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Some remarks on large deviation estimates for controlled - - PowerPoint PPT Presentation
Some remarks on large deviation estimates for controlled - - PowerPoint PPT Presentation
Some remarks on large deviation estimates for controlled semi-martingales March 9, 2012 at NCTS Hideo Nagai Division of Mathematical Science for Social Systems, Graduate School of Engineering Science, Osaka University, Toyonaka, 560-8531,
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Robust version of large deviation estimates dP ζ dP
- FT
:= e
∫ T
0 ζ∗ sdWs−1 2
∫ T
0 |ζs|2ds
W ζ
t := Wt −
∫ t
0 ζsds : B.M. under P ζ
dXt = λ(Xt)dW ζ
t + {β(Xt) + λ(Xt)ζt}dt.
(1.3) J1(κ) := lim
T→∞
1 T inf
h.
sup
ζ.
log P ζ
(
1 T {FT(X., h.) + µ 2
∫ T
0 |ζs|2ds} ≤ κ
)
.
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Motivated examples ”Market model” Riskless asset: (1.4) dS0(t) = r(Xt)S0(t)dt, S0(0) = s0. Risky assets: (1.5)
dSi(t) = Si(t){αi(Xt)dt + ∑n+m
k=1 σi k(Xt)dW k t },
Si(0) = si, i = 1, ..., m Factors: (1.6) dXt = ˜ β(Xt)dt + ˜ λ(Xt)dWt, X(0) = x,
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Total wealth: Vt =
m
∑
i=0
Ni
tSi t
Ni
t : Number of the shares
hi
t = Ni
tSi t
Vt
: Portfolio proportion i = 0, 1, 2, . . . , m. ht = (h1
t , . . . , hm t )
dVt Vt = r(Xt)dt + h(t)∗(α(Xt) − r(Xt)1)dt + h(t)∗σ(Xt)dWt, log VT = log V0 +
∫ T
0 {−1
2h∗
sσσ∗(xs)hs + h∗ sˆ
α(Xs) + r(Xs)}dt +
∫ T
0 h∗ sσ(Xs)dWs,
ˆ α(x) = α(x) − r(x)1.
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Asymptotics of the minimizing probability : (1.7) J0(κ) := lim
T→∞
1 T inf
h∈H(T) log P(1
T log VT(h) ≤ κ). concerns the problem of down-side risk minimization for the given target growth rate κ. Setting n + m = M, n = N, and f(x, h) = −1 2h∗σσ∗(x)h + h∗ˆ α(x) + r(x), ϕ(x, h) = σ∗(x)h, we arrive at the above problem (1.2) with S(x) = σσ∗(x) = δ∗δ(x). Complete market case: n = 0, m = N = M, Xi
t = log Si t
dXi
t = {αi(Xt) − 1
2(σσ∗(Xt))ii} +
m
∑
j=1
σi
j(Xt)dW j t ,
Xi
0 = xi
regarded as factors and Si
t = exieXi
t satisfies (1.5) with si = exi.
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Related contributions
- Upside chance maximization
Pham ’03; Hata-Sekine ’05,’10; Hata-Iida ’06; Sekine ’06 Knispel ’12
- Downside risk minimization for specified models
Hata - N. - Sheu ’10, AAP; Hata ’11 APFM,
- Under partial information ( for hidden Markov models)
- Y. Watanabe, ’11, Ph.D. thesis, Osaka Univ.
- Under partial information (for Linear Gaussian Models)
- N. ’11 May, QF
- Full information (General factor models)
- N. : to appear in AAP; Hata and Sheu: on going
- With constraints
Sekine: to appear in FS
- Robust downside risk minimization for one factor models
- T. Kagawa ’12 master thesis, Osaka Univ.
