Non smooth analysis method in optimal investment BSDE approach

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R E S E A R C H

Open Access

Non-smooth analysis method in optimal

investment-BSDE approach

Helin Wu

1

_{, Yong Ren}

2

_{and Feng Hu}

3*

*_{Correspondence:} [email protected]

3_{School of Statistics, Qufu Normal} University, Qufu, China Full list of author information is available at the end of the article

Abstract

In this paper, the investment process is modeled by backward stochastic diﬀerential equation. We investigate a necessary condition for optimal investment problem by the method of non-smooth analysis. Furthermore, some applications of our result are given.

MSC: 60H10; 93E20; 49J52

Keywords: Backward stochastic diﬀerential equation (BSDE); Non-smooth analysis; Optimal investment

1 Introduction

Single-period mean-variance portfolio selection was ﬁrst introduced and studied by Markowitz [14]. Markowitz’s pioneer work [14] has laid down the foundation for modern ﬁnancial portfolio theory. In 2000, Li and Ng [11] extended the Markowitz model to the dynamic setting. From then on, various continuous time Markowitz models were studied in much of the literature (see, e.g., [1,12,13,21]).

Generally, there are two approaches, i.e. the forward (primal) approach and the back-ward (dual) approach, employed to solve the above problem in the continuous time case. The first approach is inspired by indefinite stochastic linear quadratic control (see, e.g., [3,20]) and builds the relationship between the mean-variance problem and a family of indefinite stochastic linear quadratic optimal control problems (see, e.g., [12,13,21]). The second approach is first studied by Bielecki et al. in [1], which is the generalization of the well-known risk neutral computational approach in the discrete time case (see, e.g., [9,17, 18]). The backward approach includes two steps: the first step is to compute the optimal terminal wealth; the second step is to compute the replicating portfolio strategy corre-sponding to the obtained optimal terminal wealth. As shown in [1], the optimal terminal wealth is first obtained using the Lagrange multiplier method and then the optimal repli-cating portfolio strategy is obtained by solving a backward stochastic differential equation (BSDE for short). Along this line, in 2008, Ji and Peng [10] used Ekeland’s variational prin-ciple to obtain a necessary condition for portfolio optimization problem by the backward approach. In this paper, we study optimal investment problem by the method of non-smooth analysis, which makes more general portfolio optimization problem be inside the scopes of our consideration.

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Now the framework of optimal investment problem that we study are introduced. Let (Ω,F,P) be a probability space and (Wt)t≥0be ad-dimensional standard Brownian motion

with respect to ﬁltration (Ft)t≥0generated by Brownian motion and allP-null subsets, i.e.,

Ft=σ{Ws;s≤t} ∨N,

whereN is the set of allP-null subsets. Fix a real numberT> 0. Denote F := (Ft)t∈[0,T]

Here, letL2₍_F

T) denote the space ofFT-measurable random variablesξ:Ω→R

satis-fyingE[ξ2_{] <}_∞;_H2

T(Rn) denote the space of F-predictable processesϕ:Ω×[0,T]→Rn

satisfyingE[₀T|ϕt|2dt] <∞andS2T(R) denote the space of F-progressively measurable

RCLL processesϕ:Ω×[0,T]→RsatisfyingE[sup₀_≤_t_≤_T|ϕt|2] <∞.

Given an initial capitalx, the investment process can be modeled by the following one-dimensional BSDE on [0,T]:

yt=ξ+

T

t

g(s,ys,zs) ds–

T

t

zs·dWs (1)

with y0≤x, whereξ ∈L2(FT) andg(ω,t,y,z) :Ω×[0,T]×R×Rd→Ris a function

uniformly satisfying the Lipschitz condition, i.e., there exists a positive constantMsuch that for all (y1,z1), (y2,z2)∈R×Rd

g(ω,t,y1,z1) –g(ω,t,y2,z2)≤M

|y1–y2|+|z1–z2|

(A.1)

and

g(·, 0, 0)∈H_T2(R). (A.2)

By Pardoux and Peng [15], we know that: Suppose thatgsatisﬁes (A.1) and (A.2). Then, for anyξ∈L2₍_F

T), BSDE (1) has a unique pair of adapted processes (yt,zt)∈ST2(R)×HT2(Rd).

Usually, we call{yt}a wealth process and{zt}a portfolio process.

Assume that the investor has initial wealthx, he invests it in the ﬁnancial market ac-cording to BSDE (1). DenoteE_0,g_T(ξ) :=y0. Then his investment strategy is determined by

all available terminal values for him, i.e.,

A(x) :=ξ∈L2(FT)|E0,gT(ξ)≤x

.

