Upload
makosantoniou
View
58
Download
1
Embed Size (px)
Citation preview
Portfolio Optimization and Stochastic Control.
Closed-Form Solutions to the HJB Equation through Lie Symmetries.
Solomon M. Antoniou
SKEMSYS
Scientific Knowledge Engineering
and Management Systems
37 Κoliatsou Street, Corinthos 20100, Greece [email protected]
Αφιερωμένο στη Γκέλλυ
Dedicated to Gelly
Abstract
We introduce a solution procedure for the Hamilton-Jacobi-Bellman (HJB)
equation associated to a portfolio optimisation problem. The HJB equation is
derived under the principle of Dynamic Programming. The solution method relies
on determining the underlying Lie point symmetries of the equation. On finding
the symmetries, we consider a linear combination of the infinitesimal generators,
and the two invariants that they determine. Using the transformations based on the
two invariants, we convert the HJB equation to an ordinary non-linear differential
equation, which can in turn be converted to an Abel differential equation. Abel’s
differential equation admits a closed form solution. What follow are twelve
families of solutions.
Keywords: Stochastic Control, Hamilton-Jacobi-Bellman equation, Portfolio
Optimization, Nonlinear PDEs, Lie Symmetries, Closed-Form Solutions.
2
1. Introduction
Stochastic Control (Bertsekas [6], [7], Evans [19], Fleming and Soner [20],
Ingersoll [28], Øksendal [42], Pham [46], Soner [49], Touzi [52]) is an
optimisation procedure used in many areas of financial applications, such as
Portfolio Optimization (Benth and Karlsen [5], Korn and Korn [31], Korn
[32], Kraft [33])
Option Pricing (Baron and Jensen [3])
Consumption-Investment Strategies (Davis, Panas and Zariphopoulou [15],
Elliott and Kopp [18], Karatzas, Lehoczky, Sethi and Shreve [29],
Karatzas, Lehoczky and Shreve [30] and Zariphopoulou [53])
Stochastic Interest Rates (Lewis [36])
Bond Pricing (Kraft [33]) and Bond Portfolio Optimization (Puhle [47])
Asset Allocation (Munk [41])
Risk (Bielecki and Pliska [8], Dmitrasinovic-Vivodic, Lari-Lavassani, Li
and Ware [16])
DSGE models (Papanicolaou [45])
Pairs Trading (Mudchanatongsuk, Primbs and Wong [40])
Pension Funds (Boulier, Trussant and Florens [13], Gerrard, Haberman and
Vigna [21])
Insurance (Hipp, [24])
Debt Crises (Stein, [50])
Various applications (Bucci et al. [14])
The basic equation used in the optimization process is the Hamilton-Jacobi-
Bellman (HJB) equation, derived under the principle of Dynamic Programming
(Bellman [4]). Despite successful predictions of the HJB equation for such
models, there is no available general solution procedure. The solutions obtained so
far are based on separation of variables. It is the purpose of this paper to introduce
3
a solution procedure (and to find some general solutions) based on Lie
Symmetries.
The Lie symmetry approach in solving differential equations was introduced by S.
Lie (Lie [37] and Lie [38]) and is best described in references Anderson and
Ibragimov [1], Bluman and Kumei [9], Dresner [17], Hereman [23], Ibragimov
[25], Ibragimov [26], Olver [43], Ovsiannikov [44] and Stephani [51]. This
technique was used for the first time in Finance by Ibragimov and Gazizov [27].
The same technique was also used by the author in solving the Bensoussan-
Crouhy-Galai equation, which models the stochastic equity volatility (Antoniou
[2]). There is however a growing list of papers in applying Lie symmetry methods
to Finance. For a (non-complete) set of references, see Bordag [10], Bordag and
Chmakova [11], Bordag and Frey [12], Goard [22], Leach, O’Hara and Sinkala
[35] and Silberberg [48].
The paper is organized as follows: In section 2 we consider a simple model, which
leads to the Hamilton-Jacobi-Bellman equation. In section 3 the Lie symmetry
analysis to this equation is performed. In other words we find the infinitesimal
generators of the Lie group. In section 4 we find general solutions to the equation,
by first considering a linear combination of the infinitesimal generators and then
by determining the invariants of the equation. The invariants allow us to convert
the equation into a nonlinear, second order ordinary differential equation. Using
another transformation, we convert this nonlinear equation to an Abel equation
possessing a closed-form solution. In section 5 we present our conclusions and the
twelve families of our solution.
2. A Portfolio Selection Model and the HJB Equation
In this section we consider a selection portfolio model, optimized under the
principle of Dynamic Programming, leading to an HJB equation.
The model we consider is a Brownian model, in which the money market account
and stocks are modeled by
4
dt)t(r)t(S)t(Sd 00 , 1)0(S0 (2.1)
)]t(dWσdt)t(μ[)t(S)t(Sd , 0)0(S (2.2)
The coefficients r, μ and σ are progressively measurable processes being
uniformly bounded. Let also
)t(φ : be the number of the stocks invested at time t.
