A Dual Method For Backward Stochastic Differential Equations with Application to Risk Valuation

0.32 MB·

by  Andrzej Ruszczynski

ff
A Dual Method For Backward Stochastic Di erential
Equations with Application to Risk Valuation
AndrzejRuszczyn´ski JianingYao
∗ †
7
1
January 24,2017
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2

n
a
J
Abstract

2
2 Weproposeanumericalrecipeforriskevaluationdefinedbyabackwardstochasticdiffer-
entialequation. Usingdualrepresentationoftheriskmeasure,weconverttheriskvaluationto
]
C astochasticcontrolproblemwherethecontrolisacertainRadon-Nikodymderivativeprocess.
Byexploringthemaximumprinciple,weshowthatapiecewise-constantdualcontrolprovides
O
agoodapproximationonashortinterval. Adynamicprogrammingalgorithmextendstheap-
.
h proximationto a finite time horizon. Finally, we illustrate theapplicationofthe procedureto
t
a riskmanagementinconjunctionwithnestedsimulation.
m Keywords:DynamicRiskMeasures,Forward–BackwardStochasticDifferentialEquations,Stochas-
[ ticMaximumPrinciple,RiskManagement

1
v
1 Introduction
4
3
2 The main objective of this paper is to present a simple and efficient numerical method for solving
6
0 backward stochastic differential equations with convex and homogeneous drivers. Such equations
. are fundamental modeling tools for continuous-time dynamic risk measures with Brownian filtra-
1
0 tion,butmayalsoariseinotherapplications.
7
Thekeyproperty ofdynamic riskmeasuresistime-consistency, whichallowsfordynamic pro-
1
grammingformulations. ThediscretetimecasewasextensivelyexploredbyDetlfsenandScandolo
:
v
[11], Bion-Nadal [5], Cheridito et al. [7, 8], Föllmer and Penner [12], Fritelli and Rosazza Gianin
i
X [15],FrittelliandScandolo [16],Riedel[32],andRuszczyn´ski andShapiro[35].
r Forthecontinuous-time case,Coquet,Hu,MéminandPeng[9]discovered thattime-consistent
a
dynamic risk measures, with Brownian filtration, can be represented as solutions of Backward
Stochastic Differential Equations (BSDE)[30];undermildgrowthconditions, thisistheonlyform
Rutgers University, Department of Management Science and Information Systems, Piscataway, NJ 08854, USA,
∗
Email:[email protected]
Rutgers University, Department of Management Science and Information Systems, Piscataway, NJ 08854, USA,
†
Email:[email protected]
1
possible. Specifically, the y-part solution of one-dimensional BSDE, defined below, measures the
riskofavariableξ atthecurrent timet:
T
T T
Y = ξ + g(s,Z ) ds Z dW , 0 t T, (1)
t T s s s
Z −Z ≤ ≤
t t
withthedrivergbeinginterpretedasa“riskrate.” The -measurablerandomvariableξ isusually
T T
F
afunction oftheterminalstateofacertainstochastic dynamical system.
Inspired by that, Barrieu and ElKaroui provided a comprehensive study in [3, 4]; further con-
tributionsbeingmadebyDelbean,Peng,andRosazzaGianin[10],andQuenezandSulem[31](for
amoregeneral modelwithLevyprocesses). Inaddition, application tofinancewasconsidered, for
example,in[19]. Usingtheconvergence resultsofBriand,DelyonandMémin[6],Stadje[39]finds
thedriversofBSDEcorresponding todiscrete-time riskmeasures.
Motivated by an earlier work on risk-averse control of discrete-time stochastic process [37],
Ruszczyn´skiandYao[38]formulatearisk-aversestochasticcontrolproblemfordiffusionprocesses.
Thecorresponding dynamicprogrammingequationleadstoadecoupledforward–backwardsystem
ofstochastic differentialequations.
While forward stochastic differential equations can be solved by several efficient methods, the
main challenge is the numerical solution of (1), where ξ represents the future value function. In
T
particular, Zhang [41] and Touzi et al. [40] use backward Euler’s approximation and regression.
Such an approach, however, is not well-suited for risk measurement, because it does not preserve
the monotonicity of the risk measure. Alternatively, Øksendal [28] directly attacks continuous-
timerisk-aversecontrolproblemwithjumpsbyderivingsufficientconditions. Algorithmsbasedon
maximumprinciple wereinvestigated byLudwigetal. [23].
