Accepted Manuscript

Lie symmetry analysis for a parabolic Monge-Ampère equation in theoptimal investment theory

Shaojie Yang, Tianzhou Xu

PII: S0377-0427(18)30461-8DOI: https://doi.org/10.1016/j.cam.2018.07.035Reference: CAM 11821

To appear in: Journal of Computational and AppliedMathematics

Received date : 9 March 2017Revised date : 21 July 2018

Please cite this article as: S. Yang, T. Xu, Lie symmetry analysis for a parabolic Monge-Ampèreequation in the optimal investment theory, Journal of Computational and Applied Mathematics(2018), https://doi.org/10.1016/j.cam.2018.07.035

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Lie symmetry analysis for a parabolic Monge-Ampereequation in the optimal investment theory

Shaojie Yang∗, Tianzhou XuSchool of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Abstract

In this paper, Lie symmetry analysis is performed on the parabolic Monge-Ampereequation usuyy + ryuyuyy − θu2

y = 0 arising from the optimal investment theory. Liesymmetry and optimal system of this equation are derived. In particular, based onoptimal system, symmetry reductions and invariant solutions are obtained.

Keywords: The parabolic Monge-Ampere equation; Lie symmetry analysis;Symmetry reductions; Invariant solutions

1. Introduction

In recent years, there are many researches use Lie symmetry analysis to partialdifferential equations (PDEs) which arising from physics, chemistry , economicsand other fields [1–7]. The investigation of exact analytical solutions of PDEs playan important role for a long time. Lie symmetry analysis is one of the most ef-fective methods for finding the exact analytical solutions of differential equations,and many authors used this method to find the analytical solutions of PDEs [8–12]. Lie symmetry analysis method was originally developed in the 19th centuryby the Norwegian mathematician Sophus Lie and developed in differential equa-tions since Bluman and Cole proposed similarity theory for differential equationsin 1970s [3, 7].

Over the last forty years, there was a considerable development in PDEs whicharise in mathematical finance [13–19]. It was worth pointing out that Bordag andChmakova’s pioneering paper studied the evaluation of an option hedge-cost un-der relaxation of the price-taking assumption by Lie symmetry analysis method.

∗Corresponding authorEmail address: ysjmath@163.com (Shaojie Yang)

Preprint submitted to Journal of Computational and Applied Mathematics July 21, 2018

ManuscriptClick here to view linked References

In particular, they found some analytical solutions to a nonlinear Black-Scholesequation which incorporates the feedback-effect of a large trader in case of marketilliquidity and showed that the typical solution would have a payoffwhich approx-imates a strangle, then used these solutions to test numerical schemes for solvinga nonlinear Black-Scholes equation [16].

In this paper, we consider the following parabolic Monge-Ampere equation inthe optimal investment theory [17]:

usuyy + ryuyuyy − θu2y = 0, (1.1)

where r, θ > 0 are constants. Existence of solutions to initial value problem forthis equation were showed in [18].

The rest of the paper is organized as follows. In Section 2, we recall the modelEq.(1.1) arise from optimal investment of mathematical finance theory. In Section3, vector fields and optimal system are given by employing Lie symmetry analysismethod. In Section 4, the similarity variables and analytical solutions of Eq.(1.1)are obtained by using optimal system. Finally, conclusions are presented at theend of the paper.

2. The model arise from optimal investment of mathematical finance theory

In this section, we recall the model equation presented in [17].There are n+ 1 assets continuously traded. The 0-th asset is bond, and the last

n are stocks. The price process of the i-th asset is denoted by Pi(t) and they satisfythe following system:

dP0(t) = r(t)P0(t)dt,dPi(t) = bi(t)dt + Pi(t)

∑dj=1 σi jdW j(t), 1 ≤ i ≤ n, Pi(0) = pi, 0 ≤ i ≤ n,

(2.1)

where r(t), bi(t) and σi j(t) are the interest rate, the appreciation rate, and thevolatility, respectively. W(t) = (W1(t),W2(t), ...,Wn(t)) is a d-dimensional stan-dard Brownian motion. Denote b(t) = (b1(t), b2(t), ..., bn(t)), σ(t) = (σi j(t))n×d