)t(S)t(φ 00 : be the money invested in the money market account at time t
)t(S)t(φ)t(S)t(φ)t(X 00φ : be the wealth of the investor
)t(X
)t(S)t(φ)t(π : be the proportion invested in stock
)t(π1 : be the proportion invested in the money market account.
The following proposition holds true (see for example Kraft [33], [34] or Øksendal
[42]).
Proposition. The wealth dynamics of a self-financing trading strategy is described
by the so-called wealth equation
)]t(dWσπdt))rμ(πr[()t(X)t(dX (2.3)
0x)0(X ( 0x0 )
We come now to the portfolio optimization one investor has with initial wealth
0x)0(X 0 and would like to have optimal final wealth )T(X* under an
optimal final strategy *φ . For a given initial wealth 0x0 , the investor
maximizes the following utility functional
))]Τ(X(U[Esup π
Aπ
(2.4)
where
π:π{A admissible and ]))Τ(X(U[E π}
and )t(Xπ is given by (2.3):
)t(dWσ)t(πdt)])t(r)t(μ()t(π)t(r[)t(X)t(dX ππ
5
0π x)0(X
The function )x(U , defined for positive real numbers, is called the utility
function, which is strictly concave, strictly increasing and continuously
differentiable, and satisfies the conditions
)x(Ulim)0(U0x
, 0)x(Ulim)(Ux
Let us define the value function
))]Τ(X(U[Esup)x,t(G πx,t
Aπ
where
]x)t(X|))Τ(X(U[E))]Τ(X(U[E ππx,t
)x(U)x,t(G
Suppose now that there is an optimal strategy *π . The corresponding HJB
equation, which maximizes the final wealth, derived under the principle of
Dynamic Programming, is given by
0)x,t(Gx2
1x])r(r[)x,t(Gsup)x,t(G xx
222xt
(2.5)
for any admissible strategy π and terminal condition )x(U)x,T(G .
We consider the case where r, μ and σ are all constants.
Pointwise maximization over π yields the first order condition
0)t(π)x,t(Gxσx)rμ()x,t(G xx22
x (2.6)
which implies that the optimal strategy is given by
)x,t(Gx
)x,t(G
σ
rμ*π
xx
x2
(2.7)
Since the derivative of (2.6) with respect to π is )x,t(Gxσ xx22
, the candidate
*π is the global maximum if 0Gxx .
Substituting *π into the HJB equation (2.5), yields a PDE for )x,t(G :
6
0)x,t(G
)x,t(Gk
2
1)x,t(Gxr)x,t(G
xx
2x2
xt (2.8)
where σ
rμk
.
The traditional way of solving equation (2.8) is by separation of variables. In fact,
considering that (Merton [39], Øksendal [42])
γ1γ )t(fxγ
1)x,t(G (2.9)
equation (2.8) yields the first order differential equation
)t(fK)t(f (2.10)
where
2k
)1(2
1r
1K (2.11)
The above differential equation (2.10) has the solution )tT(Ke)t(f , and
satisfies the condition 1)T(f . Therefore solution (2.9) is compatible with the
condition γxγ
1)x(U)x,T(G . From (2.9) and (2.7) we determine the value
function and the optimal strategy respectively. For the type of solution we have
obtained, we can check that 0Gxx for 1γ .
In section 4 we provide our own method of solution, based on Lie Symmetry
Analysis.
3. Lie Symmetry Analysis of the HJB Equation
We now consider the HJB equation (2.8) written in the equivalent form
0uk2
1uuxruu 2
x2
xxxxxt (3.1)
where we have changed the notation of the function from )x,t(G to the more
traditional notation )t,x(u .
7
We are going to perform a Lie symmetry analysis of this equation. By this we
mean that we shall determine the infinitesimal generators of the Lie symmetry
group. These would then be used to determine the invariants, needed to convert
the partial differential equation into an ordinary differential equation.
We introduce the following concepts and notation (Olver [43]). Consider the base
space M, which is the Cartesian product UX of a 2-dimensional space X of
independent variables X)t,x( by a 1-dimensional space U of dependent
variables Uu . Let )1(U be the space of the first derivatives of the functions
)t,x(u with respect to x and t: )1(
tx U)u,u( . Let also )2(U be the space of the
second derivatives of the functions )t,x(u with respect to x and t:
)2(ttxtxx U)u,u,u( . Since the differential equation (3.1) is second order, we
introduce a second order jet bundle )2(M by considering the Cartesian product
)2()1()2( UUMM
The coordinates of the space )2(M are labeled by
)2(
ttxtxxtx M)u,u,u,u,u,u,t,x(z
In the space )2(M , equation (3.1) can be expressed as 0)u,t,x(Δ )2( where
2x
2xxxxxt
)2( uk2
1uuxruu)u,t,x(Δ (3.2)
Let ΔL be the solution manifold of (3.1):
)2()2(
Δ M}0Δ|Mz{L
A symmetry group ΔG of equation 0Δ is defined by
}LL:g|)M(Diffg{G ΔΔ)2(
Δ
We want to determine a subgroup of )M(Diff )2( , compatible with the structure of
ΔL . We shall first find the symmetry Lie algebra )Μ(Diff)Μ(Diff )2()2(Δ and
then use the main Lie theorem to determine ΔG . Let us denote by V
8
( )Μ(DiffV ) an element of a vector field on M (the generator of Lie
symmetries), defined by
u
)u,t,x(φt
)u,t,x(τx
)u,t,x(ξV
(3.3)
where )u,t,x(τ),u,t,x(ξ and )u,t,x(φ are smooth functions of their arguments.