Ourideaistoderivearecursivemethodbasedonrisk-aversedynamicprogramming,sothatthe
approximation becomesatime-consistent coherent riskmeasureindiscrete time.
Thepaperisorganized asfollows. Insection 2,wequickly introduce theconcept ofadynamic
risk measure and review its properties. In section 3 werecall the dual representation of a dynamic
risk measure and formulate an equivalent stochastic control problem. The optimality condition for
thedualcontrolproblem, aspecialformofamaximumprinciple, arederivedinsection4. Sections
5 and 6 estimate the errors introduced by using constant processes as dual controls. In section 7
we present the whole numerical method with piecewise constant dual controls and analyze its rate
of convergence. Finally, in section 8, weillustrate the efficacy ofour approach on atwo-stage risk
managementmodel.
2 The Risk Evaluation Problem
Givenacompletefilteredprobabilityspace(Ω, ,P)withfiltration generatedbyd-dimen-
t t [0,T]
sionalBrownianmotion W ,weconsideFr thefollowingstoch{Fast}ic∈ differentialequation:
t t [0,T]
{ }∈
dX = b(t,X) dt+σ(t,X) dW, X = x, t [0,T], (2)
t t t t 0
∈
withmeasurable b : [0,T] Rn Rn,andσ : [0,T] Rn Rn Rd. Weintroduce thefollowing
× → × → ×
notation.
2
E[ ]:= E[ ];
t t
• · ·|F
2(Ω, ,P;Rn): the set of Rn-valued -measurable random variables ξ such that ξ 2 :=
t t
• L F F k k
E[ ξ2] < ;forn = 1,wewriteit 2(Ω, ,P);
| | ∞ L F
2,n[t,T]: thesetofRn-valuedadaptedprocessesYon[t,T],suchthat Y 2 := E T Y 2ds <
• H k k 2,n[t,T] t t | s|
;forn= 1wewriteit 2[t,T];1 H hR i
∞ H
k([t,T] Rn)thespaceoffunctions f : [t,T] Rn Rwhosehasderivativeuptok-th,in
• Cb × × →
themeanwhile, allthosederivatives arecontinuous andbounded; for f : Rn R,wedenote
7→
byCk(RN);
b
(B): thespaceofLipschitzcontinuous functions f : B R.
L
• C →
Wemakefollowingassumptions aboutthedriftandvolatility terms.
Assumption2.1. (i) b(,0) + σ(,0) 2[0,T];
| · | | · | ∈ H
(ii) Thefunctions b,σ 1([0,T] Rn),theconstantC > 0denotes theLipschitzconstants
∈ Cb ×
b(t,x ) b(t,x ) + σ(t,x ) σ(t,x ) C x x a.s.
1 2 1 2 1 2
| − | | − |≤ | − |
b(t,x ) + σ(t,x ) C x , a.s..
1 1 1
| | | | ≤ | |
(iii) The dimension of Brownian motion and the state process coincide, i.e., n = d, and we also
assume
1
σ(t,x)σ (t,x) , (t,x) [0,T] Rd.
⊤
≥ CI ∀ ∈ ×
Ourintention istoevaluateriskofaterminalcostgenerated bytheforwardprocess(2):
ρ Φ(X ) , (3)
0,T T
(cid:2) (cid:3)
where Φ (Rn) is bounded, and ρ is a dynamic risk measure consistent with the
∈ CL s,t 0 s t T
filtration t t [0,T]. Wereferthereader(cid:8)to[(cid:9)29≤]≤fo≤racomprehensivediscussiononriskmeasurement
{F}∈
andfiltration-consistent evaluations.
Special role in the dynamic risk theory is played by g-evaluations which defined by one-
dimensional backwardstochastic differentialequations ofthefollowingform:
dY = g(t,Y,Z)dt Z dW, Y = Φ(X ), t [0,T], (4)
t t t t t T T
− − ∈
g
withρ Φ(X )]definedtobeequaltoY. ThedrivergisjointlyLipschitzin(y,z),andtheprocess
t,T T t
g(,0,0)(cid:2) 2[0,T].