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Large deviation estimates for controlled semi-martingale (1.1) dXt = λ(Xt)dWt + β(Xt)dt, X0 = x ∈ RN, (1.2) J(κ) := lim
T→∞
1 T inf
h.
log P
(1
T FT(X., h.) ≤ κ
)
. FT(X., h.) = F0 +
∫ T
0 f(Xs, hs)ds +
∫ T
0 ϕ(Xs, hs)∗dWs
f(x, h) := −1 2h∗S(x)h + h∗g(x) + U(x), ϕ(x, h) = δ(x)h, S(x) : RN → Rm ⊗ Rm, g(x) : RN → Rm, δ(x) : RN → RM ⊗ Rm,
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Risk-sensitive control and its H-J-B equation Assume that F0 = 0 Consider (2.1) ˆ χ(θ) = lim
T→∞
1 T inf
h∈A(T) J(x; h; T),
θ < 0, where (2.2) J(x; h; T) = log E[eθ{∫ T
0 f(Xs,hs)ds+∫ T 0 ϕ(Xs,hs)∗dWs}],
and h ranges over the set A(T) of all admissible investment strate- gies defined by A(T) = {h ∈ H(T); E[eθ ∫ T
0 h∗ sδ∗(Xs)dWs−θ2 2
∫ T
0 h∗ sδ∗δ(Xs)hsds] = 1}.
Then, we shall see that (2.1) could be considered the dual problem to (1.2).
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Assumptions (2.3) λ, β, S, g, δ are smooth and globally Lipschitz, U is smooth c4|x|2 − c5 ≤ U(x); U, |DU| ≤ c6|x|2 + c7 (2.4) c2|ξ|2 ≤ ξ∗λλ∗(x)ξ ≤ c3|ξ|2, c2, c3 > 0, ξ ∈ Rn, (2.5) δ∗δ(x) ≥ cδI, cδ > 0 (2.6) c0δ∗δ(x) ≤ S(x) ≤ c1δ∗δ(x), x ∈ RN, c0, c1 > 0
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Note that, when setting Qθ := S(x) − θδ∗δ(x), θ ≤ 0, Qθ satisfies (2.7) (c0 − θ)δ∗δ(x) ≤ Qθ(x) ≤ (c1 − θ)δ∗δ(x) and (2.8) θQ−1
θ
(x) ≤ θ c1 − θ(δ∗δ(x))−1, θ c0 − θ(δ∗δ(x))−1 ≤ θQ−1
θ
(x) Moreover, (2.9) c0 c0 − θI ≤ I + θδQ−1
θ
δ∗ ≤ I
- holds. Indeed, (2.7) follows directly from (2.6) and thus (2.8) is
- btained from (2.7). The lefthand side of (2.9) is seen since
θ c0 − θI ≤ θ c0 − θδ(δ∗δ)−1δ∗ ≤ θδQ−1
θ
δ∗, which follows from (2.8). The right hand side of (2.9) is obvious.
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Value function (2.13) v(t, x) = inf
h.∈A(T−t) log E[eθ{∫ T−t f(Xs,hs)ds+∫ T−t ϕ(Xs,hs)∗dWs}].
Under P h(A) = E[eθ ∫ T
0 h∗ sδ∗(Xs)dWs−θ2 2
∫ T
0 h∗ sδ∗δ(Xs)hsds : A],
Xt satisfies dXt = {β(Xt) + θλδ(Xt)ht}dt + λ(Xt)dW h
t ,
X0 = x with B. M. W h
t defined by
W h
t := Wt − γ
∫ t
0 δ(Xs)hsds
(2.14) v(t, x) = inf
h.∈A(T) log Eh[eθ ∫ T−t {f(Xs,hs)+θ
2h∗ sδ∗δ(Xs)hs}ds]
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The H-J-B equation :
∂v ∂t + 1 2tr[λλ∗D2v] + 1 2(Dv)∗λλ∗Dv
+ infh{[β + θλδh]∗Dv + θf(x, h) + θ2
2 |ϕ(x, h)|2)} = 0,
v(T, x) = 0 which is written as (2.15)
∂v ∂t + 1 2tr[λλ∗D2v] + β∗ θDv + 1 2(Dv)∗λ(I + θδQ−1 θ
δ∗)λ∗Dv +θ
2g∗Q−1 θ
g + θU = 0, v(T, x) = 0, where βθ = β + θλδQ−1
θ
g, Qθ = S − θδδ∗.