The infinitesimal generators of ΔGg will have the structure of (3.3) and will
form an algebra )Μ(DiffΔ . The algebra )Μ(Diff )2(Δ will be spanned by the
vectors Vpr )2( , the second prolongation of V, defined by
tt
tt
xt
xt
xx
xx
t
t
x
x)2(
uφ
uφ
uφ
uφ
uφVVpr
The symmetries are determined by the equation (Olver [42], Theorem 2.31)
0)]u,t,x(Δ[Vpr )2()2( (3.4)
as long as
0)u,t,x(Δ )2( (3.5)
We implement next the equation 0)]u,t,x(Δ[Vpr )2()2( . We have that
0)]u,t,x(Δ[Vpr )2()2(
0]uk2
1uuxruu[Vpr 2
x2
xxxxxt)2(
or
xxt
x2
xxx
xxx uφ)ukuxr(φ)uur(ξ
0)uxru(φ xtxx (3.6)
The coefficients xφ , tφ and xxφ are calculated to be (Olver [43], Example
2.38)
txutx2xuxxux
x uuτuτuξu)ξφ(φφ
2tutxuttuxtt
t uτuuξu)τφ(uξφφ
9
2xxuuutxxxxxxuxx
xx u)ξ2φ(uτu)ξφ2(φφ
xxxut2xuu
3xuutxxu u)ξ2φ(uuτuξuuτ2
xtxuxxtuxxxuxtx uuτ2uuτuuξ3uτ2
The functions ξ,τ and φ have the following dependence
)t(ττ , )t,x(ξξ και )u,t,x(φφ
Equation (3.6) thus becomes
}u)ξφ(φ{)ukuxr()uur(ξ xxuxx2
xxxxx
txxtuxxxtt uu)τφ(u}uξφ{
2xuuxxxxuxx uφu)ξφ2(φ{
txxxu u}u)ξ2φ(
2xuuxxxxuxxx uφu)ξφ2(φ{uxr
0}u)ξ2φ( xxxu (3.7)
We now have to take into account the condition 0)u,t,x(Δ )2( . We therefore
substitute tu by xx
2x2
xu
uk
2
1uxr in the previous equation (3.7):
}u)ξφ(φ{)ukuxr()uur(ξ xxuxx2
xxxxx
xx
2x2
xxxtuxxxttu
uk
2
1uxru)(u)u(
2xuuxxxxuxx uφu)ξφ2(φ{
xx
2x2
xxxxuu
uk
2
1uxr}u)2(
2xuuxxxxuxxx uφu)ξφ2(φ{uxr
0}u)ξ2φ( xxxu (3.8)
10
We equate all the coefficients of the function u and their derivatives in the
previous equation (after multiplying by xxu ) to zero:
Coefficient of 4
x )u( :
0φk2
1uu
2 (3.9)
Coefficient of 3x )u( :
0)ξφ2(k2
1xxxu
2 (3.10)
Coefficient of 2x )u( :
0φk2
1xx
2 (3.11)
Coefficient of xxxuu :
0φk x2 (3.12)
Coefficient of xx2
x u)u( :
0)t(τk2
1 2 (3.13)
Coefficient of 2xx )u( :
0φφxr tx (3.14)
Coefficient of 2
xxx )u(u :
0))t((xrr xt (3.15)
Equations (3.9)-(3.15) are the determining equations of the Lie symmetries of
equation (3.1). We are now going to solve the system of equations (3.9)-(3.15).
From equation (3.9) we get that 0φuu , which means that φ is a linear function
with respect to u:
)t,x(βu)t,x(α)u,t,x(φ (3.16)
From equation (3.12) we get that 0φx . Therefore, using (3.16), we have
11
0βuα xx
from which we get 0αx and 0βx . Therefore the functions α and β
introduced in (3.16) are independent of x. On the other hand, since 0φx , we
get from (3.14) that 0φ t . Therefore 0βuα tt , from which we have that
0α t and 0β t , which means that functions α and β are constants. We then
obtain that
21 aua)u,t,x(φ (3.17)
Equation (3.11) then becomes an identity.