· ∈ H
As proved in [9] every F-consistent nonlinear evaluation that is dominated by a g-evaluation
with g = µy + νz with some ν,µ > 0 is in fact a g-evaluation for some g; the dominance is
| | | |
understood asfollows:
ρ [ξ+η] ρ [ξ] ρν,µ[η]
0,T − 0,T ≤ 0,T
forallξ,η 2(Ω, ,P).
T
∈ L F
1Whenthenormisclearfromthecontext,thesubscriptsareskipped.
3
Proposition 2.2. Forall0 t T andallξ,ξ (Ω, ,P),thefollowingproperties hold:
′ 2 T
(i) Generalized constan≤tpr≤eservation: Ifξ ∈ L(Ω, F,P),thenρg [ξ] = ξ;
(ii) Timeconsistency: ρg [ξ] = ρg [ρg [ξ]∈],Lfo2ralFl0t s t; t,t
s,T s,t t,T ≤ ≤
(iii) Localproperty: ρg [ξ +ξ ]= ρg [ξ]+ ρg [ξ ],forallA .
t,T 1A ′1Ac 1A t,T 1Ac t,T ′ ∈ Ft
From now on, shall focus exclusively ong-evaluations asdynamic risk measures, and weshall
skipthesuperscript ginρg.
Theevaluation ofriskisequivalent tothesolution ofadecoupled forward–backward systemof
stochastic differential equations (2)–(4). Animportant virtueofthissystemisitsMarkovproperty:
ρ Φ(X ) = v(t,X), (5)
t,T T t
(cid:2) (cid:3)
wherev: [0,T] Rn R. Wehave
× →
v(t,x) = ρx Φ(Xt,x) , (t,x) [0,T] Rn, (6)
t,T T ∈ ×
(cid:2) (cid:3)
where Xt,x isthesolutionofthesystem(2)restartedattimetfromstate x:
s
{ }
dXt,x = b(s,Xt,x) ds+σ(s,Xt,x)dW , s [t,T], Xt,x = x, (7)
s s s s ∈ t
and ρx Φ(Xt,x) is the (deterministic) value of Yt,x in the backward equation (4) with terminal
t,T T t
condition(cid:2) Φ(Xt,x(cid:3)).
T
Numericalmethodsforsolvingforwardequationsareverywellunderstood(see,e.g.,[18]). We
focus, therefore, onthebackward equation (4). Sofar, alimitednumber ofresults areavailable for
thispurpose. ThemostprominentistheEulermethodwithfunctional regression (see,e.g.,[24,25,
40, 41, 42]). Our intention is to show that for drivers satisfying additional coherence conditions, a
muchmore effective method canbe developed, which exploits time-consistency, duality theory for
riskmeasures, andthemaximumprinciple instochastic control.
3 The Dual Control Problem
Wefurther restrict the risk measures under consideration tocoherent measures, bymaking the fol-
lowingadditional assumption aboutthedriverg.
Assumption3.1. Thedrivergsatisfiesforalmostallt [0,T]thefollowingconditions:
∈
(i) gisdeterministic andindependent ofy,thatis,g :[0,T] Rd R,andg(,0) 0;
× → · ≡
(ii) g(t, )isconvex forallt [0,T];
· ∈
(iii) g(t, )ispositively homogeneous forallt [0,T].
· ∈
g
Undertheseconditions, onecanderivefurtherpropertiesoftheevaluationsρ []fort [0,T],
t,T · ∈
inadditiontothegeneralpropertiesofF-consistentnonlinearexpectationsstatedinProposition2.2.
Theorem3.2. Supposegsatisfies Assumption3.1. Thenthedynamicriskmeasure ρ has
t,r 0 t r T
{ } ≤≤ ≤
thefollowingproperties:
4
(i) TranslationProperty: forallξ (Ω, ,P)andη (Ω, ,P),
2 r 2 t
∈ L F ∈ L F
ρ [ξ+η] = ρ [ξ]+η, a.s.;
t,r t,r
(ii) Convexity: forallξ,ξ (Ω, ,P)andallλ L (Ω, ,P)suchthat0 λ 1,
′ 2 r ∞ t
∈ L F ∈ F ≤ ≤
ρ [λξ+(1 λ)ξ ] λρ [ξ]+(1 λ)ρ [ξ ], a.s..
t,r ′ t,r t,r ′
− ≤ −
(iii) PositiveHomogeneity: forall ξ (Ω, ,P)and allβ L (Ω, ,P)such thatβ 0, we
2 r ∞ t
∈ L F ∈ F ≥
have
ρ [βξ] = βρ [ξ], a.s..
t,r t,r
Itfollowsthat under Assumption 3.1,the operators ρ constitute afamilyofcoherent
t,r 0 t r T
{ } ≤≤ ≤
conditional measuresofrisks.