In this paper, we study the following problem: if there existsξ∗∈A(x), which is the optimal objective of problem

min

ξ∈A(x)ρ(ξ), (2)

what is the necessary condition forξ∗? Here,ρis a function deﬁned onL2₍_F

T) and

rep-resents a risk measure. We also assume thatρsatisﬁes the Lipschitz condition.

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2 Some results about non-smooth analysis

In this section, we give some results about non-smooth analysis that are used in this paper. The following deﬁnitions, lemmas and propositions can be found in [4,5].

Suppose thatXis a Banach space, andX∗is its dual space. Obviously,L2₍_F

T) is a Banach

space with the normξ:= (E[ξ2_])12 _{for any}_ξ∈_L2₍F_T_{), and}_L2₍F_T_{) is also its dual space.}

Deﬁnition 1 Suppose that f is a function deﬁned on X, Given an x ∈ X, if

lim sup_y_∈_X_,_y_→_x_,_t_↓₀f(y+tv)–f(y)

t ∈Rfor anyv∈X, denote

fo(x;v) := lim sup y∈X,y→x,t↓0

f(y+tv) –f(y)

t . (3)

We callfo(x;v) the generalized directional derivative off atx.

Remark1 Suppose that a functionf :X→Rsatisﬁes Lipschitz condition, i.e., for anyx1,

x2∈X,

_f(x1) –f(x2)≤Mx1–x2

holds for someM> 0, depending onf. By Deﬁnition1, we have:

(i) Obviously,fo₍_x_;_v_{) is sub-linear on}_X_{, then, by the Hahn–Banach theorem, the}

gener-alized directional derivative set off atx

∂of(x) :=ζ∈X∗|ζ(v)≤fo(x;v),∀v∈X (4)

is nonempty and weak star compact inX∗.

(ii) Iff(x) attains minimum value at some pointx0∈X, i.e.,

f(x)≥f(x0), ∀x∈X,

the Fermat condition

0∈∂of(x0)

holds.

Given a setC⊂X, the distance functiondC:X→Ris deﬁned as

dC(x) :=inf y∈Cy–x,

for a givenx∈X.

Lemma 1(Exact penalization) Suppose that f is a Lipschitz function with coeﬃcient M deﬁned on X,x∗∈C⊂X and f takes its minimum value at x∗on C.Then,for anyMˆ ≥M,

g(x) :=f(x) +MdCˆ (x)attains minimum value at x0on X.On the contrary,ifMˆ >M and C

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Remark2 From Lemma1, we know that, supposing thatf is a Lipschitz function with coeﬃcientMdeﬁned onXandCis a closed subset ofXand lettingg(y) :=f(y) +MdCˆ (y), for anyy∈X, whereMˆ >M, then, if there existsx∗∈Csuch that

fx∗=min

x∈Cf(x), (5)

then we have

gx∗=min x∈Xg(x) =f

x∗. (6)

Deﬁnition 2 Assumex∈C. Givenv∈X, ifdo_C(x;v) = 0, thenvis said to be a contingent derivative atx. Denote

TC(x) :=

v∈X|do_C(x;v) = 0.

TC(x) is the set of contingent derivatives atx. By polarity, the normal derivative set is deﬁned as

NC(x) :=ζ∈X∗|ζ(v)≤0,∀v∈TC(x).

Proposition 1 Suppose that x∈C,then we have

NC(x) =cl

λ≥0

λ∂odC(x)

,

where cl means the weak star closure.

Proposition 2 Given an x∈X,suppose that h is Lipschitz in a neighborhood of x and

0 /∈∂oh(x).Let

C:=y∈X|h(y)≤h(x). (7)

Then we have

NC(x)⊂

λ≥0

λ∂oh(x). (8)

Remark3 Given anx∈X, suppose thathis Lipschitz in a neighborhood of xand 0 /∈

∂oh(x). LetC:={y∈X|h(y)≤h(x)}. We also assume thatf is a Lipschitz function with coefficient Mdefined onX, andf takes its minimum value atxonC. By Definition1, Remark2, Definition2and Proposition2, then the Fermat condition

0∈∂of(x) +NC(x) (9)

holds, i.e., there exist a non-negative constantλ,ζ ∈∂o_f₍_x_{) and}_η_∈_∂o_h₍_x_{) such that}

ζ+λη= 0

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Definition 3 A functionf defined onXis called strictly differentiable atx∈X, if there existsx∗∈X∗such that

lim y∈X,y→x,t→0+

f(y+tv) –f(y)

t =

x∗,v, (10)

for anyv∈X.