We now get from (3.13) that 0)t(τ . Therefore )t(τ is a constant:
3a)t(τ (3.18)
We get from (3.10), because of (3.17), that 0ξxx . Therefore )t,x(ξ is a linear
function with respect to x:
)t(gx)t(f)t,x(ξ (3.19)
where )t(f and )t(g are functions to be determined.
Using the previous expression, equation (3.15) becomes
0))t(g)t(gr(x)t(f
from which we get , since this equation should be true for any x, that
0)t(f and 0)t(g)t(gr
We thus have
4a)t(f (3.20)
and
tr5 ea)t(g (3.21)
Therefore the function )t,x(ξ takes the form
tr
54 eaxa)t,x(ξ (3.22)
Relations (3.22), (3.18) and (3.17) allow us to write down the generator of the
symmetries:
12
u
)aua(t
ax
)eaxa(V 213tr
54
(3.23)
Therefore we have the following
Theorem 1. The Lie algebra of the infinitesimal transformations of the HJB
equation (3.1) is spanned by the five vector fields
u
uX1
(3.24)
u
X2
(3.25)
t
X3
(3.26)
x
xX4
(3.27)
x
eX tr5
(3.28)
4. General Solutions to the HJB Equation
In order to find a solution to the HJB equation (3.1), we have to determine the
invariants of the equation. For this purpose we consider a linear combination of
the above found generators of the symmetries.
We consider the combination
21543 ΧρΧνΧμXλX
In other words we consider the generator
u
)ρuν(x
)eμxλ(t
tr
(4.1)
where the coefficients λ, μ, ν and ρ will be determined on the basis of finding
closed form solutions to the equation. The parameter appearing in (4.1) not to
be confused with appearing in the definition of k after equation (2.8).
Based on (4.1), we consider the system
13
ρuν
du
eμxλ
dx
1
dt
tr
(4.2)
In order to solve the first equation
treμxλ
dx
1
dt
(4.3)
of the above system, we have to consider two cases: rλ and rλ . Details are
given in the Appendix.
4.1. Case 1. Solution of the HJB equation in the rλ case.
If rλ , equation (4.3) has the general solution
t)λr(tλ1 e
λr
μxeC
(4.4)
Therefore one invariant of the HJB equation (3.1) is given by
t)λr(tλ eλr
μxey
(4.5)
The equation
ρuν
du
1
dt
has the general solution
)ρuνln(tνC2 .
Therefore another invariant is given by )y(υυ , where
)ρυe(ν
1u tν (4.6)
Using (4.5) and (4.6), we find that the partial derivatives of the function u
transform as
tν
yt)λr(
t eυeλr
μyλ
ν
1υu
(4.7)
yt)λr(
x υeν
1u (4.8)
14
yyt)λ2r(
xx υeν
1u (4.9)
Using (4.7)-(4.9), the original HJB equation (3.1) is transformed into
0)υ(k2
1υυy)λr(υυν 2
y2
yyyyy (4.10)
which is a nonlinear, second order ordinary differential equation.
Under the transformation
υ
υ)y(w
y (4.11)
equation (4.10) takes on the form
0wy)λr(wk2
1νw]wy)λr(ν[ 322
y
(4.12)
which is an Abel differential equation.
Introducing the notation
2k2
1νγ and λrδ ( 0 ) (4.13)
equation (4.12) can be written as
wyδν
wyδwγw
32
y
(4.14)
which, in turn, under the substitution
)y(Z
1)y(w (4.15)
takes on the form
y)y(Z
y)y(ZZy
(4.16)
Introducing the function )y(V by
)y(Vy)y(Z (4.17)
we transform equation (4.16) into the equation
15
δVν
δVγVyV y
which is equivalent - after separating variables - to the equation
y
1V
V)(V
Vy2
(4.18)
4.1.I. Let 0 , i.e. 2k2
1 ( 0 ). Equation (4.18) can be written as
y
1V
VV
Vy2
(4.19)
Denoting by 4D 2 the discriminant of the trinomial VV2 , we
have the following:
4.1.Ia For 0D , i.e. 4rr or 4rr , equation (4.19) admits a
solution given implicitly by
Ayln|V|ln)(
|V|ln)(
(4.20)
where and are the two real roots of the trinomial, given by
2
D and
2
D
respectively and A is an arbitrary constant.
4.1.Ib For 0D , i.e. 4rr , equation (4.19) admits a solution given
implicitly by
Aylnp
Varctan
p)p)Vln((
2
1 22
(4.21)
where and p are two parameters defined by
2 and
2
Dp
respectively and A is an arbitrary constant.
16
4.1.Ic For 0D , i.e. 4 , equation (4.19) admits a solution given implicitly
by
Ayln|2V|ln2V
12
(4.22)
where A is an arbitrary constant.
4.1.II. Let 0 , i.e. 2k2
1 . We shall distinguish two cases: and .
4.1.IIa If , equation (4.18) becomes
y
1V
V
Vy2
In this case we have to consider two cases: and . In the former case
we may set either 2 or 2 , 0 .