Particularly important forusisthedualrepresentation oftheriskmeasure ρ ,whichis
t,T t [0,T]
basedonthedualrepresentation ofthedriverg: (cid:8) (cid:9)∈
g(t,z) = maxνz,
ν At
∈
whereA = ∂ g(t,0)isabounded,convex,andclosedsetinRn. Thefollowingstatementspecializes
t z
theresultsofBarrieuandElKaroui[3]toourcase.
Theorem3.3. Suppose Assumption3.1issatisfied. Then
ρg Φ(Xt,x) = sup E Γ Φ(Xt,x) , (8)
t,T T t,T T
(cid:2) (cid:3) µ∈At,T (cid:2) (cid:3)
where isthespaceofA -valuedadaptedprocesseson[t,T],andtheprocess Γ satisfies
At,T s t,s s [t,T]
thestochastic differential equation: (cid:8) (cid:9) ∈
dΓ = µ Γ dW , s [t,T], Γ = 1. (9)
t,s s t,s s t,t
∈
Moreover, asolutionµˆ oftheoptimalcontrolproblem (8)–(9)exists.
Thefollowinglemmaprovides ausefulestimate.
Lemma 3.4. A constant C exists, such that for all 0 t < s T and all Γ that satisfy (9), we
t,s
≤ ≤ { }
have
Γ 1 2 C(s t). (10)
t,s
k − k ≤ −
Proof. UsingItôisometry, weobtainthechainofrelations
s s s
Γ 1 2 = µ Γ 2dr µ 2 Γ 2dr µ 2 1+ Γ 1 2 dr. (11)
t,s r t,r r t,r r t,r
k − k Z k k ≤ Z k k k k ≤ Z k k k − k
t t t
(cid:0) (cid:1)
IfK isauniformupperboundonthenormofthesubgradientsofg(r,0)wededucethat Γ 1 2
t,r
ψ ,r [t,T],whereψsatisfiestheODE: dψr = K2(1+ψ ),withψ = 0. Consequently,k − k ≤
r ∈ dr r t
Γ 1 2 ψ = eK2(r t) 1. (12)
t,r r −
k − k ≤ −
Theconvexity oftheexponential function yieldsthepostulated bound. (cid:3)
5
Thedualrepresentationtheoremallowsustotransformtheriskevaluationproblemtoastochas-
ticcontrolproblem. Ourobjectivenowistoapproximatetheevaluationofriskonashortinterval∆
sothat it reduces the functional optimization problem tovector optimization. Toproceed, wehave
tofirstinvestigate thecorresponding maximumprinciple ofthecontrolproblem (8).
4 Stochastic Maximum Principle
In this section, we decipher the optimality conditions of the stochastic control problem (8)–(9).
Since only the process Γ is controlled, the analysis is rather standard. For completeness,
t,s s [t,T]
{ } ∈
werepeatsomeimportantstepshere.
Suppose µˆ isthe optimal control; then, forany µ and 0 α 1, wecanform aperturbed
∈ A ≤ ≤
controlfunction
µα = µˆ +α(µ µˆ).
−
It is still an element of , due to the convexity of the sets A . The processes Γˆ, Γ and Γα are the
s
A
stateprocesses underthecontrols µˆ,µ,andµ ,respectively.
α
Welinearize thestateequation(9)aboutΓˆ toget,for s [t,T],
∈
dηµ = µˆ ηµ+Γˆ (µ µˆ ) dW , ηµ = 0. (13)
s s s t,s s− s s t
(cid:2) (cid:3)
Itisevidentthatthisequation hasauniquestrongsolution. Denote
1
hα = Γα Γˆ ηµ, s [0,T].
s α t,s− t,s − s ∈
(cid:2) (cid:3)
Thefollowingresultjustifiestheusefulness ofthelinearized equation (13).