.Let an investor have an initial wealth y ∈ R and invest this amount in the mar-

ket described above and the wealth process Y(t) satisfies the stochastic differentialequation (SDE) as follows:

dY(t) = {r(t)Y(t) + ⟨b(t) − r(t)1, π(t)⟩}dt + ⟨π(t), σ(t)dW(t)⟩, t ∈ [0,T ],Y(0) = y,

2

(2.2)

where (π(·),N(·)) ∈ ∏[0,T ] × N[0,T ]. For this particular investor has his ownattitude to the risk versus the gain at the final time T , which is described by astrictly increasing and concave utility function g : R → R. The investor wouldlike to maximize the payoff functional J(π(t)) = E[g(Y(t))] by choosing a suitablestrategy (π(·),N(·)) ∈ ∏[0,T ] × N[0,T ]. This is so-called self-financing optimalinvestment problems:

For given initial endowment y ∈ R, find a portfolio π(·) ∈∏[0,T ], such that

J(π) = supπ(·)∈∏[0,t]

J(π(·)). (2.3)

Any π(·) ∈ Π[0,T ] satisfies (2.3) is called an optimal portfolio and the correspond-ing wealth Y(·) is again called an optimal wealth process.

In order to use the dynamic programming, we need to consider the optimalinvestment problem on the time interval [s,T ] with s ∈ [0,T ],i.e,

dY(t) = {r(t)Y(t) + ⟨b(t) − r(t)1, π(t)⟩}dt + ⟨π(t), σ(t)dW(t)⟩, t ∈ [s, T ],Y(s) = y.

(2.4)

Define J(s, y; π(·)) = Eg(Y(T ; s, y, π(·))), and

V(s, y) = supπ(·)∈∏[s,t] J(s, y; π(·)), (s, y) ∈ [0,T ) × R,V(T, y) = g(y), y ∈ R,

(2.5)

where Y(·; s, y, π(·)) is the solution of (2.4), V(s,y) is called the value function ofself-financing optimal investment problem.

Next, introducing the Hamilton for self-financing optimal investment problem℘(s, y, p,G, π) ≡ p[r(s)y + ⟨b(s) − r(s)1, π⟩] + 1

2G|σT (s)π|2,∀(s, y, p,G, π) ∈ [0,T ] × R × R × R × R,

(2.6)

and

H(s, y, p,G) = supπ∈Rn ℘(s, y, p,G, π),∀(s, y, p,G) ∈ [0,T ] × R × R × R.

(2.7)

Then we have the Hamilton-Jacobi-Bellman equation associated with self-financingoptimal investment problems

us(s, y) + H(s, y, uy(s, y), uyy(s, y)) = 0, (2.8)

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for all (s, y) ∈ [0, T ) × R, such that

(s, y, uy(s, y), uyy(s, y)) ∈ D(H), (2.9)

where D(H) = {(s, y, p,G)|H(s, y, p,G) < ∞}.Now we consider a simple case: n = d = 1, and all the functions r, b, σ are

constants with σ > 0, b − r > 0. Then from (2.8) we get

us + ryuy −(b − r)u2

y

2σuyy= 0, (2.10)

or equivalently

usuyy + ryuyuyy − θu2y = 0, (2.11)

where θ = b−rσ

.

3. Lie symmetry

In this section, we shall perform Lie symmetry analysis for Eq.(1.1). Themethod of determining Lie symmetry for a partial differential equation is standardwhich is described in [1–7].

