Tightness and convergence

Corol-lary 7.3.10 gives uniform convergence in probability. Using dominated convergence for the right-hand side of (7.11), we find limε→0E[e^−λτℓε] = exp(

−∫ℓ

0(g^′(a) + 1)^Φ

′ λ(g(a)) Φλ(g(a))da)

. For the left-hand side, it suffices to prove weak convergenceτ_ℓ^ε ⇒τℓ asε→0 for all ℓ≥0. This is done in Corollary 7.3.11 below.

Remark 7.2.3. Existence and uniqueness for (X, L) can also be concluded from classical results, cf. [DI93, suitably extended to non-bounded domains], by considering the pair (X, L) as a degenerate diffusion inR² with oblique reflection in direction (−1,+1) at a smooth boundary, see Figure 7.1b. This uses an iteration argument involving the Skorokhod-map and yields another approximation by a sequence of continuous processes.

Yet, these do not satisfy the target diffusive dynamics inside the domain, except at the limiting fixed point (unless (b, σ) are constant). In contrast, (X^ε, L^ε) adheres to the same dynamics as (X, L) between jump times, cf. (7.1) and (7.4), is Markovian and has a natural interpretation.

Remark 7.2.4. An application example for (7.12) and elastically reflected diffusions is the optimal execution for the sale of a financial asset position if liquidity is stochastic, see Chapter 4. A large trader with adverse price impact seeks to maximize expected proceeds from sellingθ risky assets in an illiquid market. Her trading strategy A(predictable, c`adl`ag, non-decreasing) affects the asset priceSt=f(Y_t^A)Stvia a volume impact process dY_t^A =−βY_t^Adt+ ˆσdBt− dAt withSt=E(σW)t for an increasing function f, and Brownian motions (B, W) with correlationρ. The gains to maximize in expectation are

G_T(A) :=

∫ T 0

e^−δtf(Y_t^A)S_tdA^c_t+ ∑

0≤t≤T

∆At̸=0

e^−δtS_t

∫ ∆At

0

f(Y_t−^A−x) dx.

The optimal strategy turns out to be the local timeLof a reflected Ornstein-Uhlenbeck processX (withb(x) :=ρσσˆ−βxand σ(x) =σ >0) at a suitable elastic boundaryg, as in (7.1)–(7.2), see Section 4.2. After a change of measure argument, one can write the expected proceeds from such strategies asE[G_∞(L)] =∫θ

0 f( g(ℓ))

E[ e^−δτℓ]

dℓ. To find the optimal free boundaryg, one can then apply (7.12) to express the proceeds as a functional of the boundaryg, and optimize over all possible boundaries by solving a calculus of variations problem. This is key to the proof in Chapter 4. The discrete local timeL^ε has a natural interpretation as the step process which approximates the continuous optimal strategyLby doing small block trades, as they would be realistic in an actual implementation, with identical (no-)action region. The approximation is asymptotically optimal for the control problem. Indeed, straightforward calculations similar to the derivation of (7.11) show thatL^εis asymptotically optimal in first order, i.e.E[G_∞(L)] =E[G_∞(L^ε)] +O(ε).

7.3 Tightness and convergence

To show convergence of (τ_ℓ^ε)ε, we will prove that the pair of c`adl`ag processes (X^ε, L^ε) forms a tight sequence inε→0. Applying weak convergence theory for SDEs by Kurtz and Protter [KP96], we show that any limit point (forε→0) satisfies (7.1) and (7.2).

Uniqueness in law for solutions of (7.1) – (7.2) will then allow to conclude Theorem 7.2.2.

7 Approximating diffusion reflections at elastic boundaries

Let (εn)_n∈N be a sequence withεn→0 and consider the sequence (X^εn, L^εn)n. To show tightness, we will apply the following criterion due to Aldous.

Proposition 7.3.1 ([Bil99, Cor. to Thm. 16.10]). Let (E,|·|) be a separable Banach space. If a sequence(Yⁿ)_n∈Nof adapted, E-valued c`adl`ag processes satisfies the following two conditions, then it is tight.

(a) The sequences (

JT(Yⁿ))

n and(Y₀ⁿ)n are tight (inR, resp.E) for anyT ∈(0,∞), with JT(Yⁿ) := sup_0<t≤T⏐

⏐Y_tⁿ−Y_t−ⁿ⏐

⏐ denoting the largest jump until time T.

(b) For any T ∈(0,∞)andε₀, η >0there existδ₀>0andn₀∈N such that for all n≥n₀, all (discrete)Yⁿ-stopping times τˆ≤T and all δ∈(0, δ₀] we have

P[

|Y_ˆτ+δⁿ −Y_τˆⁿ| ≥ε0

]≤η .

To get tightness one needs to control both jump size and, regarding (L^ε_n)n, the frequency of jumps simultaneously. As we are considering processes with jumps of size±εn→0, only the latter is not yet clear. To this end, the next lemma provides a technical bound onX^εn,L^εn, while a second lemma constricts the probability that X^εn (respectively L^εn) performs a number ofNn jumps in a time interval of fixed length.

