bounded derivatives. In the framework of Ji and Peng [8], if we take the notations of non-smooth analysis, Ji and Peng [8] obtained the following result.
Noting that when u and ϕ are both continuously differentiable, by the above Proposition, ρ(ξ) is absolutely differentiable, then there exists only one element in ∂ρ(ξ), i.e., u x (ξ) + ϕ x (ξ), where u x (·), ϕ x (·) is the corresponding derivative. At the same time, by Lemma 2.2, when the generator g of BSDE is continuously differentiable, the sub-differential of the function E g 0,T (ξ) deduced by BSDE contains only q T , then by Corollary 2.1, there exists a number λ, such that We consider a similar optimal investment problem by non-smooth analysis via BSDE approach.
in a set of variables satisfying the following constraints T (R), ξ ≥ 0, a.s., where g(t, y, z) = r(t)y + θ(t)z, r(t) and θ(t) are coefficients derived from financial market satisfying suitable measurable and integrable conditions.
In this example, since E[ξ] = c is a constant, for any b ∈ R, we take ρ(ξ) := E[ξ 2 + bξ] and it is obviously absolutely differentiable, then the sub-differential at ξ * only contains 2ξ * + b. It is easy to get ∂E g 0,T (ξ * ) = {q T }, where (ỹ t ,ỹ t ) is the solution of following BSDẼ Then, if ξ * is an optimal solution of this example, then by Theorem 2.2 or Corollary 2.1, there exists a number λ b such that

Example 3.2.
Finding an optimal ξ * in the following set Because the constraint on the expectation is not a constant, we take ρ(ξ) . We combine the constraints on the expectation and initial value of investment together to get the following new constraint Let I(x) be the subset of index satisfying Furthermore, if f i is normal, then the equality holds, where coA is the convex hull of A.
By the Lemma stated above and Theorem 2.2, we have the following theorem.
Theorem 3.2. If ξ * is an optimal solution of Example 3.2, then there exist a nonnegative number λ and a ∈ [0, 1] such that   In the classic investment problem, one often take variance as a risk measure, a mean-variance method is used in many literatures. But by Delbaen [4] or Föllmer and Schied [5], such kind of risk measure is not perfect. We often take ρ(·) as a coherent or convex risk measure in (2). In Gianin [6], when g is a sub-additive homogeneous function satisfying some usual conditions, we can define a risk measure via ρ(ξ) := E g 0,T (−ξ).
3. Suppose f is sub-additive homogeneous function satisfying usual conditions and independent of y, f z is continuously bounded, define ρ(ξ) := E f 0,T (−ξ), we want to find an optimal ξ * element in the following constrained set If ξ * is an optimal solution in this example, then we can obtain similar conditions like above examples with new sub-differential sets. By the assumptions of f , we can see obviously ∂ρ(ξ) contains only one elementȳ T , where (ȳ t ,z t ) is the solution of following BSDĒ and (y * t , z * t ) is the solution of BSDE generated by g(t, y, z) with terminal value ξ * . Thus we have the following result.
Suppose that X is a Banach space, X * is its dual space. A function f : X → R is called Lipschitzian if holds for some M > 0, where || · || is the norm in X.
The generalized directional derivative of f at x, denoted as f o (x; v), is defined as where y is a vector in X, t is a positive number. Obviously, f o (x; v) is homogeneous and sub-linear on X, then by Banach Theorem, the generalized derivative is nonempty and weak star compact in X * . By definition, the Fermat optimal principle 0 ∈ ∂ o f (x 0 ) holds when f (x) attains extreme at some point Now, we recall more results in non-smooth analysis. For more details, one can see Clark et al. [3].
Given a set C ⊂ X, the distance function d C (x) : X → R is defined as d C (x) := inf{||y − x||, y ∈ C}.
The following lemma transfers the constrained problem to the unconstrained case.
Lemma 4.1. (Exact penalization) Suppose that f is a Lipschitz function with coefficient K defined on S, x ∈ C ⊂ S and f takes its minimum value at x on C. Then, for anyK ≥ K, g(y) = f (y) +Kd C (y) attains minimum value at x on S. On the contrary, ifK > K and C is closed, then the minimum point of g on S must belong to C.
Contingent and normal derivative for a set C are defined by the distance function, see Clark et al. [3] for details.
Definition 4.1. Assume x ∈ C, if d o C (x; v) = 0, then v is said to be a contingent derivative at x ∈ X. We denote the set of contingent derivatives as T C (x). By polarity, we define the normal derivative set as N C (x) := {ζ ∈ X * |ζ(v) ≤ 0, ∀v ∈ T C (x)}.
By the above definition, we have the following proposition. Supposing that x ∈ C, then it holds that where cl means the weak star closure.
holds. For a special kind of set C, we have the following result.