First of all, let us consider a one-parameter group of infinitesimal transforma-tion:

s = s + ϵξ(s, y, u) + O(ϵ2),

y = y + ϵτ(s, y, u) + O(ϵ2),

u = u + ϵϕ(s, y, u) + O(ϵ2),

(3.1)

where ϵ ≪ 1 is a group parameter. The vector field associated with the abovegroup of transformations can be written as

V = τ(s, y, u)∂

∂s+ ξ(s, y, u)

∂

∂y+ ϕ(s, y, u)

∂

∂u. (3.2)

Thus, the second prolongation Pr(2)V is

Pr(2)V = V + ϕs ∂

∂us+ ϕy ∂

∂uy+ ϕyy ∂

∂uyy, (3.3)

4

where

ϕs = Ds(ϕ) − usDs(τ) − uyDs(ξ),ϕy = Dy(ϕ) − usDy(τ) − uyDy(ξ),ϕyy = Dy(ϕy) − usyDy(τ) − uyyDy(ξ),

and Ds and Dy denote the total derivative operator with respect to s and y.Applying the second prolongation Pr(2)V to Eq.(1.1), we find that the coeffi-

cient functions τ(s, y, u), ξ(s, y, u) and ϕ(s, y, u) must satisfy the following invari-ant condition:

Pr(2)V(∆)|∆=0 = ξruyuyy +ϕsuyy +ϕ

yryuyy − 2θϕyuy +ϕyyus +ϕ

yyryuy = 0, (3.4)

where ∆ = usuyy + ryuyuyy − θu2y = 0. Then we obtain an over determined system

of equations as follows:

ξu = ξyy = 0,ξs = −r(yξy − ξ),τs = τu = τy,

ϕs = ϕy = ϕuu.

(3.5)

Solving above Eqs.(3.5), one can get

ξ = C1y +C2,

τ = C3,

ϕ = C4u +C5,

where C1,C2,C3,C4 and C5 are arbitrary constants. Hence the Lie algebra ofinfinitesimal symmetries of Eq.(1.1) is spanned by the following vector fields:

V1 = y∂

∂y,

V2 = ers ∂

∂y,

V3 =∂

∂s,

V4 = u∂

∂u,

V5 =∂

∂u.

5

Then, all of the infinitesimal generators of Eq.(1.1) can be expressed as

V = C1V1 +C2V2 +C3V3 +C4V4 +C5V5. (3.6)

The commutation relations of Lie algebra determined by V1,V2,V3,V4,V5, areshown in Table 1. It is obvious that {V1,V2,V3,V4,V5} is commute under the Liebracket.

Table 1: The commutation table of Lie algebra

[Vi,V j] V1 V2 V3 V4 V5

V1 0 −V2 0 0 0V2 V2 0 −rV2 0 0V3 0 rV2 0 0 0V4 0 0 0 0 −V5

V5 0 0 0 V5 0

To get symmetry groups, we should solve the following ordinary differentialequations with initial problems:

dsdϵ = ξ(s, y, u),s|ϵ=0 = s,

dydϵ = τ(s, y, u),y|ϵ=0 = y,

dudϵ = ϕ(s, y, u),u|ϵ=0 = u,

then we obtain one-parameter symmetry groups gi : (s, y, u) → (s, y, u) of abovecorresponding the infinitesimal generators Vi(i = 1, 2, 3, 4, 5) are given as follows:

g1 : (s, y, u)→ (s, yeϵ , u),g2 : (s, y, u)→ (s, y + ϵers, u),g3 : (s, y, u)→ (s + ϵ, y, u),g4 : (s, y, u)→ (s, y, ueϵ),g5 : (s, y, u)→ (s, y, u + ϵ).

Thus the following theorem holds:

Theorem 3.1. If u = f (y, s) is a known solution of Eq.(1.1), then by using theabove groups gi(i = 1, 2, 3, 4, 5), the corresponding new solutions ui(i = 1, 2, 3, 4, 5)

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can be obtained respectively as follows:

u1 = f (ye−ϵ , s),u2 = f (y − ϵers, s),u3 = f (y, s − ϵ),u4 = e−ϵ f (y, s),u5 = f (y, s) − ϵ.

Using the Table 1 and the following Lie series

Ad(exp(ϵVi))V j = V j − ϵ[Vi,V j]+ϵ2

2![Vi, [Vi,V j]]− ϵ

3

3![Vi, [Vi, [Vi,V j]]]+ · · · ,

we obtain the adjoint representation in Table 2.

