Constrained LQ Problem with a Random Jump and Application to Portfolio Selection?

Yuchao DONG

Abstract This paper deals with a constrained stochastic linear-quadratic(LQ for short)optimal control problem where the control is constrained in a closed cone.The state process is governed by a controlled SDE with random coefficients.Moreover,there is a random jump of the state process.In mathematical finance,the random jump often represents the default of a counter party.Thanks to the It?o-Tanaka formula,optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations(BSDEs for short).The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions.The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.

Keywords Backward stochastic Riccati equation,Default time,Mean-variance problem

1 Introduction

Linear-quadratic problem is an important optimal control problem.The feature of such a problem is that the dynamic of the system is linear in the state and control variables and the cost functional is quadratic in both of them.It was first considered by Kalman[10](for the deterministic control of ordinary differential equations,i.e.,ODEs)and then extended to various situations,for example stochastic LQ problems.One important application of stochastic LQ optimal control theory is the continuous-time version of Markowitz’s mean-variance portfolio selection problem,which is one fundamental problem in the mathematical finance.

It is well-known that one can give in explicit forms the optimal state feedback control and the optimal value via the celebrated Riccati equation.In the deterministic case or the stochastic case with deterministic coefficients,the Riccati equation is an ODE in the space of symmetric matrixes.When the coefficients are random,the Riccati equation becomes a backward stochastic differential equation.The theory of BSDEs was pioneered by Pardoux and Peng[16].It is closely related to the optimal control theory.See Yong and Zhou[19]on this subject.For Riccati equations,the solvability is a very hard problem.Under some standard assumptions of the coefficients,it is solved by Tang[17–18]by two different approaches.For more details on this subject,see[3,7,17–18]and the references therein.

In this paper,we consider the stochastic LQ problems with a random jump.Note that similar problems have also been considered by[8,14–15].Our problem is different from theirs from two apsects.One is that our system only has at most one jump.In mathematical finance,this random jump represents the default,so sometimes we just call it the default time.In a financial market,we know that the default of one firm has usually important influences on the others.This has been shown clearly in the financial crisis.While the controlled processes considered in those papers mentioned above are driven by a Poisson random measure,their systems can have even infinitely many jumps.The other difference is that the control in our problem is constrained in a closed cone.In the mean-variance problem,this means that there are some restrictions on the trading strategy of the investor.In this paper,we shall consider the mean-variance portfolio selection problem for an investor who invests in a risky asset exposed to a counterparty risk.The investor is also not allowed to short sell.Thus we have to solve a constraint LQ problem with a random jump.We only consider the problem for the case that the state variable is scalar-valued.How to solve it in the multi-dimensional case is still a problem,but the scalar-valued case is sufficient to cover many important practical applications especially in the financial area.

To get the optimal control and the optimal value,we must first get the Riccati equation.Note that,due to the constraint,the value function is no longer quadratic with respect to the initial value.But one can easily show that the value function V is positive homogeneous since the control is constrained in a closed cone.That is

where P and N satisfy the following BSDEs:

Thus we are still able to get a system of BSDEs,sometimes called extended Riccati equation,that characterizes the optimal control and the optimal value.We can see that the BSDEs are coupled and have a random jump.Note that multi-dimensional backward Riccati equations have also been considered by K.Mitsui and Y.Tabata[15].But their equations are multidimensional because the state processes in[15]are multi-dimensional.To solve such equations,we use the method originated by Ankirchner et al[1]and further developed by Kharroubi and Lim[11].Through the decomposition of processes with respect to the progressive enlargement of filtrations,we link the BSDEs we want to solve with a family of Brownian BSDEs.By proving the solvability of the Brownian BSDEs,we are able to solve the original BSDEs.If there is no jump,the equations will be decoupled and this is the exact equation considered by Hu and Zhou[7].

The rest of the paper is organized as follows.In Section 2,we formulate the problem.In Section 3,we derive the form of the extend Riccati equations and prove its solvability in two cases.In Section 4,we give the state feedback optimal control and the optimal value via the Riccati equations.The application to mean-variance problem is in Section 5.

