Stochastic Control of Itˆo-L´evy Processes with Applications to Finance

Stochastic Control of Itˆo-L´evy Processes with Applications to Finance

Bernt Øksendal

Center of Mathematics for Applications, University of Oslo, Norway Mathematical Methods in Finance and Economy, DoSon, Vietnam,

October 2011

1 Stochastic calculus for Itˆo-L´evy processes

• In this section we give a brief survey of stochastic calculus for Itˆo-L´evy processes. We begin with a defintion of a L´evy process:

Definition 1.1. A L´evy process on a probability space (Ω,F,P) is a pro- cess,η(t)≡η(t, ω)with the following properties

(i) η(0) = 0.

(ii) ηhas stationary, independent increments.

(iii) η is stochastically continuous⇒ η has a c`adl`ag (i.e. left continuous with right limits) version. This version will be used from now on.

The jump ofηat timetis∆η(t) = η(t)−η(t−).

• The jump measureN([0, t], U)gives the number of jumps ofηup to timet with jump size in the set U ⊂ R0 ≡ R\ {0}. If we assume thatU ⊂ R0, thenU contains only finitely many jumps in any finite time interval.

• The L´evy measureν(.)ofηis defined by

ν(U) = E[N([0,1], U)]. (1) andN(dt,dζ)is the differential notation of the random measureN([0, t], U).

• LetN˜()denote the compensated jump measure ofη, defined by

N˜(dt,dζ)≡N(dt,dζ)−ν(dζ)dt (2)

• For convenience we shall from now on impose the following additional in- tegrability condition onν(.) :

Z

R

z²ν(dz)<∞, (3)

which is equivalent to the assumption that for allt ≥0

E[η²(t)]<∞ (4) .

This condition still allows for many interesting kinds of L´evy processes. In particular, it allows for the possibility that a L´evy process has the following property:

Z

R

(1∧ |z|)ν(dz) = ∞. (5) This implies that there are infinitely many small jumps.

• Under the assumption above the Itˆo-L´evy decomposition theorem states that any L´evy process has the form

η(t) = at+bB(t) + Z t

0

Z

R

zN˜(ds,dz), (6) whereB(t)is a Brownian motion, anda, bare contants.

• More generally, we study the Itˆo-L´evy processes, which are the processes of the form

X(t) = x+ Z t

0

α(s, ω)ds+ Z t

0

β(s, ω)dB(s) (7) +

Z t 0

Z

R

γ(s, ζ, ω) ˜N(ds,dζ), whereRt

0 |α(s)|ds+Rt

0 β²(s)ds+Rt 0

R

Rγ²(s, ζ)ν(dζ)ds <∞a.s., andα(t), β(t), and γ(t, ζ) are predictable processes (predictable w.r.t. the filtration F_tgenerated byη(s), fors≤t).

• In differential form we have

dX(t) =α(t)dt+β(t)dB(t) + Z

R

γ(t, ζ) ˜N(dt,dζ). (8)

• We now proceed to provide the Itˆo formula for Itˆo-L´evy processes.

LetX(t)be an Itˆo-L´evy process defined as above. Let f : [0, T]×Rbe a C^1,2function and putY(t) = f(t, X(t)).

ThenY(t)is also an Itˆo-L´evy process with representation:

dY(t) = ∂f

∂t(t, X(t))dt+∂f

∂x(t, X(t))(α(t)dt+β(t)dB(t)) (9) +1

2

∂²f

∂x²(t, X(t))β²(t)dt +

Z

R

{f(t, X(t) +γ(t, ζ))−f(t, X(t))}N˜(dt,dζ) +

Z

R

{f(t, X(t) +γ(t, ζ))−f(t, X(t))− ∂f

∂x(t, X(t))γ(x, ζ)}ν(dζ)dt, where the last term

Z

R

{f(t, X(t) +γ(t, ζ))−f(t, X(t))− ∂f

∂x(t, X(t))γ(x, ζ)}ν(dζ)dt can be interpreted as the quadratic variation of jumps.

