A Riccati Type Equation

First, we consider the following BSDE:







dY_t= h(t, Y_t, Z¹_t, · · · , Z^d_t)dt +

d

P

i=1

Zⁱ_tdW_tⁱ, 0 ≤ t ≤ T Y_T = ξ,

(80)

where ξ ∈ L²_F

T(Ω, R^m) and h ∈ L²_F(0, T ; W^1,∞(R^m × (R^m)^d; R^m)) i.e., h : [0, T ] × R^m × (R^m)^d×Ω → R^m, such that (t, ω) 7→ h(t, y, z¹, · · · , z^d; ω) is {Ft}t≥0-progressively measur-able ∀(y, z¹, · · · , z^d) ∈ R^m× (R^m)^d with h(t, 0, 0, · · · , 0; ω) ∈ L²_F(0, T ; R^m) and ∃L > 0,

|h(y, z¹, · · · , z^d) − h(y, z¹, · · · , z^d)| ≤ L |y − y| +

d

X

i=1

|zⁱ− zⁱ|

! ,

∀y, z¹, · · · , z^d, y, z¹, · · · , z^d∈ R^m, a.e. t ∈ [0, T ], a.s. (81) Denote

N [0, T ] , L²_F(Ω; C(0, T ; R^m)) × [L²_F(0, T ; R^m)]^d (82) and

k(Y, Z¹, · · · , Z^d)kN [0,T ] , E

"

sup

0≤t≤T

|Y_t|²+

d

X

i=1

Z T 0

|Z_t|²dt

#!¹₂

(83) Then, N [0, T ] is a Banach space under norm (83).

In this chapter, we present another method. It will give a sufficient condition for the unique solvability of (3). We will obtain a Riccati type equation and a BSDE associated with (3). Let us now carry heuristic derivation.

Suppose (X, Y, Z¹, · · · , Z^d) ∈ M[0, T ] is an adapted solution of (3). We assume that X and Y are related by

Y_t= P (t)X_t+ p_t, ∀t ∈ [0, T ], a.s. (84) where P : [0, T ] → R^m×nis a deterministic matrix-valued function and p : [0, T ]×Ω → R^m is an {Ft}_t≥0-adapted process. We are going to derive the equations for P and p. First of all, from (8) and the terminal condition in (3), we have

g = P (T )X_T + p_T. (85)

Let us impose

P (T ) = O, p_T = g. (86)

Since g ∈ L²_FT(Ω; R^m) and p is required to be {Ft}t≥0-adapted, we should assume that p satisfies a BSDE:







dp_t = α_tdt +

d

P

i=1

qⁱ_tdW_tⁱ, 0 ≤ t ≤ T p_T = g

(87)

with α, q¹, · · · , q^d ∈ L²_F(0, T ; R^m) being undetermined. Next, by Itbo Formula, we have dY_t = ˙P (t)X_tdt + P (t)dX_t+ dp_t

=[ ˙P (t)X_t+ P (t)(AX_t+ BY_t) + α_t]dt +

d

X

i=1

[P (t)(Aⁱ₁Xt+ B₁ⁱYt+ C₁ⁱZⁱ_t) + qⁱ_t]dW_tⁱ

={[ ˙P (t) + P (t)A + P (t)BP (t)]X_t+ P (t)Bp_t+ α_t}dt +

d

X

i=1

{[P (t)Aⁱ₁+ P (t)B₁ⁱP (t)]X_t+ P (t)C₁ⁱZⁱ_t+ P (t)B₁ⁱp_t+ qⁱ_t}dW_tⁱ (88) Now compare (88) with the second equation in (3) (note (84)), we obtain that (drift coefficient)

[ ˙P (t) + P (t)A + P (t)BP (t)]X_t+ P (t)Bp_t+ α_t = [ bA + bBP (t)]X_t+ bBp_t, (89) and (diffusion coefficient)