Lemma 2 A function f deﬁned on X is strictly diﬀerentiable at x∈X,then∂of(x) ={x∗}, where x∗can be obtained by(10).

Lemma 3 Suppose that{hi,i= 1, 2, . . . ,n}is a set of Lipschitz functions on X.Deﬁne

h(x) :=maxhi(x)|i= 1, 2, . . . ,n, ∀x∈X.

For each x∈X,let I(x)be the subset of{1, 2, . . . ,n}satisfying that hi(x) =h(x),i∈I(x).Then we have

∂oh(x)⊂co∂ohi(x)|i∈I(x)

, (11)

where coA is the convex hull of A.

3 Maximum principle for optimal investment problem

In this section, our aim is to derive a necessary condition for optimal problem (2). Just as usual, in order to use the method of non-smooth analysis, we need the following assump-tion:

the risk measureρis Lipschitz. (A.3)

The following proposition shows thatE_0,g_T is Lipschitz inL2(FT). For more details, see

[2,22,23].

Proposition 3 Suppose that g satisﬁes(A.1)and(A.2),ξi∈L2(FT),i= 1, 2,and(yit,zit)are the solutions of BSDE(1)with terminal valuesξi,respectively,then there exists a constant C> 0such that

E

sup t∈[0,T]

y2_t –y1_t2

+E

T

0

z2_t –z1_t2dt

≤CE|ξ2–ξ1|2

. (12)

Remark4 From Proposition3, we know that, supposinggsatisﬁes (A.1) and (A.2), then, for anyξ1,ξ2∈L2(FT),

_Eg

0,T(ξ1) –E0,gT(ξ2)≤C

1

2_ξ₁_–_ξ₂

holds.

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Theorem 1 Under the conditions(A.1)–(A.3),ifξ∗is an optimal objective of problem(2),

then there exist a non-negative constantλ,ζ∈∂o_ρ₍_ξ∗₎_and_η_∈_∂o_Eg

0,T(ξ∗)such that

ζ+λη= 0 (13)

holds.

In order to prove Theorem1, we need the following lemma.

Lemma 4 Suppose that g satisﬁes(A.1)and(A.2).Then,for anyξ∈L2₍_F

T),we have0 /∈ ∂oE_0,g_T(ξ).

Proof For notational simplicity, we only consider the cased= 1. We assume that there existsξ∗∈L2₍_F

T) such that 0∈∂oE0,gT(ξ∗). For anyξ∈L2(FT), leth(ξ) :=E0,gT(ξ), then hoξ∗;η≥0

holds, for any η∈L2(FT). For eachξ ∈L2(FT), suppose that (yt,zt) and (yrt,zrt) are the

solutions of the following BSDEs on [0,T]:

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Proof of Theorem1 By Lemma4, we have 0 /∈∂o_Eg

0,T(ξ), for anyξ ∈L2(FT). Hence by

Deﬁnition1, Remark2, Deﬁnition2and Proposition1, we can see that, ifξ∗is an optimal objective of problem (2), then the Fermat condition

0∈∂oρξ∗+NA(x)

ξ∗ (14)

holds.

Case1:E_0,g_T(ξ∗) =x. In this case,A(x) ={ξ∈L2(FT)|E0,gT(ξ)≤E g

0,T(ξ∗)}. Thus, by

Propo-sition2, there exist a non-negative constantλ,ζ∈∂oρ(ξ∗) andη∈∂oE_0,g_T(ξ∗) such that

ζ+λη= 0 (15)

holds.

Case2:E_0,g_T(ξ∗) =x0<x. In this case, denote

C:=ξ∈L2(FT)|E0,gT(ξ)≤x0

.

Sinceξ∗is an optimal objective of problem (2),ξ∗is also an optimal objective of problem

min ξ∈C

ρ(ξ). (16)

Thus, by Deﬁnition1, Remark2, Deﬁnition2and Proposition2again, Equation (15) also holds. The proof of Theorem1is complete.