• The Itˆo isometries state the following:

E

"

Z T 0

β(s)dB(s) ²#

= E

Z T 0

β²(s)ds

(10)

E

"

Z T 0

Z

R

γ(s, ζ) ˜N(ds,dζ) ²#

=E Z T

0

Z

R

γ²(s, ζ)ν(dζ)ds

(11)

• Martingale conditions: If the quantities of (11) are finite, then M(t) =

Z t 0

Z

R

γ(s, z) ˜N(ds,dz) (12) is a martingale fort≤T.

• The Itˆo representation theorem states that any F ∈ L²(F_T,P)has the rep- resentation

F = E[F] + Z T

0

ϕ(s)dB(s) + Z T

0

Z

R

ψ(s, ζ) ˜N(ds,dζ) (13) for suitable predictable (unique)L²-processesϕ(.)andψ(.).

Detour: Using Malliavin calculus, we get the representation ϕ(s) =E[DsF|Ft]

and

ψ(s, ζ) = E[D_s,ζF|F_s],

whereDs andDs,ζ are the Malliavin derivatives atsands, ζw.r.tB(.)and N˜(., .), respectively.

• Example:

Supposeη(t) = η₀(t) = Rt 0

R

RζN˜(ds,dζ), i.e. η(t)is a pure-jump martin- gale. We want to find the representation ofF ≡η₀²(T).

By the Itˆo formula we get d(η₀²(t)) =

Z

R

{(η0(t) +ζ)²−(η0(t))²}N˜(dt,dζ) (14) +

Z

R

{(η₀(t) +ζ)²−(η₀(t))²−2η₀(t)ζ}ν(dζ)

= Z

R

2η₀(z)ζN˜(dt,dζ) + Z

R

ζ²N˜(dt,dζ) (15) +

Z

R

ζ²ν(dζ)dt

= 2η₀(t)dη₀(t) + Z

R

ζ²N˜(dt,dζ) + Z

R

ζ²ν(dζ)dt. (16) This implies that

η²₀(T) =T Z

R

ζ²ν(dζ) + Z T

0

2η₀(t)dη₀(t) + Z T

0

Z

R

ζ²N˜(dt,dζ). (17) Note that it is NOT possible to writeF ≡ η₀²(T)as a constant + an integral w.r.t. dη₀(t).

This has an interpretation in finance: It implies that in a market withη₀(t) as the risky asset price, the claimη²₀(T)is not replicable.

So the market based on L´evy processes are typically not complete.

• Consider the following stochastic differential equation having the form:

dX(t) = b(t, X(t))dt+σ(t, X(t))dB(t) (18) +

Z

R

γ(t, X(t⁻), ζ) ˜N(dt,dζ); X(0) =x. (19) Here

b: [0, T]×Rⁿ→Rⁿ σ: [0, T]×Rⁿ→R^n×m γ : [0, T]×Rⁿ×R^l0 →R^n×l are given functions.

If these functions are Lipshitz continuous with at most linear growth, then a uniqueL²- solution to the above SDE exists.

• Example: The geometric Itˆo-L´evy process:

dX(t) = X(t⁻) [α(t)dt+β(t)dB(t) (20) +

Z

R

γ(t, ζ) ˜N(dt,dζ)

; X(0) =x >0

If γ > −1 then X(t) can never jump to the negative value and then the solution is

X(t) = xexp Z T

0

β(s)dB(s) + Z t

0

(α(s)− 1

2β²(s))ds (21) +

Z t 0

Z

R

{ln(1 +γ(s, ζ))−γ(s, ζ)}ν(ds)ds +

Z t 0

Z

R

ln(1 +γ(s, ζ)) ˜N(dt,dζ)

(22) If b(t, x) = b(x), σ(t, x) = σ(x), and γ(t, x, ζ) = γ(x, ζ), i.e. b(.), σ(.), andγ(., .)do not depend ont, the corresponding SDE takes the form

dX(t) = b(X(t))dt+σ(X(t))dB(t) (23) +

Z

R

γ(X(t), ζ) ˜N(dt,dζ) is called an Itˆo-L´evy diffusion or simply a jump-diffusion.