[P (t)Aⁱ₁+ P (t)Bⁱ₁P (t)]X_t+ P (t)C₁ⁱZⁱ_t+ P (t)B₁ⁱp_t+ qⁱ_t= Zⁱ_t, i = 1, · · · , d (90) By assuming I − P (t)C₁ⁱ to be invertible ∀t ∈ [0, T ], i = 1, · · · , d, we have from (90) that Zⁱ_t= [I − P (t)C₁ⁱ]⁻¹{[(P (t)Aⁱ₁+ P (t)B₁ⁱP (t)]Xt+ P (t)B₁ⁱpt+ qⁱ_t}, i = 1, · · · , d (91) Then, (89) can be written as

0 = [ ˙P (t) + P (t)A + P (t)BP (t) − bA − bBP (t)]X_t+ [P (t)B − bB]p_t+ α_t. (92) Now, we introduce the following differential equation for M_m×n(R)-valued function P :



 P (t) + P (t)A + P (t)BP (t) − b˙ A − bBP (t) = O, 0 ≤ t ≤ T

We refer to (93) as a Riccati type equation. Suppose (93) admits a solution P over [0, T ] such that

[I − P (t)C₁ⁱ]⁻¹ is bounded , i = 1, · · · , d, ∀t ∈ [0, T ]. (94) Then, (92) gives

α_t = [ bB − P (t)B]p_t

Combining this with (87), we see that one should introduce the following BSDE:







dp_t = [ bB − P (t)B]p_tdt +

d

P

i=1

qⁱ_tdW_tⁱ, 0 ≤ t ≤ T p_T = g.

(95)

When (93) admits solution P such that (94) holds, BSDE (95) admits a unique solution (p, q¹, · · · , q^d) ∈ N [0, T ]. In the form provided here, a proof can be found in several sources, for instance Ma & Yong (2000), pp. 15-16. From (84) and (91), the forward equation (X):



















dX_t= {[A + BP (t)]X_t+ Bp_t}dt +

d

P

i=1

{Aⁱ₁+ B₁ⁱP (t) + C₁ⁱ[I − P (t)C₁ⁱ]⁻¹[P (t)Aⁱ₁+ P (t)B₁ⁱP (t)]}X_t +B₁ⁱp_t+ C₁ⁱ[I − P (t)C₁ⁱ]⁻¹[P (t)B₁ⁱp_t+ qⁱ_t]

dW_tⁱ, 0 ≤ t ≤ T X₀ = 0

Then we can define the following:



















A(t) = A + BP (t),e

Aeⁱ₁(t) = Aⁱ₁+ B₁ⁱP (t) + C₁ⁱ[I − P (t)C₁ⁱ]⁻¹[P (t)Aⁱ₁+ P (t)B₁ⁱP (t)], i = 1, · · · , d be_t = Bp_t,

eσ_tⁱ = B₁ⁱp_t+ C₁ⁱ[I − P (t)C₁ⁱ]⁻¹[P (t)B₁ⁱp_t+ qⁱ_t], i = 1, · · · , d.

(96)

It is clear that eA and eAⁱ₁ are time-dependent matrix-valued function and eb and σeⁱ are {Ft}_t≥0-adapted processes. Further, under (94), by the Existence and Uniqueness The-orem for Stochastic Differential Equations, the following SDE admits a unique (strong) solution:







dX_t = [ eA(t)X_t+ eb_t]dt +

d

P

i=1

[ eAⁱ₁(t)X_t+eσ_tⁱ]dW_tⁱ, 0 ≤ t ≤ T, X₀ = 0.

(97)

The following Theorem gives a representation of the adapted solution of FBSDE (3).

Theorem 4. Let (93) admits a solution P such that (94) holds. Then FBSDE (3) admits a unique solution (X, Y, Z¹, · · · , Z^d) ∈ M[0, T ] which is determined by (97), (84) and (91).