2 The Model and Assumptions

In this paper,we assume throughout that(?,F,P)is a given probability space and that Wtis a k-dimensional standard Brownian motion defined on this space with W0=0.Let{Ft}be the augmentation of σ{Ws|0 ≤ s ≤ t}.In addition,let τ be a random time.Define

which is the smallest filtration containing{Ft}that makes τ a stopping time and satisfies the usual condition.

In the sequel,we shall make the following assumptions on the random time τ.For any t∈ [0,T],the conditional distribution of τ under Ftadmits a density with respect to Lebsegue measure,i.e.,there exists an-measurable positive function(ω,θ)→ αt(θ)such that

Note that for any θ≥ 0,the process{αt(θ),0 ≤ t≤ T}is a F-martingale.Moreover we assume that the family of densities satisfy αT(t)= αt(t)for all 0 ≤ t≤ T.

Remark 2.1In the finance,the random time τ usually represents the default of a counterparty.The density hypothesis is usually used in the theory of enlargement of filtrations.It was introduced in the notes of Jeulin and Yor[9]and recently adopted by El Karoui et al[4]for credit risk modelling.Note that we haveThis is related to the so-called immersion hypothesis meaning that any square integrable F-martingale is a square integrable G-martingale.

Let Lt=1{τ≤t}.Then L is a{Gt}-submartingale.We shall have the following assumption.

Assumption 2.1There exists an F-predicable bounded nonnegative process λ such that

is a martingale with respect to{Gt}.

Example 2.1Let β be a bounded nonnegative{Ft}-predictable process such that

and Θ an exponential distributed random variable that is independent of the Brownian motion W.Define the random time

Remark 2.2Let φ be a{Gt}-predictable process.Then it can be represented as

where φ0is F-predictable and φ1is P(F)? B(R)-measurable.

Consider the following controlled linear SDE:

The coeifficients A,B,C,D,E,F are{Gt}-predictable processes,and x∈R is a nonrandom scalar.Precise assumptions on these coefficients will be specified below.Let Γ?Rmbe a given closed cone.A typical example of such a cone isThe class of admissible controls is the set U:=L2([0,T]×?,P(G),Γ),i.e.,the square integrable Γ-valued{Gt}-predictable processes.The cost is given by

The optimal control problem is to minimize the cost functional over all admissible controls.Define the value function by

We have the following assumptions on the coefficients.

Assumption 2.2

By Remark 2.2,we shall have the following decompositions of the processes:

where i0is F-predictable and i1is P(F)?B(R)-measurable for i=A,B,C,D,E,F,R,Q.And

where G0is FT-measurable and G1is FT?B(R)-measurable.

3 Existence of Solutions for the Stochastic Riccati Equations

3.1 The form of the Riccati equations

In this section,we shall prove the existence of solutions for the extended stochastic Riccati equations.First of all,we shall derive the formation of the Riccati equations.Note that the admissible controls are Γ-valued and Γ is a closed cone.It means that for any u ∈ U and c ≥ 0,cu also belongs to U.Since the controlled SDE is linear and the cost functional is quadratic,it is obvious that the value function V is positive homogeneous,i.e.,V(t,cx)=c2V(t,x)for all c≥0.Hence V is of the following form:

Assume that both P and N are semimartingales with the following decompositions:

Given any u ∈ U,X is the associated solution of(2.4).By the It?o-Tanaka formula,we have

Note that X only has a jump at the time τ,i.e.,

Hence we get that

By(3.2)and It?o formula again,

where msis the local martingale part

Similarly,we also have

Combining(3.4)and(3.5)and letting s=T,

We denote that

and

Since V is the value function,the integrand should always be positive.For some admissible control u,if the integrand is zero and the local martingale part is in fact a martingale,then taking conditional expectation,we have that it will be the optimal control.Hence we must have that

Noting that Γ is a close cone,we have,thus ftshould satisfy

With a similar discussion,we see that P and N should be the solutions of the following system of BSDEs:

where

and

3.2 The solvability of the equations

We have the following definitions on the solutions of the equations.