• The generatorAof a jump-diffusionX(t)is defined by (Af)(x) = lim

t→0

E^x[f(X(t))]−f(x)

t , (24)

if the limit exists.

• The form of the generator Aof the process X(.)is given explicitly in the following lemma

Lemma 1.1. IfX(t) is a jump-diffusion andf ∈ C₀²(R), whereC₀ corre- sponds tof having compact support, then(Af)(x)exists for allxand (Af)(x) =

n

X

i=1

b_i(x)∂f

∂x_i +1 2

n

X

i=1

(σσ^T)_ij ∂²f

∂²x_i (25)

+

l

X

k=1

Z

R

{f(x+γ^(k)(x, ζ))−f(x)− ∇f(x)·γ^(k)(x, ζ)}ν_k(dζ)

• The Dynkin formula

Let X be a jump-diffusion process and let τ be a stopping time. Letf ∈ C²(R) and assume that E^xRτ

0 |Af(X(t))|dt

< ∞ and {f(X(t))}t≤τ is uniformly integrable.

Then

E^x[f(X(τ))] = f(x) +E^x Z τ

0

Af(X(t))dt

(26)

2 Application to Stochastic Control

2.1 Dynamic Programming

• Motivating example:

Suppose we have a financial market with two investment possibilities:

(i) A risk-free asset with unit priceS₀(t) = 1.

(ii) A risky asset with unit price

dS(t) = S(t−) [α(t)dt+β(t)dB(t) +

Z

R

γ(t, ζ) ˜N(dt,dζ)

, γ >−1, S(0) >0. (27) Letπ(t)denote a portfolio representing the fraction of the total wealth in- vested in the risky asset at timet.

If we assume that π(t)is self-financing, the corresponding wealth X(t) = X_π(t)satisfies the equation

dX(t) = X(t−)π(t−) [α(t)dt+β(t)dB(t) +

Z

R

γ(t, ζ) ˜N(dt,dζ)

. (28)

We call this a state equation.

• Problem:

MaximizeE[U(X_π(t))]over all π ∈ A, where A denotes as the set of all admissible portfolios andU is a given utility function.

This is a special case of the following general stochastic control problem:

State equation:

dY(t) = dYu(t)

= b(Y(t), u(t))dt+σ(Y(t), u(t))dB(t) (29) +

Z

R

γ(Y(t), u(t), ζ) ˜N(dt,dζ), Y(0) =y∈R^k. Performance functional:

J_u(y) = E^y





 Z τs

0

f(Y(s), u(s))

| {z } profit rate

ds+ g(Y(τ_s))

| {z } bequest function

1_{{τ <∞}}





, (30) where τs = inf{t ≥ 0 : Y(t) ∈ S}/ (bankruptcy time), and S is a given solvency region.

• Problem:

Findu^∗ ∈ AandΦ(y)such that Φ(y) = sup

u∈A

Ju(y) = Ju^∗(y)

Theorem 2.1. (Hamilton-Jacobi-Bellman (HJB) equation) (a) Suppose we can find a function for allϕ∈ C²(Rⁿ)such that

(i) A_vϕ(y)+f(y, v)≤0, for allv ∈ V, whereV is the set of possible control values) and

A_vϕ(y) =

k

X

i=1

b_i(y, v)∂ϕ

∂yi

+1 2

k

X

i,j=1

(σσ^T)_ij∂²ϕ

∂²yi

(31)

+X

m

Z

R

{ϕ(y+γ^(k)(y, v, ζ))−ϕ(y)− ∇ϕ(y)γ^(k)(y, v, ζ)}ν_k(dζ) (ii) lim_t→τs− =g(Y(τ_s))1_{τs<∞} + “growth conditions”

Then

ϕ(y)≥Φ(y).