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Since (2.7) (c0 − θ)δ∗δ(x) ≤ Qθ(x) ≤ (c1 − θ)δ∗δ(x) Nθ := I + θδQ−1
θ
δ∗ satisfies (2.9) c0 c0 − θI ≤ Nθ ≤ I Further, θQ−1
θ
= (δ∗δ)−1δ∗Nθδ(δ∗δ)−1 − (δ∗δ)−1
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Rewritten
∂v ∂t + 1 2tr[λλ∗D2v] + {β − λδ(δ∗δ)−1g}∗Dv
+1
2[λ∗Dv + δ(δ∗δ)−1g]∗Nθ[λ∗Dv + δ(δ∗δ)−1g]
−1
2g∗(δ∗δ)−1g + θU = 0,
v(T, x) = 0 is the H-J-B equation of the stochastic control problem: v∗(0, x; T; θ) = sup
- z. E[
∫ T
0 Φ(Ys, zs)ds]
subject to dYt = λ(Yt)dBt + {G(Yt) + λ(Yt)zt}dt, Y0 = x, G(y) = β(y) − λδ(δ∗δ)−1g Φ(y, z; θ) = −1 2z∗N−1
θ
z + g∗(δ∗δ)−1δ∗(y)z − 1 2g∗(δ∗δ)−1g(y) + θU(y).
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Convexity We can see that Φ(y, z; θ) is nothing but a linear function of θ and thus the value function of this stochastic control problem is a convex function of θ. Further, under suitable conditions we have a solution to H-J-B equation of ergodic type: (2.16) χ(θ) = 1
2tr[λλ∗D2v] + β∗ θDv + 1 2(Dv)∗λNθλ∗Dv
+θ
2g∗Q−1 θ
g + θU, and (2.17) χ(θ) = lim
T→∞
1 T v(0, x; T; θ), Then, we see the convexity of χ(θ).
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Analytical results Under the assumptions (2.3) - (2.6) H-J-B equation (2.15) has a unique solution such that v(t, x) ≤ K0 v, ∂v
∂t, ∂v ∂xi, ∂2v ∂xi∂xj, ∈ Lp(0, T; Lp loc(Rn)) ∂v ∂t ≥ −C ∂2v ∂2t, ∂2v ∂xi∂t, ∂3v ∂xi∂xj∂xk, ∂3v ∂xi∂xj∂t ∈ Lp(0, T; Lp loc(Rn))
|Dv|2 + c0
ν1(∂v ∂t + C) ≤ c(|DNθ|2 2r + |Nθ|2 2r + |D(λλ∗)|2 2r + |Dβθ|2 2r
+|βθ|2
2r + |U|2r + |DU|2r + |g|2 2r + |Dg|2 2r + 1)
x ∈ Br, t ∈ [0, T)
- cf. Bensoussan-Frehse-N ’98 AMO, N. ’03 SICON
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Verification ˆ h(t, x) := Q−1
θ
(δ∗λ∗v(t, x) + g(x)) ˆ h(T)
t
:= ˆ h(t, Xt) is the optimal strategy: v(0, x; T) = log E[eθ{∫ T
0 f(Xs,ˆ
h(T)
s
)ds+∫ T
0 ϕ(Xs,ˆ
h(T)
s
)∗dWs}]
= infh.∈A(T) log E[eθ{∫ T
0 f(Xs,hs)ds+∫ T 0 ϕ(Xs,hs)∗dWs]
Moreover, ˆ z(t, y) := Nθ{λ∗Dv(t, x) + δ(δ∗δ)−1g(x)} dYt = λ(Yt)dBt + {G(Yt) + λ(Yt)ˆ z(t, Yt)}dt v(0, x; T) = E[
∫ T