Definition 3.1We say that a pair of stochastic processes([0,T]×?,P(G))×L2([0,T]×?,P(G))×L2([0,T]×?,P(G))is a solution to BSDE(3.7)if it satisfies the equation in the It?o sense as well as the terminal condition and the constraint that R+PD′D>0.A solutionis called positive(resp.nonnegative)if P>0(resp.P≥0)and called uniformly positive if P≥c>0.These definitions extent in the obvious way to the solutions of the BSDEs defined in the rest part of the paper.

Before we solve the equation,let us emphasize some properties of h±.First,it is obvious that

Assume that p,q2+p,q3≥0,we see that

Moreover,if|p|,|q1|,|q2|,|q3|≤n,by(3.10),the infimum will be obtained in a bounded subset of Γ,hence is in fact a minimum and h±are continuous with respect to(p,q1,q2,q3)in this situation.

Note that we get a multidimensional BSDE with quadratic growth in z.In general,there may be no solution for the system.See Hu and Tang[5]for an existence result and more details on this subject.To solve the equation,we use the approach originated by Ankirchner et al[1]and further developed by Kharroubi and Lim[11]:One can explicitly construct a solution by combining solutions of an associated family of Brownian BSDEs.Fortunately,we shall see that we can solve these equations separately.To illustrate the idea,we give a simple example taken from[11].Consider the following BSDE:(

To solve it,we first solve a recursive system of Brownian BSDEs:

Define the process(Y,U)by

By It?o formula,we have

It is also easy to see that YTalso satisfies the terminal condition.Thus(Y,U)we define is a solution to(3.11).

Note that such a method is still valid in more complicate situations(see[11]and Theorem 3.1 below).We first decompose the BSDEs into two parts:The before default part and the after default part.Thus we have the following BSDEs:

where

And

where

Moreover

where

And

where

Note that we have

and

We use the following theorem from[11].

Theorem 3.1Assume that for all θ∈ R+,the Brownian BSDEs(3.12)–(3.13)admit solutionsand that the Brownian BSDEs(3.14)–(3.15)have solutionsAssume moreover that P1(θ)and N1(θ)(resp.Z1(θ)and Λ1(θ))are F?B(R+)(resp.P(F)?B(R+))-measurable.If all these solutions satisfy

and

then BSDEs(3.7)–(3.8)admit solutions(P,Z,Z),(N,Λ,Λ)∈ L∞([0,T]×?,P(G))×L2([0,T]×?,P(G))×L2([0,T]×?,P(G))given by

For the proof of this theorem,the reader can see[11,Theorem 3.1].

Remark 3.1Below,we shall prove the existence of the solutions for any given θ.Then we can choose P1and N1(resp.Z1and Λ1)as F ? B(R+)(resp.P(F)? B(R+))-measurable processes.Indeed we know(see[12])that one can construct(P1,Z1)and(N1,Λ1)as limits of solutions to Lipschitz BSDEs.From[11,Proposition C.1],we get P1and N1(resp.Z1and Λ1)as limits of F?B(R+)(resp.P(F)?B(R+))-measurable processes,hence also measurable.

We shall deal with the following two cases:

Standard case.Q ≥0,R>0 with R?1∈L∞([0,T]×?,P(G),Rm×m)and G≥ 0.

Singular case.Q ≥ 0,R≥ 0,G>0 with G?1∈L∞([0,T]×?,P(G),R)and D′D>0 with(D′D)?1∈ L∞([0,T]×?,P(G),Rm×m).

For the BSDE(3.12)(resp.(3.13)),we have the following theorem.

Theorem 3.2Under Assumption 2.2,given any θ,for the standard case,there exists a unique bounded,nonnegative maximal solution(P1(θ),Z1(θ))(resp.(N1(θ),Λ1(θ)))for(3.12)(resp.(3.13)).For the singular case,there exists a bounded,uniformly positive solution.Moreover,we have

ProofFor the proof of existence of solutions for the extended backward Riccati equations,we refer to[7,Theorems 4.1–4.2].Now we prove(3.16).