From Remark4, we know thatE_0,g_T is Lipschitz inL2(FT). By Deﬁnition1, we can see

that∂oE_0,g_T(ξ) is not empty for anyξ∈L2(FT). Indeed, we have following result. Lemma 5 Suppose thatξ∈L2₍_F

T)and g satisﬁes(A.1)and(A.2).Let(yt,zt)be the solu-tion of BSDE(1)with terminal valueξ,then for anyη∈∂o_Eg

0,T(ξ)andζ∈L2(FT),there exists(ϕt,ψt)∈∂og(t,yt,zt)such that

η,ζ=y˜0

holds,where(˜yt,zt˜)is the solution of the following BSDE on[0,T]:

˜

yt=ζ+

T

t

(ϕs˜ys+ψs˜zs) ds–

T

t

˜

zs·dWs. (17)

Proof For any givenω∈Ωandt∈[0,T], letgˆ(t,yˆ,zˆ) =go₍_t_,_yt_,_zt_;_y_ˆ_,_z_ˆ_),_∀_(ˆ_y_,_z_ˆ₎_∈_R_×_Rd_{. By}

Deﬁnition1, we can see thatgˆ(t,yˆ,zˆ) is Lipschitz, positively homogeneous and convex in (ˆy,zˆ), and hence

ˆ

g(t,yˆ,ˆz) = max

(ϕt,ψt)∈∂og(t,yt,zt)

(ϕt,ψt), (ˆy,zˆ)

. (18)

For anyζ ∈L2₍_F

T), we consider the following BSDE on [0,T]:

ˆ

yt=ζ+

T

t

ˆ

g(s,ˆys,zsˆ) ds–

T

t

ˆ

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Letfˆ(ζ) :=yˆ0. Denote

D:=η∈L2(FT)|η,ζ=y˜0,∀(ϕt,ψt)∈∂og(t,yt,zt)

, (20)

where (˜yt,zt˜) is the solution of the following BSDE on [0,T]:

˜

yt=ζ+

T

t

(ϕs˜ys+ψs˜zs) ds–

T

t

˜

zs·dWs.

By the comparison theorem of BSDE (for example, see El Karoui et al. [7]), we have ˆ

f(ζ)≥ η,ζ, for any η∈D. Sincegˆ is positively homogeneous and convex in (ˆy,zˆ), by Propositions 3.3 and 3.4 in Peng [16], we know thatfˆis positively homogeneous and con-vex inL2(FT), and hence

ˆ

f(ζ) =max

η∈Dη,ζ. (21)

By Deﬁnition1, we can obtain that

Eg

0,T

o

(ξ;ζ) =fˆ(ζ) =max η∈Dη,ζ.

This means that∂oE_0,g_T(ξ) =D. The proof of Lemma5is complete. In Ji and Peng [10], the authors assume thatgis continuously diﬀerentiable in (y,z). In this special case, we can get an explicit form of∂oE_0,g_T.

Lemma 6 Suppose thatξ∈L2(FT)and g satisﬁes(A.1)and(A.2).Let(yt,zt)be the solu-tion of BSDE(1)with terminal valueξ.If g is continuously diﬀerentiable in(y,z)∈R×Rd_, then we have

∂oE_0,g_T(ξ) ={qT}

and for anyη∈L2₍_F

T),

qT,η=˜y0, (22)

where(˜yt,˜zt)is the solution of the following BSDE on[0,T]:

˜

yt=η+

T

t

g_y(s,ys,zs)˜ys+g_z(s,ys,zs)˜zsds–

T

t

˜

zs·dWs.

Proof For any givenω∈Ω andt∈[0,T], letg˜(t,y˜,z˜) =go₍_t_,_yt_,_zt_;_˜_y_,_z_˜_),_∀_(˜_y_,_z_˜₎_∈_R_×_Rd_.

Sincegsatisfies (A.1), (A.2), and is continuously differentiable in (y,z)∈R×Rd, we know thatgis strictly differentiable in (y,z)∈R×Rd_and

˜

(9)

holds. For anyη∈L2₍_F

T), let (˜yt,zt˜) be the solution of the following BSDE on [0,T]:

˜

yt=η+

T

t

˜

g(s,ys˜,˜zs) ds–

T

t

˜

zs·dWs.

DenoteE_0,g˜_T(η) :=y˜0. By Deﬁnition1, we obtain

Eg

0,T

o

(ξ;η) =E_0,g˜_T(η). (23) Sinceg˜ is linear in (˜y,z˜), by Propositions 3.3 and 3.4 in Peng [16], we know thatE_0,g˜_T is linear inL2₍_F

T). By Proposition3, we can see thatE˜ g

0,Tis continuous inL2(FT). Thus, by

the Riesz representation theorem, there existsqT∈L2(FT) such that

Eg˜

0,T(η) =qT,η (24)

holds for any η∈L2₍_F

T). From (23) and (24), we have ∂oE0,gT(ξ) ={qT}. The proof of

Lemma6is complete.