0 Φ(Ys, ˆ
zs)ds] = sup
- z. E[
∫ T
0 Φ(Ys, zs)ds]
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H-J-B equation of ergodic type Assume that (2.18) G(x)∗x ≤ −cG|x|2 + c′
G,
cG, c′
G > 0
besides the above assumptions. Then, we have a solution w of : (2.16) χ(θ) = 1
2tr[λλ∗D2w] + β∗ θDw + 1 2(Dw)∗λNθλ∗Dw
+θ
2g∗Q−1 θ
g + θU, such that w(x) → −∞ as |x| → ∞. Moreover, such solution is unique up to additive constants. Further, χ(γ) is convex and differentiable: (EE)′ χ′(θ) = L(θ)w′ + U +1
2(λ∗Dw + δ(δ∗δ)−1g)∗{δQ−1 θ
δ∗ + θ(δQ−1
θ
δ∗)2}(λ∗Dw + δ(δ∗δ)−1g),
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where w′ = ∂w
∂θ and
L(θ)ψ = 1 2tr[λλ∗D2ψ] + β∗
θDψ + (Dw)∗λNθλ∗Dψ
Remark (2.17) χ(θ) = lim
T→∞
1 T v(0, x; T; θ), can be seen similarly to the recent work by Ichihara and Sheu. The following formula is useful to obtain our duality theorem. χ(θ)−θχ′(θ) = L(θ)(w−θw′)−1 2(Nθλ∗Dw+θδQ−1
θ
g)∗(Nθλ∗Dw+θδQ−1
θ
g)
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Duality theorem Theorem 1 For κ ∈ (χ′(−∞), χ′(0−)), we have lim
T→∞
1 T inf
h∈A(T) log P
(1
T FT(X., h.) ≤ κ
)
= − inf
k∈(−∞,κ] I(k) = −I(κ)
I(k) := sup
θ<0
{θk − χ(θ)} Moreover, for θ(κ) such that χ′(θ(κ)) = κ ∈ (χ′(−∞), χ′(0−)) take a strategy ˆ h(θ(κ),T)
t
. Then, lim
T→∞
1 T log P
(1
T FT(X., h(θ(κ),T)) ≤ κ
)
= − inf
k∈(−∞,κ] I(k) = −I(κ)
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Remark In the case of S(x) = δ∗δ(x), the problem minimizing the risk- sensitive criterion J(x, h; T) subject to Xt defined by (1.1) can be seen equivalent to the problem inf
h.
sup
ν. Eν[θ{
∫ T
0 f(Xs, hs)ds +
∫ T
0 ϕ(Xs, hs)∗dWs} −
∫ T
0 |νs|2ds],
where Xt is governed by dXt = {β(Xt) + λ(Xt)νt}dt + λ(Xt)dW ν
t ,
X0 = x, and P ν is P ν(A) = E[e
∫ T
0 ν∗ sdWs−1 2
∫ T
0 |νs|2ds : A],
W ν
t := Wt −
∫ t
0 νsds.
is the B. M. under P ν
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Indeed, in that case Qθ = S − θδ∗δ = (1 − θ)δ∗δ and so Q−1
θ
= 1 1 − θ(δ∗δ)−1, from which we obtain Nθ = I + θδQ−1
θ
δ∗ = I + θ 1 − θδ(δ∗δ)−1δ∗ and thus N−1
θ
= I − θδ(δ∗δ)−1δ∗.