We thus have the following:
4.1.IIa1 For 2 , we have to solve the equation
y
1V
V
Vy22
2
which gives upon integration
AylnV
Vln
2
1)Vln(
2
1 22
(4.23)
where A is an arbitrary constant.
4.1.IIa2. For 2 , we have to solve the equation
y
1V
V
Vy22
2
which gives upon integration
AylnV
arctan)Vln(2
1 22
(4.24)
where A is an arbitrary constant.
4.1.IIa3. For , we obtain the equation
17
y
1V
1V
1Vy2
or
y
1V
1V
1y
and by integration,
Ayln|1V|ln (4.25)
where A is an arbitrary constant.
4.1.IIb. If , denoting by 4)(D 2 the discriminant of the trinomial
V)(V2 , we have the following:
4.1.IIb1. For 04)(D 2 , equation (4.18) admits a solution given
implicitly by
Ayln|V|ln)(
|V|ln)(
(4.26)
where and are the two real roots of the trinomial, given by
2
D and
2
D
respectively and A is an arbitrary constant.
4.1.IIb2. For 04)(D 2 , equation (4.18) admits a solution given
implicitly by
Aylnp
arctanp
)p)Vln((2
1 22
(4.27)
where and p are two parameters defined by
2 and
2
Dp
respectively and A is an arbitrary constant.
4.1.IIb3. For 04)(D 2 , equation (4.18) admits a solution given
implicitly by
Ayln)Vln(V
1
(4.28)
18
where is the double root of the trinomial given by
2
and A is an arbitrary constant.
The function )y(υυ is then determined integrating (compare (4.11), (4.15) and
(4.17))
)y(Vy
1y
(4.29)
where )y(V is implicitly provided by any of the (4.20)-(4.28) equations.
Therefore we find that
y
y0)(V
dexpB)y( (4.30)
where B is a constant.
4.2. Case 2. Solution of the HJB equation in the r case.
In this case the system (4.2) is given by
u
du
exr
dx
1
dttr
(4.31)
The previous system has the general solutions:
tμexC tr1
(4.32)
and
)ρuνln(tνC2 (4.33)
Therefore in this case we have the two invariants
tμexy tr (4.34)
and
)ρυe(ν
1u tν (4.35)
19
where )y(υυ .
Using (4.34) and (4.35), we find that the partial derivatives of the function u
transform as
tν
yt eυ]μ)tμy(r[ν
1υu
(4.36)
yt)rν(
x υeν
1u (4.37)
yyt)r2ν(
xx υeν
1u (4.38)
Substituting (4.36)-(4.38) into the original equation (3.1), we find that the function
)y(υυ satisfies the equation
0)υ(k2
1υυμυυν 2
y2
yyyyy (4.39)
which is a nonlinear, second order ordinary differential equation.
Under the transformation
υ
υ)y(w
y (4.40)
equation (4.39) takes the form
32y www)w( (4.41)
which is an Abel differential equation. This equation may also be treated as an
equation with separable variables. However it will here be considered as an Abel
differential equation, since in that case we obtain simpler expressions as solutions.
In solving (4.38), we shall consider two cases: 0μ and 0μ .
4.2.I. If 0μ , we find that the general solution to the equation (4.41) is given by
Ayγ
ν)y(w
(4.42)
where A is an arbitrary constant. Integrating further υ
υ)y(w
y , we find that
20
γ
ν
)Ayγ(B)y(υ (4.43)
where B is an arbitrary constant.
4.2.II. If 0μ , equation (4.37) takes the form
wμν
wμwγw
32
y
(4.44)
4.2.IIa. If 0γ , under the substitution
)y(Y)y(w (4.45)
equation (4.44) takes the form
)νγ(μY)γν(
YγΥ
2
y
( νγ )
and by integration we obtain
AyγYln)νγ(μY)γν( 2 (4.46)
where A is an arbitrary constant.
Therefore )y(υ is the solution to the equation
)y(
y (4.47)
where )y(Υ is given implicitly by (4.46). Therefore we find that
y
y0)(
dexpB)y( (4.48)
where B is an arbitrary constant.
4.2.IIb. For 0γ (i.e. 2k
2
1ν ), equation (4.41) becomes
wμν
wμw
3
y
(4.49)
Under the substitution
21
)y(X
1w
the above equation (4.49) takes the form
XXy
which has the general solution
Ay222 (4.50)
where A is a constant.
Therefore )y(υ is the solution to the equation
)y(X
1
υ
υy (4.51)
where )y(X is given implicitly by (4.50). Hence, we find that
y
y0
)ξ(X
ξdexpB)y(υ (4.52)
where B is a constant.
5. Conclusions and Discussion.
Gathering the results of the previous section, we arrive at the following
Theorem 2. The HJB equation (3.1) admits the following twelve families of
general solutions:
Solution I.