Table 2: The adjoint representation

Ad(exp(ϵVi))V j V1 V2 V3 V4 V5

V1 V1 eϵV2 V3 V4 V5

V2 V1 − ϵV2 V2 V3 + ϵrV2 V4 V5

V3 V1 (1 − r + reϵ)V2 V3 V4 V5

V4 V1 V2 V3 V4 eϵV5

V5 V1 V2 V3 V4 − ϵV5 V5

Based on the adjoint representation, we have the following theorem:

Theorem 3.2. The optimal system of one-dimensional subalgebras of the Lie al-gebra spanned by V1,V2,V3,V4,V5 of Eq.(1.1) given by

V1 ± V3,V1 ± V4,V1 ± V3 ± V4,V2 ± V4,V3,V3 ± V4,V5.

4. Symmetry reductions and invariant solutions

In this section, making use of optimal system in Theorem 3.2, we will getderive several types of symmetry reductions and invariant solutions.

Case 1. For the infinitesimal generator V1 + V3 = y ∂∂y +

∂∂s , the similarity

variables are ξ = s − ln y, f (ξ) = u, and the group-invariant solution is u = f (ξ).

7

Substituting the group-invariant solution into Eq.(1.1), we obtain the followingreduction equation:

(1 + r) f ′ f ′′ − θ f ′2 = 0. (4.1)

Solving above reduction equation, we obtain solution of Eq.(1.1) as follows:

u = c1 + c2eθ(s−ln y)

1+r . (4.2)

Figure 1: Solution (4.2) with c1 = c2 = r = θ = 1

Case 2. For the infinitesimal generator V1 + V4 = y ∂∂y + u ∂

∂u , the similarityvariables are ξ = s, f (ξ) = u

y , and the group-invariant solution is u = y f (ξ).Substituting the group-invariant solution into Eq.(1.1), we obtain the followingreduction equation:

−θ f (s) = 0. (4.3)

Therefore, Eq.(1.1) has a solution u = 0. Obviously, the solution is not meaning-ful.

8

Case 3. For the infinitesimal generator V1 + V3 + V4 = y ∂∂y +

∂∂s + u ∂

∂u , thesimilarity variables are ξ = s − ln y, f (ξ) = u

y , and the group-invariant solution isu = y f (ξ). Substituting the group-invariant solution into Eq.(1.1), we obtain thefollowing reduction equation:

(r − θ − 1) f ′2 + (2θ − r) f f ′ + (1 − r) f ′ f ′′ + r f f ′′ − θ f 2 = 0. (4.4)

Case 4. For the infinitesimal generator V3 =∂∂s , the similarity variables are

ξ = y, f (ξ) = u, and the group-invariant solution is u = f (ξ). Substituting thegroup-invariant solution into Eq.(1.1), we obtain the following reduction equation:

rξ f ′ f ′′ − θ f ′2 = 0. (4.5)

Solving above reduction equation, we obtain solution of Eq.(1.1) as follows:

u = c1 + c2yθ+r

r . (4.6)

Figure 2: Solution (4.6) with c1 = c2 = r = θ = 1

Case 5. For the infinitesimal generator V2+V4 = ers ∂∂y+u ∂

∂u , the similarity vari-ables are ξ = s, f (ξ) = ue−ye−rs

, and the group-invariant solution is u = eye−rsf (ξ).

9

Substituting the group-invariant solution into Eq.(1.1), we obtain the followingreduction equation:

f f ′ − θ f 2 = 0. (4.7)

Solving above reduction equation, we have f = ceθs. Therefore, Eq.(1.1) hassolution as follows:

u = ceθs+ye−rs. (4.8)

Figure 3: Solution (4.8) with c = r = θ = 1

Case 6. For the infinitesimal generator V3 + V4 =∂∂s + u ∂

∂u , the similarityvariables are ξ = y, f (ξ) = ue−s, and the group-invariant solution is u = es f (ξ).Substituting the group-invariant solution into Eq.(1.1), we obtain the followingreduction equation:

f f ′′ + rξ f ′ f ′′ − θ f ′2 = 0. (4.9)

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5. Conclusions

In this paper, we study Lie symmetry analysis for a parabolic Monge-Ampereequation in the optimal investment theory . As a result, infinitesimal generator,commutation table of Lie algebra and optimal system of this equation are derived.With the help of optimal systems, symmetry reductions and invariant solutionsare obtained. It is shown that the the value function of self-financing optimalinvestment problem is a smooth and monotonically function of initial wealth y.