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On the other hand, Eν[θ{
∫ T
0 f(Xs, hs)ds +
∫ T
0 ϕ(Xs, hs)dWs} −
∫ T
0 |νs|2ds]
= Eν[
∫ T
0 {θf(Xs, hs) + θh∗ sδ∗(Xs)νs − 1
2|νs|2}ds] ≡ Eν[
∫ T
0 ψ(Xs, hs, νs; θ)ds],
where ψ(x, h, ν; θ) = −θ
2h∗(δ∗δ)h + θh∗g + θU + θh∗δ∗ν − 1 2|ν|2
= −θ
2{h − (δ∗δ)−1(g + δ∗ν)}∗(δ∗δ){h − (δ∗δ)−1(g + δ∗ν)}
+θ
2(g + δ∗ν)(δ∗δ)−1(g + δ∗ν) + θU − 1 2|ν|2
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Since infh ψ(x, h, ν; θ) = θ
2(g + δ∗ν)(δ∗δ)−1(g + δ∗ν) + θU − 1 2|ν|2
= −1
2ν∗{I − θδ(δ∗δ)−1δ∗}ν + θν∗δ(δ∗δ)−1g + θ 2g∗(δ∗δ)−1g + θU
= −1
2ν∗N−1 θ
ν + θν∗δ(δ∗δ)−1g + θ
2g∗(δ∗δ)−1g + θU
≡ Ψ(x, ν; θ) The above problem is reduced to the problem: sup
ν. Eν[
∫ T
0 Ψ(Xs, νs; θ)ds]
Thus, the H-J-B equation of the problem is
∂v ∂t + 1 2tr[λλ∗D2v] + supν{[β + λν]∗Dv + Ψ(x, ν; θ)} = 0,
v(T, x) = 0,
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written as (2.18)
∂v ∂t + 1 2tr[λλ∗D2v] + 1 2(Dv)∗λNθλ∗Dv
+(β +
θ 1−θλδ(δ∗δ)−1g)∗Dv + θ 2(1−θ)g∗(δ∗δ)−1g + θU = 0,
v(T, x) = 0, Since Q−1
θ
=
1 1−θ(δ∗δ)−1 (2.18) is seen to be identical to H-J-B
equation (2.15) of risk-sensitive control : (2.15)
∂v ∂t + 1 2tr[λλ∗D2v] + β∗ θDv + 1 2(Dv)∗λ(I + θδQ−1 θ
δ∗)λ∗Dv +θ
2g∗Q−1 θ
g + θU = 0, v(T, x) = 0 βθ = β + θλδQ−1
θ
g
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Robust version of large deviation estimates Assume that S(x) = δ∗δ. We consider the robust version: (3.1) J1(κ) := lim
T→∞
1 T inf
h.
sup
ζ.
log P ζ
(
1 T {FT(X., h.) + µ 2
∫ T
0 |ζs|2ds} ≤ κ
)
. where FT(X., h.) :=
∫ T
0 f(Xs, hs)ds +
∫ T
0 ϕ(Xs, hs)∗dWs.
Here P ζ is P ζ(A) = E[e
∫ T
0 ζ∗ sdWs−1 2
∫ T
0 |ζs|2ds : A],
and ζt is a progressively measurable process such that ζt = ζ(t, Xt) with |ζ(t, x)| ≤ CT(1 + |x|),
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Then, W ζ
t := Wt −
∫ t
0 ζsds
is the B. M. under P ζ and Xt is governed by dXt = λ(Xt)dW ζ
t + {β(Xt) + λ(Xt)ζt}dt.
Dual problem: (3.2) χ1(θ) = lim
T→∞
1 T inf
h.
sup
ζ.
log Eζ[eθ{FT (X.,h.)+µ
2
∫ T
0 |ζs|2ds}].
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The problem on a finite time horizon: u(t, x) = inf
h.∈A(T−t) sup ζ.
log Eζ[eθ{FT−t(X.,h.)+µ
2
∫ T−t
|ζs|2ds}].
Under P ζ,h(A) = Eζ[eθ ∫ T
0 h∗ sδ∗(Xs)dWs−θ2 2
∫ T
0 h∗ sδ∗δ(Xs)hsds : A],
W ζ,h
t
:= W ζ
t − θ
∫ t
0 δ(Xs)hsds
is the B.M and Xt is governed by dXt = {β(Xt) + θλδ(Xt)ht + λ(Xt)ζt}dt + λ(Xt)dW ζ,h
t
, X0 = x. The value function is written as (2.14) u(t, x) = inf
h.∈A(T) sup ζ.