Proof. First of all, a direct computation (from above) shows that the process (X, Y, Z¹, · · · , Z^d) determined by (97), (84) and (91) is an adapted solution of (3). We now prove the unique-ness. Let (X, Y, Z¹, · · · , Z^d) ∈ M[0, T ] be any adapted solution of (3). Set







Y_t = P (t)X_t+ p_t,

Zⁱ_t= [I − P (t)C₁ⁱ]⁻¹{[P (t)Aⁱ₁+ P (t)B₁ⁱP (t)]X_t+ P (t)B₁ⁱp_t+ qⁱ_t}, i = 1, · · · , d

(98)

where P and (p, q¹, · · · , q^d) are (adapted) solutions of (93) and (95), respectively. Denote Y = Y − Y and bb Zⁱ = Zⁱ− Zⁱ. By the Itbo Formula,

dYt= ˙P (t)Xtdt + P (t)dXt+ dpt

= [ bA + bBP (t) − P (t)A − P (t)BP (t)]X_tdt + P (t)

"

(AX_t+ BY_t)dt +

d

X

i=1

(Aⁱ₁X_t+ B₁ⁱY_t+ C₁ⁱZⁱ_t)dW_tⁱ

#

+ [ bB − P (t)B]p_tdt +

d

X

i=1

qⁱ_tdW_tⁱ

= [ bAX_t+ P (t)B(Y_t− Y_t) + bBY_t]dt +

d

X

i=1

{P (t)B₁ⁱ(Y_t− Y_t) + P (t)B₁ⁱ[P (t)X_t+ p_t] + P (t)(Aⁱ₁X_t+ C₁ⁱZⁱ_t) + qⁱ_t}dW_tⁱ

= [ bAX_t+ P (t)B(Y_t− Y_t) + bBY_t]dt +

d

X

i=1

{P (t)B₁ⁱ(Y_t− Y_t) + P (t)C₁ⁱZⁱ_t+ [I − P (t)C₁ⁱ]Zⁱ_t}dW_tⁱ

= [ bAX_t+ P (t)B(Y_t− Y_t) + bBY_t]dt +

d

X

i=1

{P (t)B₁ⁱ(Y_t− Y_t) + Zⁱ_t+ P (t)C₁ⁱ(Zⁱ_t− Zⁱ_t)}dW_tⁱ

Then a direct computation shows that (compare to (3))







d bY_t= [ bB − P (t)B] bY_tdt +

d

P

i=1

{[I − P (t)C₁ⁱ]bZⁱ_t− P (t)B₁ⁱYb_t}dW_tⁱ, YbT = 0.

(99)

By (94), We may set

to get the following equivalent BSDE (of (99)):







d bY_t= [ bB − P (t)B] bY_tdt +

d

P

i=1

Zeⁱ_tdW_tⁱ, Yb_T = 0.

(101)

It is clear that such a BSDE admits a unique adapted solution ( bY, eZ¹, · · · , eZ^d) ≡ (0, 0, · · · , 0) a.s. (see Ma & Yong (2000), pp. 15-16). Consequently, bZⁱ ≡ 0 a.s., i = 1, · · · , d (since Zbⁱ_t= [I − P (t)C₁ⁱ]⁻¹[eZⁱ_t+ P (t)B₁ⁱYbt], i = 1, · · · , d). Hence, by (98), we obtain







Y_t= P (t)X_t+ p_t,

Zⁱ_t = [I − P (t)C₁ⁱ]⁻¹{[P (t)Aⁱ₁+ P (t)B₁ⁱP (t)]X_t+ P (t)B₁ⁱp_t+ qⁱ_t}, i = 1, · · · , d.

(102) This means that any adapted solution (X, Y, Z¹, · · · , Z^d) of (3) must satisfy (102). Then, similar to the heuristic derivation above, we have that X has to be the solution of (97).

Hence, we obtain the uniqueness.

The following result tells us something more.

Proposition 2. Let (93) admits a solution P such that (94) holds for t ∈ [T₀, T ] (with some T₀ ≥ 0). Then, ∀ eT ∈ [0, T − T₀], linear FBSDE (3) is uniquely solvable on [0, eT ].

Proof. Let

P (t) = P (t + T − ee T ), 0 ≤ t ≤ eT . (103) Then eP satisfies (93) with [0, T ] replaced by [0, eT ] and

[I − eP (t)C₁ⁱ]⁻¹ is bounded for 0 ≤ t ≤ eT , i = 1, · · · , d (104)

Thus, Theorem 4.1 applies.