Lemma4.1.
lim sup hα 2 = 0. (14)
α 00 s Tk sk
→ ≤ ≤
Proof. Wefirstprovethat
lim sup Γα Γˆ 2 = 0. (15)
α 0t s T k t,s− t,sk
→ ≤ ≤
Wehave
d Γα Γˆ = µαΓα µˆ Γˆ dW = (µα µˆ )Γˆ +µα(Γα Γˆ ) dW . (16)
t,s − t,s s t,s− s t,s s s − s t,s s t,s − t,s s
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
ByItôisometry,
r
Γα Γˆ 2 = (µα µˆ )Γˆ +µα(Γα Γˆ ) 2 ds
k t,r − t,rk Z k s − s t,s s t,s − t,s k
t
r r
2 (µα µˆ )Γˆ 2 ds+2 µα(Γα Γˆ ) 2 ds
≤ Z k s − s t,sk Z k s t,s− t,s k
t t
r r
2 (µα µˆ )Γˆ 2 ds+K Γα Γˆ 2 ds,
≤ Z k s − s t,sk Z k t,s− t,sk
t t
6
where K is a constant. Since the first integral on the right hand side converges to 0, as α 0, the
→
Gronwallinequality yields(15).
Wecannowprove(14). Combining(16)and(13),weobtainthestochasticdifferentialequation
forhα:
1
dhα = (µˆ +α(µ µˆ ))Γα µˆ Γˆ µˆ ηµ Γˆ (µ µˆ ) dW
s (cid:26)α s s− s t,s − s t,s − s s − t,s s − s (cid:27) s
(cid:2) (cid:3)
1
= µˆ Γα Γˆ +(µ µˆ ) Γα Γˆ µˆ ηµ dW
(cid:26)α s t,s− t,s s− s t,s− t,s − s s(cid:27) s
(cid:2) (cid:3) (cid:2) (cid:3)
= µˆ hα+(µ µˆ ) Γα Γˆ dW .
(cid:26) s s s − s t,s− t,s (cid:27) s
(cid:2) (cid:3)
Sincetheprocesses µˆ and µ arebounded, Itôisometryyieldsagain
s s
{ } { }
r r
hα 2 K hα 2 ds+K Γα Γˆ 2 ds,
k rk ≤ Z k sk Z k t,s − t,sk
t t
where K isconstant. BytheGronwallinequality, using(15),wegetthedesiredresult. (cid:3)
Theconvergence resultabovedirectly leadstothefollowingvariational inequality.
Lemma4.2. Foranyµ wehave
∈ A
E ξ ηµ 0. (17)
T T ≤
(cid:2) (cid:3)
Proof. Sinceµˆ istheoptimalcontrol,
E ξ Γα Γˆ 0.
T t,T − t,T ≤
(cid:2) (cid:0) (cid:1)(cid:3)
Lemma4.1leadsto
1
limE ξ Γα Γˆ = E ξ ηµ 0,
α 0 Tα t,T − t,T T T ≤
→ h (cid:0) (cid:1)i (cid:2) (cid:3)
asrequired. (cid:3)
We now express the expected value in (17) as an integral, to obtain a pointwise variational
inequality (the maximum principle). To this end, we introduce the following backward stochastic
differential equation(theadjointequation):
dp = k µˆ ds+k dW , p = ξ , s [t,T], (18)
s s s s s T T
− ∈
withξ = Φ Xt,x . Byconstruction, E ξ ηµ = E pˆ ηµ . ApplyingtheItôformulatotheproduct
T T T T T T
µ
process psηs(cid:0),we(cid:1)obtain (cid:2) (cid:3) (cid:2) (cid:3)
d(p ηµ) = k ηµ+ pˆ µˆ ηµ+Γˆ (µ µˆ ) dW +k Γˆ (µ µˆ ) ds.
s s s s s s s t,s s s s s t,s s s
− −
(cid:16) (cid:2) (cid:3)(cid:17)
Iffollowsthat
T
E ξ ηµ = E k Γˆ (µ µˆ )ds . (19)
T T (cid:20) Zt s t,s s− s (cid:21)
(cid:2) (cid:3)
7
We can summarize our derivations in the following version of the maximum principle. We define
theHamiltonian H : R Rn Rn R:
× × →
H(γ,ν,κ) = kγν.