log Eζ,h[eθ ∫ T−t
{f(Xs,hs)+θ
2h∗ sδ∗δ(Xs)hs+µ 2|ζs|2}ds]
SLIDE 30
The H-J-B equation:
∂u ∂t + 1 2tr[λλ∗D2u] + 1 2(Du)∗λλ∗Du
+ infh supζ{[β + λζ + θλδh]∗Du + θf(x, h) + θ2
2 |δh|2 + θµ 2 |ζ|2} = 0,
u(T, x) = 0 It is written as
∂u ∂t + 1 2tr[λλ∗D2u] + 1 2(Du)∗λNθλ∗Du
− 1
2θµ{ θ 1−θδ(δ∗δ)−1g + Nθλ∗Du}∗R−1 θ,µ{ θ 1−θδ(δ∗δ)−1g + Nθλ∗Du}
+(β +
θ 1−θλδ(δ∗δ)−1g)∗Du + θU + θ 2(1−θ)g∗(δ∗δ)−1g = 0
u(T, x) = 0, where Rθ,µ := I + 1 µ(1 − θ)δ(δ∗δ)−1δ, R−1
θ,µ = I −
1 1 + µ − µθδ(δ∗δ)−1δ
SLIDE 31
It is also written as
∂u ∂t + 1 2tr[λλ∗D2u] + 1 2(λ∗Du + δ(δ∗δ)−1g)∗Nθ(λ∗Du + δ(δ∗δ)−1g)
− 1
2θµ{ θ 1−θδ(δ∗δ)−1g + Nθλ∗Du}∗R−1 θ,µ{ θ 1−θδ(δ∗δ)−1g + Nθλ∗Du}
+(β − λδ(δ∗δ)−1g)∗Du + θU − 1
2g∗(δ∗δ)−1g = 0
u(T, x) = 0,
SLIDE 32
As the above remark, introduce a new probability measure P ζ,ν : P ζ,ν(A) = Eζ[e
∫ T
0 νsdW ζ s −1 2
∫ T
0 |νs|2ds : A],
and the B.M under this measure: W ζ,ν
t
:= W ζ
t −
∫ t
0 νsds.
Then, the game counterpart of the current problem is inf
h.
sup
ν,ζ
Eζ,ν[θ{
∫ T
0 (f(Xs, hs)+µ
2|ζs|2)ds+
∫ T
0 ϕ(Xs, hs)∗dWs}−
∫ T
0 |νs|2ds]
dXt = {β(Xt) + λ(Xt)(ζt + νt)}dt + λ(Xt)dW ζ,ν
t
, X0 = x,
SLIDE 33
Then, Eζ,ν[θ{
∫ T
0 (f(Xs, hs) + µ
2|ζs|2)ds +
∫ T
0 h∗ sδ(Xs)∗dWs} − 1
2
∫ T
0 |νs|2ds]
= Eζ,ν[θ{
∫ T
0 (f(Xs, hs) + µ 2|ζs|2 + h∗ sδ(Xs)∗(ζs + νs))ds} − 1 2
∫ T
0 |νs|2ds]
and infh[θ{f(x, h) + µ
2|ζ|2 + ϕ(x, h)∗(ζ + ν)} − 1 2|ν|2]
= θ
2{g + δ∗(ζ + ν)}∗(δ∗δ)−1{g + δ∗(ζ + ν)} + θU + θµ 2 |ζ|2 − 1 2|ν|2
≡ Ξ(x, ζ, ν; θ) Thus, the problem is reduced to I(0, x; T; θ) := sup
ζ.ν.
Eζ,ν[
∫ T
0 Ξ(Xs, ζs, νs; θ)ds].