Lemma 7.3.2 (Upper bound). Fix a time horizon T ∈(0,∞) and η ∈(0,1]. Then there exists a constantM ∈R such thatP[∃n:g(L^ε_Tⁿ−ε_n)> M]≤η, with the domain of definition for the functiong being extended byg(−x) :=g(0) for−x <0.

Proof. Consider a continuous (b, σ)-diffusion Y that starts at time t = 0 at g(0).

For n ∈ N and k = 0,1,2, . . ., let ℓ(n, k) := kε_n. By induction over k, using comparison for diffusion SDEs, cf. [KS91, Theorem 5.2.18], one obtains that (a.s.) X_t^εn ≤ Y_t for t ∈ [[0, τ_ℓ(n,k)^εn [[ for all k ≥ 1, and hence X^εn ≤ Y on [0,∞) (a.s.) because limk→∞τ_ℓ(n,k)^εn = ∞ for any n by Lemma 7.2.1. Therefore, on the event {∃n : g(L^ε_Tⁿ −ε_n) > M} we have sup_t∈[0,T]Y_t ≥ M, and hence H^g(0)→M ≤ T. Thus P[∃n : g(L^ε_Tⁿ−ε_n)> M] ≤P[H^g(0)→M ≤T]. Now the claim follows because lim_M→∞P[H^g(0)→M ≤T] = 0.

Lemma 7.3.3(Frequency of jumps). FixT ∈(0,∞), ε0, η >0, and setNn:=⌈ε0/εn⌉.

Then there existsδ >0 andn0∈N such that for every bounded stopping timeτˆ≤T we haveP[

J_{ˆτ ,δ}^εn ≥Nn

]≤η for alln≥n0, where J_{τ ,δ}^ε_ˆⁿ := inf{k|L^ε_τˆⁿ+kεn≥L^ε_τ+δˆⁿ } is the number of jumps ofX^εn, respectivelyL^εn, in time]]ˆτ ,τˆ+δ]].

Proof. We will first find an estimate for the jump count probability for arbitrary but fixed δ >0,n∈N,Nn ∈Nand ˆτ≤T. Only in part 2) of the proof we will consider (Nn)_n∈N as stated, to study the limitn→ ∞. More precisely, we will show in part 1) that, given Fτˆ, for everyλ >0 there existkn,λ∈ {0,1, . . . , Nn−1} s.t. forxn :=g(L^ε_τˆⁿ+εnkn,λ),

P[

J_{τ ,δ}^ε_ˆⁿ ≥Nn

⏐

⏐Fτˆ

]≤e^λδ

(Φ_λ(x_n−ε_n) Φλ(xn)

)Nn−1

. (7.13)

1) In this part, fix arbitraryδ >0,n∈N, Nn ∈N and ˆτ ≤T. We enumerate the jumps and estimate the sum of excursion lengths by δ. Let ℓk := L^ε_τˆⁿ+kεn be the (discrete) local time at thek-th jump after time ˆτ. IfX^εn has at leastNn jumps in the

124

7.3 Tightness and convergence

with the last equality being in distribution conditionally onFτˆ, forHk being conditionally independent and distributed asH^{g(ℓk−1)−εn→g(ℓk)}. Clearly, ℓk isFτˆ-measurable. By the Laplace transform (7.8) ofHk and the Markov inequality, we get forλ >0

P[ To investigate the limitn→ ∞, first observe that by Taylor expansion

logΦλ(x−εn) Φ_λ(x) =−εn

Φ^′_λ(x)

Φ_λ(x)+εnr(x, εn),

where r(·, ε_n)→0 converges uniformly on compacts forε_n →0. Since ˆτ+δ≤T+δ is bounded, Lemma 7.3.2 yields a constantM ∈Rsuch that P[

∃n:x_n > M]

≤ ^η₂ for thex_n from above. On the event{∀n:x_n ∈I} with compactI:= [g(0), M], we have uniform convergence ofr(xn, εn) and thereby get

lim sup where κ^x_ℓ is the inverse local time at constant level xof a (b, σ)-diffusion Z^x starting atx, and A^x(t) is the occupation timeA^x(t) :=∫t

→0. By compactness ofI and Dini’s theorem there existsλ=λ_ε0,η,M such that forδ:= 1/λwe have

7 Approximating diffusion reflections at elastic boundaries

on the event{xn ≤M for alln}. By equation (7.13) andP[∃n:xn > M]≤η/2, this completes the proof.

Using the preceding two lemmas, we will first prove tightness of (L^εn)n and of (X^εn)n

separately. Tightness of the pair (X^εn, L^εn)n is handled afterwards.

Lemma 7.3.4 (Tightness of the local time approximations). The sequence (L^εn)n of c`adl`ag processes defined by (7.4)and (7.5)satisfies Aldous’ criterion and thus is tight.