C)(V
dexpeB)t,x(u
y
y
t
0
(5.1)
t)λr(tλ e
λr
μxey
( 0μ ) , r (5.2)
where )y(V is a function defined implicitly by
Ayln|)y(V|ln)(
|)y(V|ln)(
(5.3)
22
The parameters and are defined by
2
D and
2
D (5.4)
respectively, where λrδ ( 0 ) and 4D 2. The parameter is such
that 2k2
1 ( 0 ) and the parameter satisfies either one of the inequalities
4rr or 4rr .
Solution II
C)(V
dexpeB)t,x(u
y
y
t
0
(5.5)
t)λr(tλ eλr
μxey
( 0μ ) , r (5.6)
where )y(V is a function defined implicitly by
Aylnp
)y(Varctan
p)p))y(Vln((
2
1 22
(5.7)
The parameters and p are defined by
2 and
2
Dp (5.8)
respectively, where λrδ ( 0 ) and 4D 2.
The parameter is such that 2k2
1 ( 0 ) and the parameter takes values
within the range 4rr .
Solution III.
C)ξ(Vξ
ξdexpeB)t,x(u
y
y
tν
0
(5.9)
t)λr(tλ e
λr
μxey
( 0μ ) , r (5.10)
23
where )y(V is a function defined implicitly by
Ayln|2)y(V|ln2)y(V
12
(5.11)
The parameter is defined by λrδ ( 0 ) and satisfies the equation
4 . The parameter is such that 2k2
1 ( 0 ).
Solution IV
C)ξ(Vξ
ξdexpeB)t,x(u
y
y
tν
0
(5.12)
t)λr(tλ eλr
μxey
( 0μ ) , 0 , (5.13)
where )y(V is a function defined implicitly by
Ayln)y(V
)y(Vln
2
1))y(Vln(
2
1 22
(5.14)
where 2ωνδ , λrδ ( 0 ) and 2k2
1 .
Solution V
C)(V
dexpeB)t,x(u
y
y
t
0
(5.15)
t)λr(tλ eλr
μxey
0μ , 0 , (5.16)
where )y(V is a function defined implicitly by
Ayln)y(V
arctan))y(Vln(2
1 22
(5.17)
The parameter is defined through 2 where λrδ ( 0 ) and
2k2
1
24
Solution VI
C)(V
dexpeB)t,x(u
y
y
t
0
(5.18)
t)λr(tλ eλr
μxey
0μ , 0 , (5.19)
where )y(V is a function defined implicitly by
Ayln|1)y(V|ln (5.20)
The parameters satisfy , λrδ ( 0 ) and 2k2
1 .
Solution VII
C)(V
dexpeB)t,x(u
y
y
t
0
(5.21)
t)λr(tλ e
λr
μxey
( 0μ ) (5.22)
where )y(V is a function defined implicitly by
Ayln|)y(V|ln)(
|)y(V|ln)(
(5.23)
The parameters are given by
2
D,
2
D and 04)(D 2 ,
and 0 , where 2k
2
1νγ and λrδ ( 0 )
Solution VIII
C)(V
dexpeB)t,x(u
y
y
t
0
(5.24)
t)λr(tλ e
λr
μxey
( 0μ ) (5.25)
25
where )y(V is a function defined implicitly by
Aylnp
)y(Varctan
p)p))y(Vln((
2
1 22
(5.26)
The parameters are given by
2,
2
Dp , 04)(D 2 , and 0
where 2k2
1νγ and λrδ ( 0 )
Solution IX
C)(V
dexpeB)t,x(u
y
y
t
0
(5.27)
t)λr(tλ eλr
μxey
( 0μ ) (5.28)
where )y(V is a function defined implicitly by
Ayln))y(Vln()y(V
1
(5.29)
The parameters are given by
2, 04)(D 2 , and 0
where 2k
2
1νγ and λrδ ( 0 )
Solution X
C)Ayγ(eB)t,x(u γ
ν
tν (5.30)
where
trexy (5.31)
26
A, B, C are arbitrary constants and ν is a free parameter with 0ν , and
2k2
1νγ with 0γ .
Solution XI
C)(
dexpeB)t,x(u
y
y
t
0
(5.32)
where )y(Y is a function defined implicitly by
Ayγ)y(Yln)νγ(μ)y(Y)γν( 2 ( 0γ ) (5.33)
and
tμexy tr ( 0μ ) (5.34)
A, B, C are arbitrary constants and μ, ν are free parameters with 0μ , 0ν , and
2k2
1νγ with 0γ .
Solution XII
C)ξ(X
ξdexpeB)t,x(u
y
y
tν
0
(5.35)
where )y(X is a function defined implicitly by
Ay2)y(2)y(2 (5.36)
and
tμexy tr ( 0μ ) (5.37)
provided that 2k
2
1ν ( 0γ ). A, B, C are arbitrary constants and μ, ν are free
parameters with 0μ , 0ν .■
Only one solution that has been dealt with in the literature so far. It is the solution
(5.30)-(5.31). The equation (2.9) is a member of this family of solutions when we
27
choose to place 0CA and 1B , and an obvious adjustment (and renaming)
of the parameters, using the same boundary condition
γxγ
1)x(U)T,x(u
The other solutions appear for the first time in this paper. It is obvious that the
parameters and the various constants need to be adjusted to match the appropriate
boundary conditions. It remains to be seen if the general solutions found in this
article are to have any practical interest in Finance and Stochastic Control.