Theorem4.3. Foralmostall s [t,T],withprobability 1,
∈
H(Γˆ ,µˆ ,k )= maxH(Γˆ ,ν,k ).
t,s s s t,s s
ν A
∈
Proof. Foranyµ ,wedefinetheset
∈ A
= (ω,s) Ω [t,T] : k Γˆ (µ µˆ )> 0 .
s t,s s s
G { ∈ × − }
Weconstruct anewcontrolµ :
∗
∈ A
µ , (ω,s) ,
µ = s ∈ G
∗s 
µˆ , otherwise.
 s
Themeasurability andadaptedness ofµ canbeeasilyverified. Itfollowsfrom(17)and(19)that
∗
T
E k Γˆ (µ µˆ )ds 0.
(cid:20) Zt s t,s ∗s − s (cid:21) ≤
Bytheconstruction ofµ ,
∗
k Γˆ (µ µˆ )ds P(dω) 0.
" s t,s ∗s − s ≤
G
Sincetheintegrandispositiveon ,theproductmeasureof mustbezero. (cid:3)
G G
5 Regularity of the Integrand in the Adjoint Equation
Wealsomakeastronger assumption aboutthedrift anddiffusion termsoftheforwardsystem, and
abouttheterminalcostfunction.
Assumption5.1. Thefunctions b,σ,Φ 2([0,T] Rn),and
∈ Cb ×
1
σ(s,x) σ(t,x) C s t 2
| − | ≤ | − |
forall s,t [0,T]andall x Rn.
∈ ∈
Consider the forward–backward system (7) and (18). The key to our further estimates is the
followingregularity resultabouttheintegrand k intheadjointequation (18).
t
{ }
Lemma5.2. AconstantC exists, suchthatforall0 t < s T,andall x Rn,
≤ ≤ ∈
1
ks kt C s t 2. (20)
k − k ≤ | − |
8
Proof. Thequasilinearparabolicpartialdifferentialequationcorrespondingtotheforward–backward
system(7)–(18)hasthefollowingform(see,e.g. [26,sec. 8.2]),
1
u(t,x)+u (t,x)b(t,x)+ tr u (t,x)σ(t,x)σT(t,x) +u (t,x)σ(t,x)µˆ = 0, (21)
t x xx x t
2
(cid:0) (cid:1)
with the boundary condition u(T,x) Φ(x). Due to the linearity of the driver of (18), the terms
≡
withu canbecollapsed. Thentheequation (21)istheFeynman-Kacequation for
x
u(s,X˜t,x) = E Φ X˜t,x , s [t,T], (22)
s T Fs ∈
(cid:2) (cid:0) (cid:1)(cid:12)(cid:12) (cid:3)
where (cid:12)
dX˜t,x = b(s,X˜t,x)+σ(s,X˜t,x)µˆ ds+σ(s,X˜t,x)dW , s [t,T], X˜t,x = x.
s s s s s s ∈ s
(cid:2) (cid:3)
Ma and Yong [26] consider it on page 195 in formula (1.12). Under Assumption 5.1, the equation
(21)hasaclassical solution u(, ),andthentheprocess
· ·
k = u (s,X˜t,x)σ(s,X˜t,x), s [t,T], (23)
s x s s ∈
isthesolution oftheadjoint equation (18). By[26,Prop. 8.1.1], aprocess H 2,n n[t,T]exists,
×
suchthattheprocessG = u (s,X˜t,x)satisfiesthefollowingn-dimensional BS∈DEH:
s x s
T T
G = Φ (X˜t,x)+ b (s,X˜t,x)+σ (s,X˜t,x)µˆ G +σ (s,X˜t,x)H ds H dW . (24)
t x T Z x s x s s s x s s −Z s s
t (cid:16)(cid:2) (cid:3) (cid:17) t
Weobtainthefollowingestimate:
k k = u (s,X˜t,x)σ(s,X˜t,x) u (t,x)σ(t,x)
k s − tk k x s s − x k
u (s,X˜t,x)σ(s,X˜t,x) u (s,X˜t,x)σ(t,x)+u (s,X˜t,x)σ(t,x) u (t,x)σ(t,x) (25)
≤ k x s s − x s x s − x k
u (s,X˜t,x) σ(s,X˜t,x) σ(t,x) + σ(t,x) G G .