(b) Suppose we fory∈ S can findv =u(y)b such that Abu(y)ϕ(y) +f(y,u(y)) = 0b

andbu(y)is an admissible feedback control (Markov control), i.e.u(y)b meansu(Yb (t)), thenu(y)b is an optimal control and

ϕ(y) = Φ(y).

Sketch of Proof:

The growth condition implies that we, by an approximation argument, can use the Dynkin Lemma withf =ϕandτ =τ_sfor any given controlu∈ A.

This gives (ifτ_s <∞)

E^y[ϕ(Y(τ_s))] = ϕ(y) +E^y Z τs

0

Aϕ(Y(t))dt

(32)

≤_(Aϕ+f≤0) ϕ(y)−E^y Z τs

0

f(Y(t), u(t))dt

(33) This implies

ϕ(y) ≥ E^y Z τs

0

f(Y(t), u(t))dt+g(Y(τ_s))

(34)

= J_u(y), for all u∈ A, (35)

which means that

ϕ(y) ≥ sup

u∈A

J_u(y) = Φ(y). (36)

This proves(a).

To prove (b), observe that if we get the equality above, i.e. Aϕ+f = 0, then we get

ϕ(y) = J

bu(y).

Hence

Φ(y)≤ϕ(y) = J

bu(y)≤Φ(y).

• To illustrate this, let us return to the optimal portfolio problem SupposeU(x) = ln(x)

The problem is to maximizeE[lnX_π(T)].

Put

dY(t) =

dt dX(t)

=

1

X(t)π(t)α(t)

dt+

0

X(t)π(t)β(t)

dB(t) (37) +

0

X(t)π(t) Z

R

γ(t, ζ) ˜N(dt,dζ) (38) and

A_πϕ(t) = ∂ϕ

∂t −xπ∂ϕ

∂x + 1

2x²π²∂²ϕ

∂x² (39)

+ Z

R

{ϕ(t, x+γ(t, ζ))−ϕ(t, x)− ∂ϕ

∂x(t, x)γ(t, ζ)}ν(dζ) Heref = 0andg(t, x) = lnx.

We guess that ϕ(x) = lnx+κ(t), whereκ(t)is a deterministic function, and we maximizeA_πϕover allπ.

Then we find, if assuming α(t), β(t), and γ(t, z)are deterministic (other- wise the system would not be Markovian), that the optimal portfolio π^∗ is the solution of the equation

π^∗(t)β²(t) +π^∗(t) Z

R

γ²(t, ζ)ν(dζ)

1 +π^∗(t)γ(t, ζ) = α(t). (40) In particular, ifν = 0andβ²(t)6= 0, then

π^∗(t) = α(t) β²(t).

Remark The assumption thatα(t),β(t), andγ(t, z)are deterministic func- tions are indispensible conditions when employing the dynamic program- ming techniques in solving the stochastic control problems. This is the limitation of the dynamic programming approach to solving the stochastic control problems. On the other hand, as we shall see, the backward stochas- tic differential equations (BSDE) approach does not requireα(t),β(t), and γ(t, z)being deterministic.

3 Risk Minimization

3.1 Introduction

• A convex risk measure is a mapρ:L^p(F_T, P)→R; p∈[1,∞]

(i) (Convexity): ρ(λF + (1−λ)G)≤λρ(F) + (1−λ)ρ(G) (ii) (Monotonicity): F ≤G⇒ρ(F)≥ρ(G)

(iii) (Translation invariance): ρ(F +α) =ρ(F)−α ifa∈R i.e. reducing the risk accordingly

Remark

We may regard ρ(F) as the cash position amount we need to add to the position F in order to make it ”acceptable”. i.e. ρ(F +ρ(F)) = 0

(ρ(F)is acceptable ifρ(F)≤0)

One can prove that basically any risk convex measureρcan be represented as follows:

•

ρ(F) = sup

Q∈℘

{E^Q(−F)−ζ(Q)} (41) for some family℘of measureQP and for some convex penalty function ζ : ℘ → R We refer to F¨ollmer and Schied (2010) for more information about risk measures.