By Theorem1and Lemma6, we immediately obtain the following theorem.

Theorem 2 Suppose thatρsatisfies(A.3),and g satisfies(A.1), (A.2),and is continuously differentiable in(y,z).Assume thatξ∗is an optimal objective of problem(2),let(y∗_t,z∗_t)be the solution of BSDE(1)with terminal valueξ∗.Then there exist a non-negative constant λandζ∈∂oρ(ξ∗)such that

ζ+λq∗_T= 0

holds,where q∗_Tcan be obtained by(22).

4 Some examples

In this section, we give three examples for our obtained result. In the sequel, suppose that

gsatisﬁes (A.1), (A.2), and is continuously diﬀerentiable in (y,z).

ProblemA. Minimize

Eϕ(ξ)–c

subject to

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

E[u(ξ)] =c,

Eg

0,T(ξ) =x,

Eg

0,T(ξ)≥0, ξ∈L2(FT),

whereϕ:R→Ris Lipschitz and continuously diﬀerentiable, andu:R→Rhas bounded continuous derivativeu.

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ProblemB. Minimize

Eϕ(ξ)–c

subject to

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

E[u(ξ)]≤c,

Eg

0,T(ξ)≤x,

Eg

0,T(ξ)≥0, ξ∈L2₍_F

T),

whereϕ:R→Ris Lipschitz and continuously diﬀerentiable, andu:R→Rhas bounded continuous derivativeu.

In order to study the optimization problems A and B, we need the following lemma.

Lemma 7 If u:R→R has bounded continuous derivative u,then the function ρ(ξ) =

E[u(ξ)]for anyξ∈L2₍_F

T)is strictly diﬀerentiable,and∂oρ(ξ) ={u(ξ)}.

By the Fubini theorem and the dominated convergence theorem, we can easily prove Lemma7. So we omit it.

Theorem 3 If the optimal objectiveξ∗of ProblemAexists,there exist two constantsλ1

andλ2such that

ϕξ∗+λ1u

ξ∗+λ2q∗T= 0

holds,where∂oE_0,g_T(ξ∗) ={q∗_T}.

Proof Denoteρ(ξ) :=E[ϕ(ξ)] –c, for anyξ∈L2₍_F

T) and

U:=ξ∈L2(FT)|h(ξ)≤0

,

whereh(ξ) :=max{E[u(ξ)] –c,c–E[u(ξ)],E_0,g_T(ξ) –x,x–E_0,g_T(ξ), –E_0,g_T(ξ)}. If the optimal

objectiveξ∗of Problem A exists, then, by Lemmas6and7, we have∂oρ(ξ∗) ={ϕ(ξ∗)},

∂oE[u(ξ∗)] ={u(ξ∗)}and∂oE_0,g_T(ξ∗) ={q∗_T}.

By Theorem2and Lemma3, there exist two constantsλ1andλ2such that

ϕξ∗+λ1u

ξ∗+λ2q∗T= 0

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Theorem 4 If the optimal objectiveξ∗of ProblemBexists,then there exist three constants

In the classic investment problem, one often takes the variance as a risk measure and the mean-variance method is used in much of the literature. But by Delbaen [6], and Föllmer and Schied [8], such a kind of risk measure is not perfect. We often takeρ(·) as a coherent or convex risk measure in (2). In Rosazza Gianin [19], the author proved that ifgis a sub-additive and positively homogeneous function satisfying (A.1) and (A.2), deﬁneρ(ξ) :=

Eg

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ProblemC. Supposef is sub-additive and homogeneous function satisfying (A.1) and

Theorem 5 If the optimal objectiveξ∗of ProblemCexists,then there exist three constants

λ≥0and a,β∈[0, 1]satisfyingα+β≤1,such that

The proof of Theorem5is very similar to that of Theorem4. So we omit the proof.

Acknowledgements

The authors would like to thank the anonymous referees for their constructive suggestions and valuable comments.

Funding

The work of Helin Wu is supported by the Scientiﬁc and Technological Research Program of Chongqing Municipal Education Commission (KJ1400922). The work of Yong Ren is supported by the National Natural Science Foundation of China (11871076). The work of Feng Hu is supported by the National Natural Science Foundation of China (11801307) and the Natural Science Foundation of Shandong Province (ZR2016JL002 and ZR2017MA012).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, China.}2_{Department of}

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Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 5 June 2018 Accepted: 4 December 2018

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