SLIDE 34
Further, we can see that Ξ(x, ζ, ν; θ) =
θµ 2 ζ∗(I + 1 µδ(δ∗δ)−1δ∗)ζ − 1 2ν∗N−1 θ
ν + θν∗δ(δ∗δ)−1δ∗ζ + θg∗(δ∗δ)−1δ∗ζ + θg∗(δ∗δ)−1δ∗ν + θU − θ
2g∗(δ∗δ)−1g
The H-J-B equation for I(0, x; T; θ) :
∂u ∂t + 1 2tr[λλ∗D2u] + supζ,ν{[β + λ(ζ + ν)]∗Du + Ξ(x, ζ, ν; θ)} = 0,
u(T, x) = 0,
SLIDE 35
After tedious calculation we can see that it is (3.3)
∂u ∂t + 1 2tr[λλ∗D2u] + 1 2(Du)∗λKθ,µλ∗Du + β∗Du
−
1−θµ 1+µ−θµg∗(δ∗δ)−1δ∗λ∗Du + θU + θµ 2(1+µ−θµ)g∗(δ∗δ)−1g = 0
u(T, x) = 0, where Kθ,µ := (1 − 1 θµ){I − 1 − θµ 1 + µ − θµδ(δ∗δ)−1δ∗} with (1 − 1 θµ)( µ 1 + µ − θµ)I ≤ Kθ,µ ≤ (1 − 1 θµ)I
SLIDE 36
It is identical to the equation deduced in the above:
∂u ∂t + 1 2tr[λλ∗D2u] + 1 2(Du)∗λNθλ∗Du
− 1
2θµ{ θ 1−θδ(δ∗δ)−1g + Nθλ∗Du}∗R−1 θ,µ{ θ 1−θδ(δ∗δ)−1g + Nθλ∗Du}
+(β +
θ 1−θλδ(δ∗δ)−1g)∗Du + θU + θ 2(1−θ)g∗(δ∗δ)−1g = 0
u(T, x) = 0, where Rθ,µ := I + 1 µ(1 − θ)δ(δ∗δ)−1δ, R−1
θ,µ = I −
1 1 + µ − µθδ(δ∗δ)−1δ
SLIDE 37
Then, K−1
θ,µ :=
θµ θµ − 1{I + 1 − θµ µ δ(δ∗δ)−1δ}
∂u ∂t + 1 2tr[λλ∗D2u] + (β − λδ(δ∗δ)−1g)∗Du
+1
2[λ∗Du + K−1 θ,µ µ 1+µ−θµδ(δ∗δ)−1g]∗Kθ,µ
×[λ∗Du + K−1
θ,µ µ 1+µ−θµδ(δ∗δ)−1g] + θU + θµ 2(1−θµ)g∗(δ∗δ)−1g = 0
u(T, x) = 0,
SLIDE 38
Convexity Since Ξ is a linear function of θ the value I(x, 0; T; θ) is a convex function of θ. Further, under suitable conditions we have a solution to H-J-B equation of ergodic type: (3.4) χ1(θ) = 1
2tr[λλ∗D2u] + 1 2(Du)∗λKθ,µλ∗Du + β∗Du
−
1−θµ 1+µ−θµg∗(δ∗δ)−1δ∗λ∗Du + θU + θµ 2(1+µ−θµ)g∗(δ∗δ)−1g
and χ1(θ) = lim
T→∞
1 T I(0, x; T; θ). Thus we see that χ1(θ) is convex.
SLIDE 39
Analytical results Under the assumptions (2.3) - (2.6) H-J-B equation (3.3) has a solution such that u(t, x) ≤ K0 u, ∂u
∂t , ∂u ∂xi, ∂2v ∂xi∂xj, ∈ Lp(0, T; Lp loc(Rn)) ∂u ∂t ≥ −C ∂2u ∂2t , ∂2u ∂xi∂t, ∂3u ∂xi∂xj∂xk, ∂3u ∂xi∂xj∂t ∈ Lp(0, T; Lp loc(Rn))
|Du|2 + c0
ν1(∂v ∂t + C) ≤ c(|DKθ,µ|2 2r + |Kθ,µ|2 2r + |D(λλ∗)|2 2r + |Dβ|2 2r
+|β|2
2r + |U|2r + |DU|2r + |g|2 2r + |Dg|2 2r + 1)
x ∈ Br, t ∈ [0, T)