For the standard case,we know that(see[7]),there exists a constant c1which only depends on the bound of the coefficents A,B,C,D,R,G,such that

Thus the norm is uniformly bounded in θ.By(3.10),one can find two constants C1,C2>0 such that

By the boundness and non-negativity of P and the inequality(3.17),taking expectation,we get that

with the constant c2independent of θ.Hence we finish the proof for the standard case.

For the singular case,there will be a constant c3>0 independent of θ such that

In this case,we have

Following the same argument as above,we prove the theorem.

Now we show the existence of the solution to(3.14)and(3.15).We only proof it for(3.14),since the proof is same for(3.15).

Theorem 3.3Under Assumptions 2.1 and 2.2,for the standard case there exists a bounded,nonnegative solution(P0,Z0)to the BSDE(3.14).And it will be uniformly positive in the singular case.

ProofFor the standard case,let us first consider the following BSDE:

This is a linear BSDE with bounded coefficients and withand G0≥0.Hence there exists a unique nonnegative,bounded solution(P′,Z′).Denote by c1>0 the upper bound for P′.Now consider the following BSDE:

where the function F is defined by

whereas g1:R+→[0,1]is a smooth truncation function satisfying g1(x)=1 for x∈[0,c1],and g1(x)=0 for x∈[2c1,+∞).Note that F satisfies the hypothesis(H1)of[12]thanks to the role of the truncation function g1.According to[12],there is a bounded maximal solution(P,Z)to BSDE(3.19)(see[12,p.565]and Theorem 2.3 for its definition and proof).Now asand(P′,Z′)is the only,hence maximal,bounded solution to(3.18),we get that P≤P′≤c1.Moreover,noting that G≥0,Q≥0 andwe conclude that P≥0 since(0,0)is a solution to(3.19)with Q0=0,G0=0 and F(t,p,q)replaced byThis proves that(P,Z)is a bounded nonnegative solution to(3.14).

For the singular case,we consider the following BSDE:

where

This is the BSDE studied in[6,13].By[6,Lemma 4.1],there exists a unique bounded,uniformly positive solutionDenote by c2the lower bound foreP.Now,let us consider the following BSDE:

This means that(P,Z)is actually a bounded,uniformly positive solution to the BSDE(3.14).

Combining Theorems 3.1–3.3,we show that there exist bounded solutions for the system of BSDE(3.7)(resp.(3.8)).

Theorem 3.4Under Assumptions 2.1 and 2.2,either in the standard case or the singular case,there exists a bounded,nonnegative solution(resp.for the BSDE(3.7)(resp.(3.8)).The solution will be uniformly positive in the singular case. Furthermore,we have that

and

4 Solve the Constrained LQ Problem

In this section we give the optimal control for the LQ problem by the solutions to the system of BSDEs for both standard and singular case.Define

Note that the minimizers are achievable due to the discussion in the above section and Γ is closed.By the definition,ξ+and ξ?also have the following decompositions:

Theorem 4.1In both the standard and singular cases,let?,P(G))×L2([0,T]×?,P(G))×L∞([0,T]×?,P(G))be the bounded,nonnegative solutions to BSDEs(3.7)and(3.8)(uniformly positive in singular case).Then the following state feedback control

is the optimal control for the LQ problem.Moreover,the value function is

ProofNow consider the state feedback control:

By the lemma that follows,this equation has a c`adl`ag(left limit right continuous)solution.Let(u,X)be any admissible control and the corresponding state process and(u?,X?)the state feedback control(4.1)and the state process.Following the discussion in Section 3,we see that the Lebesgue integrands in(3.6)are always positive.Define the following stopping time κn:

Obviously,κnis an increasing sequence of stopping time and converging to T almost surely.Hence taking integration from t to κnand then taking conditional expectation in(3.6),we have

Letting n→∞and noting that the processes P and N are quasi-left continuous,from the dominated convergence theorem,we have

We are now going to prove that u?∈ L2([0,T]×?,P(G)).Once we prove this,the analysis above shows that

because the Lebesgue integrand in(3.6)is identically zero.