• Returning to the financial market above, suppose we want to minimize the risk of the terminal wealth, rather then maximizing the expected utility.

Then the problem is to minimize ρ(X_π(T)) over all possible admissible portfoliosπ ∈A.

Hence we want to solve the problem

π∈infA(sup

Q∈℘

{EQ[−X_π(T)]−ζ(Q)}) (42) This is an example of a stochastic differential game. Heuristically, this can be interpreted as ”to perform as good as possible under the worst possible scenario”.

Thus, we have to solve a stochastic differental game (of zero sum type) under the worst case scenario.

• This is a special case of the following problem: We have 2 players and 2 types of controls. Suppose the stateY(t) = Y_u(t)has the form

dY(t) = b(Y(t), u₀(t))dt+σ(Y(t), u₀(t))dB(t) +

Z

R

γ(Y(t), u(t, ζ), ζ) ˜N(dt,dζ) ;Y(0) =y (43)

• Hereu= (u₀, u₁), withu₀ = (θ₀, π₀), u₁ = (θ₁, π₁) andθ= (θ₀, θ₁),π = (π₀, π₁),

whereθis the control of player 1, andπis the control of player 2

Remark

(θ₁, π₁)are the controls for the jump component. In the examples where we do not take jumps into account(θ1, π1)will not appear.

3.2 Performance functional

• We define the performance functional as follows:

Jθ,π =E^y[ Z τs

0

f(Y(t), θ(t), π(t))dt+g(Y(τs))1τs<∞] (44)

Problem: Find Φ(y)andθ^∗ ∈Θ, π^∗ ∈Π s.t. Φ(y) = inf

π∈Π(sup

θ∈Θ

J_θ,π(y)) =J_θ^∗_,π^∗ (45) Remark

This type of problem (a stochastic differential game problem) is not solvable by the classical Hamilton-Jacobi-Bellman (HJB) equation. We need a new tool, namely the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, which in this setting goes as follows:

Theorem 3.1. HJBI equation for zero-sum games (Mataramvura and Øksendal (2008))

Suppose we can find a functionϕ∈C²(S)T

C( ¯S)(continuous on boundary) and a Markov control pairθ(y),ˆ π(y)ˆ such that (y is the control variable)

(i) A_θ,ˆπ(y)ϕ(y) +f(y, θ,π(y))ˆ ≤0; ∀θand∀y∈ S (ii) Aθ(y),πˆ ϕ(y) +f(y,θ(y), π)ˆ ≥0; ∀πand∀y∈ S (iii) Aθ(y),ˆˆ π(y)ϕ(y) +f(y,θ(y),ˆ π(y)) = 0ˆ ; ∀y∈ S (iv) lim

t→τs

ϕ(Y_θ(t)) =g(Y_θ,π(τ_s))1_τs≤∞

(+) growth conditions Then

ϕ(y) = Φ(y) = inf

π (sup

θ

J_θ,π(y)) = sup

θ

(infπ J_θ,π(y))

= inf

π Jθ,πˆ (y) = sup

θ

J_θ,ˆπ(y)

=Jθ,ˆˆπ(y) Proof: Similar to HJB

To apply this to our risk minimization problems, we parametrize the family℘of measuresQ P as follows:

For eachθ₀(t), θ₁(t, ζ)define:

dZ_θ(t) = Z_θ(t⁻)[θ₀(t)dB(t) +R

Rθ₁(t, ζ) ˜N(dt,dζ)]; Z_θ(0) >0, θ₁ >−1 i.e.