In any case, the Hamilton-Jacobi-Bellman equation associated to an optimization
problem seems to have a rich mathematical structure and thus deserves further
investigation.
Appendix. Solution of equation (4.3).
Equation (4.3) can be written as
0dt)ex(xd tr (A.1)
Using standard methods, we find that equation (A.1) admits te as an integrating
factor. Therefore we have to find the general solution of the complete first order
differential equation
0dt)eex(dxe t)r(tt (A.2)
We suppose that (A.2) admits a solution of the form
C)t,x(U (A.3)
Since
0dxx
Udt
t
UdU
(A.4)
and on comparing this equation with (A.2), we obtain the following system of
differential equations
t)r(t eex
t
U
(A.5)
28
tex
U
(A.6)
Integrating (A.5) with respect to t, we need to distinguish two cases:
r and r .
A1. If r , we obtain from equation (A.5) the solution
)x(er
exU t)r(t
(A.7)
where )x( is a function to be next determined. From (A.7)
)x(ex
U t
(A.8)
(comparing (A.8) to (A.6)) we find that 0)x( , i.e. 1C)t( . Therefore, using
(A.7) and (A.3), we obtain the following solution of (4.3) when r :
Cer
ex t)r(t
(A.9)
which is equation (4.4).
A2. If r , we get from (A.5)
text
U (A.10)
which upon integration with respect to t gives us
)x(texU t (A.11)
where )x( is a function to be determined next.
From (A.11)
)x(ex
U t
(A.12)
(comparing (A.12) to (A.6)), we get 0)x( , i.e. 2C)t( . Therefore using
(A.11) and (A.3), we get the following solution of (4.3) when r :
2tr Ctex (A.13)
which is equation (4.34).
29
References
[1] R. L. Anderson and N. H. Ibragimov: “Lie-Bäcklund Transformations
in Applications ”. SIAM, Philadelphia 1979
[2] S. M. Antoniou: “A General Solution to the Stochastic Equity Volatility
Equation” SKEMSYS report
[3] E. N. Baron and R. Jensen: “A stochastic control approach to the pricing
of options”
Mathematics of Operations Research 15 (1990) 49-79
[4] R. Bellman: “Dynamic Programming”
Princeton University Press 1957
[5] F. E. Benth and K. H. Karlsen: “A Note on Merton’s portfolio selection
problem”
Stochastic Analysis and Applications 23 (2005) 687-704
[6] D. P. Bertsekas: “Dynamic Programming and Optimal Control”
Athena Scientific, Belmont Mass. 1995
[7] D. P. Bertsekas: “Dynamic Programming and Stochastic Control”
Academic Press, Inc. Orlando, FL 1976
[8] T. R. Bielecki and S. R. Pliska: “Risk sensitive asset management with
transaction costs”.
Finance and Stochastics 4 (2000) 1-33
[9] G. Bluman and S. Kumei: “Symmetries and Differential Equations ”
Springer Verlag, 1989
[10] L. A. Bordag: “Symmetry reductions of a nonlinear model of financial
derivatives”, arXiv:math.AP/0604207
[11] L. A. Bordag and A. Y. Chmakova: “Explicit solutions for a nonlinear
model of financial derivatives”
Int. Journal of Theoretical and Applied Finance 10 (2007) 1-21
30
[12] L. A. Bordag and R. Frey: “Nonlinear option pricing models for illiquid
markets: scaling properties and explicit solutions”
arXiv:0708.1568v1 [math.AP] 11 Aug 2007
[13] J-F Boulier, E Trussant and D. Florens: “A dynamic model for pension
funds management”
Proceedings of the 5th
AFIR Int. Colloquium 1995 (pp. 361-384)
[14] A. Bucci, C. Colapinto, M. Forster and D. La Torre: “On Human
capital and Economic Growth with Random Technology Shocks”
Preprint, State University of Milan
[15] M. H. A. Davis, V. G. Panas and T. Zariphopoulou: “European Option
Pricing with Transaction Costs”
SIAM Journal on Control and Optimization 31 (1993) 470-493
[16] G. Dmitrasinovic-Vivodic, A. Lari-Lavassani, X. Li and A. Ware:
“Dynamic Portfolio Selection under Capital-at-Risk”
University of Calgary preprint
[17] L. Dresner: “Applications of Lie’s Theory of Ordinary and Partial
Differential Equations” IOP Publishing, 1999
[18] R. E. Elliott and P. E. Kopp: “Mathematics of Financial Markets”
Springer 2005
[19] L. C. Evans: “Partial Differential Equations”
Graduate Studies in Mathematics, Vol. 19, AMS 1997
[20] W. H. Fleming and H. M. Soner: “Controlled Markov Processes and
Viscosity Solutions”.