The above proposition tells that if (93) admits a solution P satisfying (94), FBSDE (3) is uniquely solvable over any [0, eT ] (with eT ≤ T ). Then in this case, by Theorem 1, the solvability (3) of FBSDE over [0, eT ] admits a solution ∀g ∈ L²_F

Te(Ω; R^m), of which a necessary condition is

det



[ O I ]e^At



 O

I







> 0, 0 ≤ t ≤ T. (105)

Therefore, by Theorem 3, compare (105) and (29), we see that the solvability of Riccati type equation (93) is only sufficient condition for the solvability of (3).

We have seen that (105) is necessary condition for (93) having a solution P satisfying (94). The following result gives the inverse of this.

Theorem 5. Let (105) hold. Then (93) admits a unique solution P which has the fol-lowing representation: Moreover, it holds

I−P (t)C₁ⁱ = Consequently, if in addition to (105), (30) holds, then (94) holds and the linear FBSDE (3) is uniquely solvable with the representation given by (97), (84) and (91).

Proof. We can easily check that (106) is a solution of (93). You can find in Ma & Yong (2000), pp. 49-50.

Uniqueness is obvious since (93) is a terminal value problem with the right hand side of the equation being locally Lipschitz.

I − P (t)C₁ⁱ = I +

Finally, an easy calculation shows (by (105), (30))

det(I − P (t)C₁ⁱ) =

det

[ O I ]e^{A(T −t)}C₁ⁱ

det



[ O I ]e^{A(T −t)}



 O

I









> 0, ∀t ∈ [0, T ], i = 1, · · · , d,

and I − P (t)C₁ⁱ is a continuous function on [0, T ], ∀i = 1, · · · , d, hence (94) holds. Then we complete proof.

5. Conclusion

This study proposed some extension of Ma & Yong (2000). We give the sufficient and necessary conditions of the linear FBSDE (3), and modify their work to prove the closeness of R(K) (with bA = O). Then we find the connection between a Riccati type equation and the linear FBSDE.

There are at least three more possible extensions of our method for future research.

First, we can add nonzero Z_is term in drift coefficient and derive the sufficient and nec-essary conditions. Second, one can prove or give a counterexample of the closeness in general case (not just special case in this study). Third, someone can consider some spe-cial nonlinear cases (like quadratic form) and discuss the solvability of the FBSDE. This work may be can apply in the mathematical finance like mean-variance portfolio selection and consider in optimal control and LQ problem (like Zhou and Li (2000)) in the future.

Bibliography

Bjork, T. (2009). Arbitrage theory in continuous time (3rd ed.). New York: Oxford.

Karatzas, I., & Shreve, S., E. (1998). Brownian motion and stochastic calculus (2nd ed.).

New York: Springer.

Karoui, N. E., Peng, S., & Quenez, M. C. (1997). Backward stochastic differential equa-tions in finance. Mathematical Finance, Vol. 7., No. 1, 1-71.

Lax, P. D., (2002). Functional Analysis. Wiley.

Ma, J., & Yong, J. (2000). Forward-backward stochastic equations and their applications.

Springer.

Oksendal, B. (2003). Stochastic differential equations (6th ed.) Springer.

Pham, H. (2000). Continuous-time stochastic control and optimization with Financial Application. Springer.

Zhou, X. Y., & Li, D. (2000). Continuous-time mean-variance portfolio selection: a stochastic LQ framework, Applied Mathematics Optimization, 19-33.