Z_θ(t) =Z_θ(0) exp Z t

0

θ₀(s)dB(s)− 1 2

Z t 0

θ²₀(s)ds+ Z t

0

Z

R

ln(1 +θ₁(s, ζ)) ˜N(ds,dζ) +

Z t 0

Z

R

{ln(1 +θ₁(s, ζ))−θ₁(s, ζ)}ν(dζ)ds] (46)

SupposeQP onF_T and putZ(T) = ^dQ_dP

• DefineZ(t) = ^d(Q|F_d(P|F^t)

t) , then Z(t) =E[Z(T)|F_t]

• HenceZ(t)is a martingale. If we also assume thatP QthenZ(t)>0 andZ(t)can be written in the form above for appropriate choice ofθ₀(t), θ₁(t, z)

• This gives thatdQ=Z_θ(T)dP, for someθ, for all suchQ

• If we restrict ourselves to the family ℘of measuresQ = Q_θ forθ ∈ Θthe risk minimization problem gets the form:

π∈Πinf(sup

θ∈Θ

{EQθ[−X_π(T)]−ζ(Q₀)}) = inf

π∈Π(sup

θ∈Θ

{E[−Z_θ(T)X_π(T)]−ζ(Q₀)})

• For example,ζ(Q_θ) = Rτs

0 λ(Y(s), θ(s)) ds, then this problem is a special case of the zero-sum stochastic differential game.

Extension of HJBI to non-zero sum games

In this case we have two performance functionals, one for each player:

J_θ,π⁽ⁱ⁾ =E^y Z τs

0

fi(Y(t), θ(t), π(t))dt+gi(Y(τs))1τs<∞

; i= 1,2 (47) (In the zero-sum game we haveJ⁽²⁾ =−J⁽¹⁾)

• The pair (θ,ˆ π) is called a Nash equilibrium ifˆ (i) J_θ,ˆ⁽¹⁾_π ≤J⁽¹⁾_ˆ

θ,ˆπ(y), ∀θ ∈Θ (ii) J⁽²⁾_ˆ

θ,π(y)≤J⁽²⁾_ˆ

θ,ˆπ(y), ∀π ∈Π

• Remark This is related to the Nash prisoner dilemma, and it is not a very strong equilibrium: One can sometimes obtain a better result for both play- ers at points which are not Nash equilibria.

Theorem 3.2. (Mataramvura and Øksendal (2008))

Suppose∃ϕ_i ∈ C²(S), and a Markovian control(ˆθ,π) such that:ˆ

(i) A_θ,ˆπ(y)ϕ₁(y) +f₁(y, θ,ˆπ(y))≤Aθ(y),ˆˆ π(y)ϕ₁(y) +f₁(y,θ(y),ˆ π(y))ˆ ;∀θ (ii) Aθ(y),πˆ ϕ2(y) +f2(y,θ(y), π)ˆ ≤Aθ(y),ˆˆ π(y)ϕ2(y) +f2(y,θ(y),ˆ π(y))ˆ ;∀π (iii) lim

t→τ_s⁻

ϕ_i(Y_θ,π(t)) =g(Y(τ_s))1_τs<∞

(+) growth condition

Then(ˆθ,π)is a Nash equilibrium andˆ ϕ₁(y) = sup

θ∈Θ

J₁^θ,ˆπ(y) =J₁^θ,ˆˆπ(y) andϕ₂(y) = sup

π∈Π

J₂^θ,πˆ (y) =J₂^θ,ˆˆπ(y).

4 Stochastic Control 2

4.1 Introduction

• Consider a controlled Itˆo-L´evy process of the form

dX(t) =b(X(t), u(t), ω)dt+σ(t, X(t), u(t), ω)dB(t) +

Z

R

γ(t, X(t), u(t), z, ω) ˜N(dt,dz) (48) Hereb(t, x, u, ω)is a givenF_t-adapted process, for eachxandu

and similarly withσandγ. So this system is not Markovian

• The performance functional has the form:

J(u) = E[RT

0 f(t, X(t), u(t), ω)dt+g(X(T), ω)]

T > 0is a fixed constant.