- cf. Bensoussan-Frehse-N ’98 AMO, N. ’03 SICON
SLIDE 40
Verification ˆ ζ(t, x) :=
−1 µ(1−θ)R−1 θ,µδ(δ∗δ)−1g + −1 θµ R−1 θ,µNθλ∗Du
=
−1 1+µ−µθδ(δ∗δ)−1g + −1 θµ (I − 1−θµ 1+µ−θµδ(δ∗δ)−1δ∗)λ∗Du
ˆ ν(t, x) := Nθ{θδ(δ∗δ)−1(δ∗ˆ ζ + g) + λ∗Du} =
θµ 1+µ−θµδ(δ∗δ)−1g + (I − 1−θµ 1+µ−θµδ(δ∗δ)−1δ∗)λ∗Du
dXt = [β(Xt)+λ(Xt){ˆ ζ(t, Xt)+ˆ ν(t, Xt)}]dt+λ(Xt)dW ζ,ν
t
, X0 = x, u(0, x; T) = supζ.ν. Eζ,ν[
∫ T
0 Ξ(Xs, ζs, νs; θ)ds]
= Eˆ
ζ,ˆ ν[
∫ T
0 Ξ(Xs, ˆ
ζs, ˆ νs; θ)ds] . ˆ ζs = ˆ ζ(s, Xs), ˆ νs = ˆ ν(s, Xs)
SLIDE 41
˜ h(t, x) := Q−1
θ
(δ∗λ∗Du + δ∗ˆ ζ + g) =
µ (1+µ−θµ)(δ∗δ)−1g + θµ−1 θ(1+µ−θµ)(δ∗δ)−1δ∗λ∗Du]
dXt = {β(Xt) + λ(Xt)ˆ ζ(t, Xt)}dt + λ(Xt)dW ζ
t ,
X0 = x. ˜ ht := ˜ h(t, Xt) ˜ ζt = ˆ ζ(t, Xt) u(0, x; T) = infh.∈A(T) supζ. log Eζ[eθ{FT (X.,h.)+µ
2
∫ T
0 |ζs|2ds}]
= log E˜
ζ[eθ{FT (X.,˜ h.)+µ
2
∫ T
0 |˜
ζs|2ds}]
SLIDE 42
H-J-B equation ergodic type Assume that (2.18) G(x)∗x ≤ −cG|x|2 + c′
G,
cG, c′
G > 0
besides the above assumptions. Then, we have a solution w1 of : (3.4) χ1(θ) = 1
2tr[λλ∗D2w1] + β∗Dw1 + 1 2(Dw1)∗λKθ,µλ∗Dw1
−
1−θµ 1+µ−θµg∗(δ∗δ)−1δ∗λ∗Dw1 + θU + θµ 2(1+µ−θµ)g∗(δ∗δ)−1g,
such that w1(x) → −∞ as |x| → ∞. Moreover, such solution is unique up to additive constants.
SLIDE 43
Further, χ1(γ) is convex and differentiable: (EE)′ χ′
1(θ)
= L1(θ)w′
1 + µ(θµ−1) 2θ(1+µ−θµ)2(Dw1)∗λδ(δ∗δ)−1δ∗λ∗Dw1
+
1 2µθ2(Dw1)∗λ(I − 1−θµ 1+µ−θµδ(δ∗δ)−1δ∗)λ∗Dw1 + U
+
µ2 (1+µ−θµ)2g∗(δ∗δ)−1δ∗λ∗Dw1 + µ(1+µ) 2(1+µ−θµ)2g∗(δ∗δ)−1g
where L1(θ)ψ = 1 2tr[λλ∗D2ψ] + β∗
θ,µDψ + (Dw1)∗λKθ,µλ∗Dψ,
where βθ,µ = β − 1 − θµ 1 + µ − θµλδ(δ∗δ)−1g
SLIDE 44
Useful formulae ˆ ζ1(x) :=
−1 µ(1−θ)R−1 θ,µδ(δ∗δ)−1g + −1 θµ R−1 θ,µNθλ∗Dw1
=
−1 1+µ−µθδ(δ∗δ)−1g + −1 θµ (I − 1−θµ 1+µ−θµδ(δ∗δ)−1δ∗)λ∗Dw1
ˆ ν1(x) := Nθ{θδ(δ∗δ)−1(δ∗ˆ ζ1 + g) + λ∗Dw1} =
θµ 1+µ−θµδ(δ∗δ)−1g + (I − 1−θµ 1+µ−θµδ(δ∗δ)−1δ)λ∗Dw1
Then, χ1(θ) =
1 2tr[λλ∗D2w1] + {β + λ(ˆ
ζ1 + ˆ ν1)}∗Dw1 + Ξ(x, ˆ ζ1, ˆ ν1; θ) = L1(θ)w1 + Ξ(x, ˆ ζ1, ˆ ν1; θ) and χ1(θ) − θχ′
1(θ) = L1(θ)(w1 − θw′ 1) − 1
2ˆ ν∗
1ˆ
ν1
SLIDE 45