Appendix

Lemma 6. Consider the following SDE:















dX_t= (A_tX_t+ B_tY_t)dt +

d

P

i=1

(Aⁱ_1,tX_t+ B_1,tⁱ Y_t+ C_1,tⁱ Zⁱ_t)dW_tⁱ, dYt= bBtYtdt +

d

P

i=1

Zⁱ_tdW_tⁱ, X₀ = 0, Y₀ = 0,

0 ≤ t ≤ T,

where











A, Aⁱ₁ : [0, T ] × Ω → M_n(R) B, B₁ⁱ, C₁ⁱ : [0, T ] × Ω → M_n×m(R) B : [0, T ] × Ω → Mb _m(R)

all the processes are {Ft}_t≥0-progressively measurable, ∃M ≥ 0 with kA_tk, kAⁱ_1,tk, kB_tk, kB_1,tⁱ k, kC_1,tⁱ k ≤ M , i = 1 · · · , d, a.e. t ∈ [0, T ], a.s. And ∃bb > 0 such that k bB_tk > bb a.e. t ∈ [0, T ], a.s.

Then there exists c > 0 (independent of (Z₁, · · · , Z^d)), ∀(Z₁, · · · , Z^d) ∈ H such that E[|X_T|²] ≤ cE[|Y_T|²].

Proof. By the Itbo Formula,

d|Yt|² = 2Yt· dYt+ dhYit

= 2Y_t· bB_tY_t+

d

X

i=1

|Zⁱ_t|²

! dt + 2

d

X

i=1

Y_t· Zⁱ_tdW_tⁱ, hence,

E[|Y_t|²] = E

"

Z t 0

2Y_s· bB_sY_s+

d

X

i=1

|Zⁱ_s|²ds

#

≥ 2bbE

Z t 0

|Y_s|²ds

 +

d

X

i=1

E

Z t 0

|Zⁱ_s|²ds



. (108)

Similarly, by the Itbo Formula,

d|Xt|² = 2Xt· dX_t+ dhXit

=

"

2X_t· (A_tX_t+ B_tY_t) +

d

X

i=1

|Aⁱ_1,tX_t+ B_1,tⁱ Y_t+ C_1,tⁱ Zⁱ_t|²

# dt

+ 2

d

XX · (Aⁱ X + Bⁱ Y + Cⁱ Zⁱ)dWⁱ,

taking expectation we get and use the inequality αa²+ 1

By Gronwall Inequality and (108), E[|X_t|²] ≤ eM (2+αM +3M d)t

Take α sufficient small, we obtain E[|X_t|²] ≤ eM (2+αM +3M d)t 1

, we complete the argument.

The following case is that we want to show in the Chapter 3.

Lemma 7. Consider the following SDE:



the matrices are defined by (4). Then there exists C > 0 (independent of (Z1, · · · , Z^d)),

∀(Z₁, · · · , Z^d) ∈ H such that

E[|X_T|²] ≤ CE[|Y_T|²]. (109) Proof. We use the General Itbo Formula in e^{− bBt}Y_t,

d(e^{− bBt}Y_t) = − bBe^{− bBt}Y_tdt + e^{− bBt}dY_t = e^{− bBt}

d

X

i=1

Zⁱ_tdW_tⁱ.

Now, we consider e^te^{− bBt}Y_t and use the Itbo Formula again, d(e^te^{− bBt}Y_t) = e^te^{− bBt}Y_tdt + e^td(e^{− bBt}Y_t)

= e^te^{− bBt}Y_tdt + e^te^{− bBt}

d

X

i=1

Zⁱ_tdW_tⁱ

Let bY_t= e^te^{− bBt}Y_t, and bZⁱ_t= e^te^{− bBt}Zⁱ_t, we derive the SDE:















dX_t= (AX_t+ Be^−te^BtbYb_t)dt +

d

P

i=1

(Aⁱ₁X_t+ B₁ⁱe^−te^Btb Yb_t+ C₁ⁱe^−te^Btb Zbⁱ_t)dW_tⁱ, d bYt= bYtdt +

d

P

i=1

Zbⁱ_tdW_tⁱ, X0 = 0, bY0 = 0,

0 ≤ t ≤ T,

by Lemma 6, ∃c > 0 such that

E[|XT|²] ≤ cE[| bYT|²] ≤ ce^2Te^{2k bBkT}E[|YT|²] Take C = ce^2Te^{2k bBkT}, we prove (109).

在文檔中 線性正倒向隨機微分方程與里卡蒂方程 (頁 37-48)