• Problem:

Findu^∗ ∈ Aso thatsup

u∈A

J(u) =J(u^∗)

4.2 The Maximum Principle Approach

Define the Hamiltonian as follows:

H(t, x, u, p, q, r(.)) =f(t, x, u)+b(t, x, u)·p+σ(t, x, u)·q+R

Rγ(t, x, u, ζ)r(ζ)ν(dζ) Herer(.)is a spatial function

• The backward stochastic differential equation (BSDE) in the adjoint pro- cessesp(t), q(t), r(t, ζ)is defined as follows:







dp(t) =−^∂H_∂x(t, X(t), u(t), p(t), q(t), r(t,·))dt+q(t)dB(t) +R

Rr(t, ζ) ˜N(dt,dζ); 0≤t ≤T p(T) =g⁰(X(T)) (First order of optimality)

(49)

This BSDE is easy to solve due to its linearity.

Theorem 4.1. (The Mangasarian (sufficient) maximum principle)

Supposeuˆ∈ A, with correspondingX(t) =ˆ X_uˆ(t),p(t),ˆ q(t),ˆ r(t,ˆ ·).Suppose the

functions x → g(x)and(x, u) → H(t, x, u,p(t),ˆ q(t),ˆ r(t,ˆ ·))are concave and that

maxv∈V H(t,X(t), v,ˆ p(t),ˆ q(t),ˆ r(t,ˆ ·)) = H(t,X(t),ˆ u(t),ˆ p(t),ˆ q(t),ˆ r(t,ˆ ·)), (50) for allt, whereV is the set of all possible control values.

Moreover, suppose that some growth conditions are satisfied.

Thenuˆis an optimal control.

Let us apply this to the optimal portfolio problem:

MaximizeE[U(X_π(T))]over all admissible portfoliosπwhere, dX(t) =X(t⁻)π(t⁻)[α(t, ω)dt+β(t, ω)dB(t) +

Z

R

γ(t, ζ, ω) ˜N(dt,dζ)], (51) In this case we have

H=xπα(t)·p+xπβ(t)·q+ Z

R

xπγ(t, z)r(z)ν(dz) (52) b(t, x, u) =xπα(t), σ(t, x, u) = xπβ(t)

(53) The BSDE becomes







dp(t) =−π(t){α(t)p(t) +β(t)q(t) +R

Rγ(t, z)r(z)ν(dz)}dt+q(t)dB(t) +R

Rr(t, z) ˜N(dt,dz); 0≤t ≤T p(T) =U⁰(X(T))

(54) H gives intuition into the choice ofπ

∵πappears linearly inH,∴we guess that the coefficient ofπmust be0Otherwise one could makeH arbitrary big by choosingπsuitably. Therefore

α(t)p+β(t)q+ Z

R

γ(t, z)r(z)ν(dz) = 0 (matchdp(t)) (55) (dp(t) =q(t)dB(t) +R

Rγ(t, z)r(z)ν(dz)

p(T) =U⁰(X(τ)) (56)

Solution Using Malliavin calculus we get:







p(t) =E[R|Ft]

q(t) =E[DtR|Ft] whereR=U⁰(X(T)) r(t, ζ) =E[Dt,ζR|Ft]

(57)

Substituting this back into (55) we get:

α(t)E[R|F_t] +β(t)E[D_tR|F_t] + Z

R

γ(t, ζ)E[D_t,ζR|F_t]ν(dζ) = 0

This is a Malliavin-type differential equation in the unknown random variableR.