For n-dimensional case, let r =√

y21 + · · · + y2

n, Eq.(2.8) becomes Eq.(2.11), sodiscuss the case n = d = 1 that involve the general case. The solutions which weobtain can be used to self-financing optimal investment problem and to check onthe accuracy and reliability of numerical algorithm of Hamilton-Jacobi-Bellmanequation (2.8).

Acknowledgements

The authors would like to thank the referees for their valuable comments andsuggestions. This work is supported by the Graduate Student Science and Tech-nology Innovation Activities of Beijing Institute of Technology (NO.2017CX10058).

References

[1] Ibragimov, Ranis N. Applications of Lie group analysis in geophysical fluiddynamics , Higher Education Press, 2011.

[2] P.Olver, Applications of Lie groups to differential equations, Springer,NewYork,1986.

[3] G.W.Bluman, Symmetry and integration methods for differential equations,Springer, NewYork,2002.

[4] Ovsiannikov LV . Group analysis of differential equations. Academic Press,1982 .

[5] Grigoriev YN , Ibragimov NH , Kovalev VF , Meleshko SV . Symmetryof integro-differential equations: with applications in mechanics and plasmaphysica. Springer, 2010 .

[6] Ganghoffer JF , Mladenov I . Similarity and symmetry methods. Springer,2014 .

11

[7] Bluman GW , Cheviakov AF , Anco SC . Applications of symmetry methodsto partial differential equations. Springer, 2010 .

[8] H.Z.Liu, J.B.Li, Q.X.Zh, Lie symmetry analysis and exact explicit solutionsfor general Burgers’ equation, J. Comput. Appl. Math. 228 (2009)1-9.

[9] H.Z.Liu, J.B.Li, L.Liu, Lie group classifications and exact solutions for twovariable-coefficient equations, Appl. Math. Comput. 215(2009) 2927-2935.

[10] S.J.Yang, C.C.Hua, Lie symmetry reductions and exact solutions of a cou-pled KdV-Burgers equation, Appl. Math. Comput. 234(2014)579-583.

[11] Gao B, Tian H. Symmetry reductions and exact solutions to theill-posed Boussinesq equation , International Journal of Non-LinearMechanics.72(2015)80-83.

[12] Gao B, Zhang Z, Chen Y. Type-II hidden symmetry and nonlinear self-adjointness of Boiti-Leon-Pempinelli equation, Communications in Nonlin-ear Science and Numerical Simulation. 19(2014)29-36.

[13] D. Bell, S. Stelljes, Arbitrage-free option pricing models, J. Aust. Math. Soc.87 (2009) 145-152.

[14] Y.-K. Kwok, Mathematical models of financial derivatives, second ed,Springer, 2008.

[15] J. Ahna, S. Kang, Y.H. Kwon, A Laplace transform finite difference methodfor the Black-Scholes equation, Mathematical and Computer Modelling. 51(2010)247-255.

[16] L.A. Bordag, A.Y. Chmakova, Explicit solutions for a nonlinear model offinancial derivatives, International Journal of Theoretical and Applied Fi-nance. 10(2007) 1-21.

[17] J.Yong, Introduction to mathematical finance, in: J.Yong, R. Cont (Eds.),Mathematical Finance-Theory and Applications.

[18] Songzhe, Lian. Existence of solutions to initial value problem for a parabol-ic Monge-Ampere equation and application, Nonlinear Analysis: TheoryMethods Applications. 65(2006)59-78.

12

[19] Andriopoulos, K, et al. On the systematic approach to the classification of d-ifferential equations by group theoretical methods, Journal of Computationaland Applied Mathematics. 230(2009)224-232.

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