≤ k x s kk s − k k kk s − tk
Thefirsttermontherighthandsideof (25)canbebounded withthehelpofAssumption5.1:
σ(s,X˜t,x) σ(t,x) σ(s,X˜t,x) σ(s,x) + σ(s,x) σ(t,x)
k s − k ≤ k s − k k − k
C1 s t 21 +C2 X˜s x C3 s t 21,
≤ | − | k − k≤ | − |
whereC ,C ,andC aresomeuniversalconstants. Itfollowsfrom(24)that
1 2 3
r r
G G = b (s,X˜ )+σ (s,X˜ )µˆ G +σ (s,X˜ )H ds+ H dW .
r t x s x s s s x s s s s
− −Z Z
t (cid:16)(cid:2) (cid:3) (cid:17) t
Therefore, thesecondtermontherighthandsideof (25)canbeboundedas G G 2 C s t.
s t 4
k − k ≤ | − |
Integrating theseestimatesinto(25),weobtain(20)withauniversal constantC. (cid:3)
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6 Error Estimates for Constant Controls on Small Intervals
To reduce an infinite dimensional control problem to a finite dimensional vector optimization, we
partition the interval [0,T] into N short pieces of length ∆ = T/N, and develop a scheme for
evaluating the risk measure (3) by using constant dual controls on each piece. We denote t = i∆,
i
fori= 0,1,...,N.
For simplicity, in addition to Assumption 3.1, we assume that the driver g does not depend on
time,andthusallsets A = ∂g(0)arethesame. Wedenotethemwiththesymbol A;asweshallsee
t
lateronthisisnotamajorrestriction.
If the system’s state at time t is x, then the value of the risk measure (6) is then the optimal
i
valueofproblem (8). Bydynamicprogramming,
v(ti,x) = ρtxi,ti+1 v ti+1,Xttii+,x1 .
(cid:2) (cid:0) (cid:1)(cid:3)
The risk measure ρtxi,ti+1[·] is defined by problem (8), with terminal time ti+1 and the function Φ(·)
replaced by v(ti+1, ·). Equivalently, it is equal to Yttii,x, in the corresponding forward–backward
systemontheinterval[ti,ti+1]:
dXti,x = b(s,Xti,x) ds+σ(s,Xti,x)dW , Xti,x = x, (26)
s s s s ti
−dYsti,x = g(Zsti,x)ds−Zsti,x dWs, Yttii+,x1 = v ti+1,Xttii+,x1 . (27)
(cid:0) (cid:1)
Under Assumption 5.1, the function v(, ) is the classical solution of the associated Hamilton–
· ·
Jacobi–Bellman equation:
1
v(t,x)+v (t,x)b(t,x)+ tr v (t,x)σ(t,x)σT(t,x) +g v (t,x)σ(t,x) = 0, (28)
t x xx x
2
(cid:0) (cid:1) (cid:0) (cid:1)
withtheterminalcondition v(T,x) = Φ(x).
Supposeweuseaconstantcontrol intheinterval[ti,ti+1]:
µs := µˆti = argmax ktiν, ∀s ∈ [ti,ti+1], (29)
ν A
∈
where(p,k)solvetheadjoint equationcorresponding to(18):
dps = −ksµˆsds+ksdWs, s∈ [ti,ti+1], pti+1 = v ti+1,Xttii+,x1 . (30)
(cid:0) (cid:1)
Westill use Γˆ to denote the state evolution under the optimal control, while Γ is the process under
control µ defined in (29). It is well-known that the value function v(, ) of the system (26)–(27) is
· ·
in 2([0,T] Rn); see, for example, [42, Thm. 2.4.1]. Therefore, the bounds developed insection
Cb ×
5remainvalidfortheprocesses (p,k)in(30).
Our objective is to show that a constant C exists, independent of x, N, and i, such that the
approximation errorontheithintervalcanbebounded asfollows:
0 ≤ E v ti+1,Xttii+,x1 Γˆti,ti+1 −Γti,ti+1 ≤C∆23. (31)
(cid:2) (cid:0) (cid:1)(cid:0) (cid:1)(cid:3)
The fact that wedo not know k will not be essential; later, we shall generate even better constant
ti
controls bydiscrete-time dynamicprogramming.
Wecannowderivesomeusefulestimates fortheconstant controlfunction (29).
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