Springer, 1993
[21] R. Gerrard, S. Haberman and E. Vigna: “Optimal Investment Choices
Post Retirement in a Defined Contribution Pension Scheme”.
City University, Cass Business School preprint
[22] J. Goard: “New Solutions to the Bond-Pricing Equation via Lie’s Classical
Method”, Mathematical and Computer Modeling 32 (2000) 299-313
31
[23] W. Hereman: “Review of Symbolic Software for the Computation of Lie
Symmetries of Differential Equations”
Euromath Bulletin 2, No.1 Fall 1993
[24] C. Hipp: “Stochastic Control with Application in Insurance”.
University of Karlsruhe report
[25] N. H. Ibragimov: “Elementary Lie Group Analysis and Ordinary
Differential Equations” Wiley, New York
[26] N. H. Ibragimov (ed.): “CRC Handbook of Lie Group Analysis of
Differential Equations” Vol.1 1994, Vol.2 1995, Vol.3 1996
CRC Press, Boca Raton, FL.
[27] N. H. Ibragimov and R. K. Gazizov: “Lie Symmetry Analysis of
differential equations in finance”.
Nonlinear Dynamics 17 (1998) 387-407
[28] J. E. Ingersoll, Jr. “Theory of Financial Decision Making”
Savage: Rowman & Litterfield 1987
[29] I. Karatzas, J. P. Lehoczky, S. Sethi and S. E. Shreve: “Explicit solution
of a general consumption/investment problem”
Math. Oper. Res. 11 (1986) 261-294
[30] I. Karatzas, J. P. Lehoczky and S. E. Shreve: “Existence and uniqueness
of multi-agent equilibrium in a stochastic, dynamic consumption/
investment model”
Math. Oper. Res. 15 (1990) 80-128
[31] R. Korn and E. Korn: “Option Pricing and Portfolio Optimization”
AMS, 2001
[32] R. Korn: “Optimal Portfolios: Stochastic Models for Optimal Investments
and Risk Management in Continuous Time”
World Scientific, 1997
[33] H. Kraft: “Optimal Portfolios with stochastic short rates and defaultable
assets”, Springer, 2004
32
[34] H. Kraft: “Financial Mathematics I: Stochastic Calculus, Option Pricing,
Portfolio Optimization”
Lecture Notes, University of Kaiserslautern, August 2005
[35] P.G.L. Leach, J.G. O’Hara and W. Sinkala: “Symmetry-based solution
of a model for a combination of a risky investment and a riskless
investment”.
J. of Math. Analysis and Applications 334 (2007) 368-381
[36] A. L. Lewis: “Option Valuation under Stochastic Volatility”
Finance Press, second printing 2005
[37] S. Lie: “Gesammelte Werke”, Vols. 1-7,
F. Engel and P. Hergard (Eds.) Teubner, Leipzig 1899
[38] S. Lie: “Vorlesungen über Differentialgleichungen mit bekannten
infinitesimalen Transformationen”
Teubner, Leipzig 1912
[39] R. C. Merton: “Continuous-time Finance”
Blackwell 1990
[40] S. Mudchanatongsuk, J. A. Primbs and W. Wong: “Optimal Pairs
Trading”. Stanford University preprint
[41] C. Munk: “Dynamic Asset Allocation”
Lecture Notes, University of Southern Denmark, March 2008
[42] B. Øksendal: “Stochastic Differential Equations”
Springer, 2005
[43] P. J. Olver: “Applications of Lie Groups to Differential Equations ”
Graduate Texts in Mathematics, vol.107, Springer Verlag, N.Y. 1993
[44] L. V. Ovsiannikov: “Group Analysis of Differential Equations”
Academic Press, New York 1982
[45] D. Papanicolaou: “Investment-Specific Technological Change and Asset
Prices”
Kellogg School of Management, Northwestern University preprint 2008
33
[46] H. Pham: “Continuous-time Stochastic Control and Optimization with
Financial Applications”
Springer 2009
[47] M. Puhle: “Bond Portfolio Optimization”
Springer 2008
[48] G. Silberberg: “Discrete Symmetries of the Black-Scholes Equation”
Proceedings of 10th
Int.Conference in Modern Group Analysis.
[49] H. M. Soner: “Stochastic Optimal Control in Finance”
Lectures at Pisa, 2003
[50] J. L. Stein: “Stochastic Optimal Control, International Finance and Debt
Crises”
Oxford University Press 2006
[51] H. Stephani: “Differential Equations: Their Solutions Using Symmetries ”
Cambridge University Press, 1989
[52] N. Touzi: “Stochastic Optimal Control Problems, Viscosity Solutions,
and Application to Finance”
Lectures at Pisa, 2002
[53] T. Zariphopoulou: “Optimal investment and consumption models
with non-linear stock dynamics”
Math. Methods Oper. Res. 50 (1999) 271-296