This type of Malliavin differential equation is discussed in (Øksendal, B. and Sulem, A. (2009))

The general solution of this equation isR=R_c,θ(T), where R_c,θ(t) = cexp

Z t 0

θ₀(s)dB(s)− 1 2

Z t 0

θ²₀(s)ds +

Z t 0

Z

R

ln(1 +θ₁(s, ζ)) ˜N(ds,dζ) +

Z t 0

Z

R

{ln(1 +θ₁(s, ζ))−θ₁(s, ζ)}ν(dζ)ds

(58) i.e. dR_c,θ(t) = R_c,θ(t⁻)h

θ₀(t)dB(t) +R

Rθ₁(t, ζ) ˜N(dt,dζ)i

for arbitrary constantc∈R, and anyθ₀(t), θ₁(t, z)satisfying the equation:

α(t) +βθ0(t) + Z

R

γ(t, z)θ1(t, z)ν(dz) = 0 (59) (ifc = 1,R is the Radon-Nikodyn derivative, thus it is a martingale) Recall that the risky asset price is given by:

dS(t) = S(t⁻)

α(t)dt+β(t)dB(t) + Z

R

γ(t, ζ) ˜N(dt,dζ)

(60) So (59) is saying that if we defineQ_θ bydQ_θ =R_1,θ(T)dP thenQ_θ is an equiva- lent martingale measure for the market.

(RecallR=U⁰(X(T)),Ris the terminal value ofR_c,θ(t))

For simplicity, assume thatν= 0from now on (i.e., that there are no jumps).

Then (59) becomes:

α(t) +β(t)θ₀(t) = 0 i.e.,

θ₀(t) = −α(t) β(t)

SinceR_c,θ(T)=U⁰(X(T))we haveX(T) =I(R_c,θ(T)) whereI = (U⁰)⁻¹ (I is the inverse ofU⁰)

Now thatθ₀is known, what aboutc?

Recall the equation forX(t):

(dX(t) =X(t)π(t) [α(t)dt+β(t)dB(t)]

X(T) =I(R_c,θ(T)); θ =θ₀ =−^α(t)_β(t) (61) Consider the BSDE

(dX(t) =π(t)^α(t)_β(t)Z(t)dt+Z(t)dB(t)

X(T) = I(R_c,θ(T)) (62)

The solution of this linear BSDE is X(t) = 1

Γ(T)E[I(R_c,θ(T))Γ(T)|F_t] (63) wheredΓ(t) =−Γ(t)^α(t)_β(t)dB(t); Γ(0) = 1

Now putt= 0and take expectation to get

X(0) =x=E[I(R_c,θ(T)Γ(T)]

This determines the constantcand hence the optimal terminal wealthX(T).

Remark

The advantage of this approach is that it applies to a general non-Markovian set- ting, which is inaccessible for dynamic programming.

Moreover, this approach can be extended to case when the agent has only partial information to her disposal, which means that her decisions must be based on an information flow which is a subfiltration ofF. More information can be found in the references below.

References

[1] F¨ollmer, H. and Schied, A. (2011), “Stochastic Finance,” Third Edition,De Gruyter.

[2] Mataramvura, S. and Øksendal, B (2008), “Risk minimizing portfolios and HJBI equations for stochastic differential games,”Stochastics,80, 317–337.

[3] Øksendal, B. and Sulem, A. (2007), “Applied Stochastic Control of Jump Diffusions,” Second Edition,Springer.

[4] Øksendal, B. and Sulem, A. (2009), “Maximum principles for optimal control of forward-backward stochastic differential equations with jumps,”

SIAM J. Control Optimization,48, 2845–2976.

[5] Øksendal, B. and Sulem, A. (2011a), “Portfolio optimization under model uncertainty and BSDE games”Quantitative Finance(to appear)

Preprint, University of Oslo 2011(http://www.duo.uio.no/sok/work.html?WORKID=16779) [6] Øksendal, B. and Sulem, A. (2011b), “Forward-backward SDE games and

stochastic control under model uncertainty” (to appear)

Preprint, University of Oslo 2011(http://www.duo.uio.no/sok/work.html?WORKID=16779)