In the standard case,denote by c the constant such that R≥cIn.Then we have

This implies that u?∈ L2([0,T]× ?,P(G)).For the singular case,construct a sequence of stopping time as follows

We rewrite the equation(4.2)as a kind of BSDE with a random terminal time:

Denote by

Then as in the standard estimation for the BSDE,we have

Appealing to Fatou’s lemma,we conclude that X?,z ∈ L2([0,T]× ?,P(G)).This in turn implies that u?∈ L2([0,T]×?,P(G)).

Lemma 4.1The equation(4.2)has a c`adl`ag solution.

ProofBefore the proof,let us illustrate the meaning of such a SDE.First,the dynamic of X is governed by a Brownian SDE.Then at the random τ= θ,a jump of X is induced.The size of the jump is related to X and θ the time that the jump happens.After the jump,X still evolves according to a Brownian SDE,but the coefficients of the SDE may be changed based on the jump time.So we can solve the SDE by decomposing it into two parts:The before default part and the after default part.We shall rewrite the SDE(4.2)into the following form:

where the coefficients are

with some F-predictable processand P(F)×B(R+)-measurable processThis is also true for the other coefficients.We shall use similar notations for the decompositions.

Now consider the following SDEs:

and

Each SDE has a unique continuous F-adapted solution(see[7,Lemma 5.1]).Then it is obvious that the processis a solution to(4.3),hence a solution to(4.2).

5 Application to Portfolio Selection

For simplicity,we consider a financial market consisting of a bank account and one stock.We suppose that the Brownian motion W is one dimensional and F is the filtration generated by it and satisfying the usual condition.The value of the bank count,S0(t),satisfies an ordinary differential equation:

where rtis deterministic.The dynamic of the risky asset is affected by other firms,the counterparties,which may default at some random time denoted by τ.When the default happens,it may induce a jump in the asset price and change the dynamic of the stock.But this asset still exists and can be traded after the default of the counterparties.More precisely,let the process Ltand the filtration G be what we defined in Section 2.Before the default,the stock price is governed by the following SDE:

where the coefficients are F-measurable.We denote bythe price of the stock after the default if the default time is at time θ.At the default time τ,the price has a jump

After the default,there is a change of regime in the coefficients depending on the default.For example,if a downward jump on the stock price is induced at default time τ= θ,the rate of the return b1(θ)should be smaller than the rate of return b0before the default,and this gap should be increasing when the default occurs early.The stock price is still governed by an SDE for default time τ= θ:

Denoting by b and σ the G-predictable processesandwe rewrite the price process S as

Consider now an invest strategy that can trade continuously in this market.This is mathematically quantified by a G-predictable process π called self- financed trading strategy.It represents the money invested in the stock at time t.By Remark 2.2,we know that it has the formThen the wealth process X is given by

where X0is the wealth process in the default-free market,governed by

and X1(θ)is the wealth process after the default at time τ= θ,governed by

Thus we can rewrite the wealth process as follows:

We assume that the coefficients satisfies Assumption 2.2 and the admissible control is the set of all square-integrable Γ-valued G-predictable processes with Γ =R+.Note that we only allow Γ-valued processes,which means that the investor cannot short sell the stock.

Mathematically,it can be formulated as the following problem parameterized by z:

ProofWe first prove the “if” part.Define

Condition(5.2)implies that the measure of M is non-zero.Consider the following control:

Taking expectation,we have

To handle the constraint E[XT]=z,we apply Lagrange multiplier technique.Define

We first solve the following unconstrainted problem:

where P and N is the solutions of the following BSDEs:

Thus we get a contradiction which implies that

Thus

This implies that

AcknowledgementsThe author would like to thank his advisor,Prof.Shanjian Tang from Fudan University,for the helpful comments and discussions.The author would also thank the referees of this paper for helpful comments.