Proof. Part (a) of Proposition 7.3.1 is clear, as the initial valueL^ε₀ⁿ=ε_n is deterministic andJ_T(L^εn)≤ε_n. For part (b), considerT, η, ε₀ >0 and any boundedL^εn-stopping time ˆτ≤T. The event|L^ε_τ+δˆⁿ −L^ε_τˆⁿ| ≥ε₀means thatL^εnperforms at leastN_n :=⌈ε₀/ε_n⌉ jumps in the stochastic interval ]]ˆτ ,τˆ+δ]]. Lemma 7.3.3 yields somen₀ andδ₀=δ₀(ε₀) such that Aldous’ criterion is satisfied for all n ≥ n₀. Hence, (L^εn)_n is tight by Proposition 7.3.1.

Next we show boundedness of (X^εn)n, needed for Lemma 7.3.6 to prove tightness.

Lemma 7.3.5 (Bounding the diffusion approximations). Let T ∈ (0,∞) and η > 0.

Then there existsM ∈Rsuch that P[sup_t∈[0,T]|X_t^εn|> M]< η for alln∈N.

Proof. By Lemma 7.3.2, for everyn ∈N the process X^εn on [0, T] is bounded from above by a constantM with probability at least 1−η/2. It remains to show that it is also bounded from below with high probability. To this end, we will construct a process Y that is a lower bound for allX^εn and then argue forY.

For ˆε:= sup_nε_n consider a (b, σ)-diffusionY which is discretely reflected by jumps of size−ˆεat a constant boundaryc:=g(0)−ˆε, withY₀=y:=g(0)−2ˆε. SuchY is a special case of (7.4)–(7.5), for a constant boundary function: dY_t=b(Y_t) dt+σ(Y_t) dW_t−L^Y_t with L^Y_t := ∑

0≤s≤t∆L^Y_t and ∆L^Y_t := ˆε1{Yt−=c}. Let τ_k^Y := inf{t > 0| L^Y_t > kˆε}

be the k-th hitting time of Y at the boundary c. Thus on all intervals [[τ_k^Y, τ_k+1^Y [[, Y is a continuous (b, σ)-diffusion starting in y. Now for fixed n, ε := εn, note that X_τ^εε

mε =g((m−1)ε)−ε≥c≥Yτ_mε^ε by monotonicity of g. As τ_mε^ε → ∞ for m→ ∞ by Lemma 7.2.1, induction over the inverse (discrete) local timesτ_mε^ε ,m∈N, yields X^ε≥Y on [[τ_k^Y, τ_k+1^Y ]] ifX_τ^εY

k

≥Y_τY

k by comparison results [KS91, Thm. 5.2.18]. Since X₀^ε≥Y0, the latter follows by induction overk. Asτ_k^Y → ∞fork→ ∞by Lemma 7.2.1, we getX^εn≥Y on [0,∞) for alln. So it suffices to showP[inf_t∈[0,T]Yt<−M]< η/2 for someM, which directly follows from the c`adl`ag property ofY.

Lemma 7.3.6 (Tightness of the reflected diffusion approximations). The sequence (X^εn)n of c`adl`ag processes from (7.4)and (7.5) satisfies Aldous’ criterion and thus is tight.

Proof. Condition (a) of Proposition 7.3.1 holds. To verify part (b), letη >0,T ∈(0,∞), and ˆτ≤T be a stopping time. By Lemma 7.3.5,|X_τ^ε_ˆⁿ|is with a probability of at least 1−η/4 bounded by some constant M (not depending onnand ˆτ). Let us consider the events{X_τ+δ^ε_ˆⁿ ≤X_τ^ε_ˆⁿ−ε₀} ∪ {X_ˆτ+δ^εn ≥X_τˆ^εn+ε₀}={|X_ˆτ+δ^εn −X_ˆτ^εn| ≥ε₀} separately.

1) First consider{X_τ+δ^ε_ˆⁿ ≤X_τ^ε_ˆⁿ−ε₀}. Forξ:=X_τ^ε_ˆⁿ we construct a reflected process Y^ξ such thatY_t^ξ ≤X_τ+t^ε_ˆⁿ for allt≥0. We then estimate P[X_τ+δ^ε_ˆⁿ ≤X_τ^ε_ˆⁿ−ε0] by means ofP[Y_δ^x≤x−ε0] in (7.15), uniformly for allnlarge enough. We estimate the latter in (7.16) using the probability of a down-crossing in timeδof intervals [x−ε0, x−2ˆε] by a

126

7.3 Tightness and convergence continuous diffusion. Covering⋃

x[x−ε0, x−2ˆε] by finitely many intervals [yk, yk+1] in (7.17) then allows us to chooseδ >0 sufficiently small.

To this end, choose ˆε≤ε₀/4 andn large enough such thatε_n ≤ε, and let (Yˆ _t^ξ)_t≥0 be the (b, σ)-diffusion w.r.t. the Brownian motion (Wτ+tˆ −Wτˆ)t≥0 withY₀^ξ =ξ−2ˆε, which is discretely reflected by jumps of size −ˆε at a constant boundary at level ξ−ε.ˆ More precisely, dY_t^ξ = b(Y_t^ξ) dt+σ(Y_t^ξ) dW_τ+tˆ −K_t^ξ with (discrete) local timeK_t^ξ:=∑

0≤s≤t∆K_s^ξ for ∆K_t^ξ := ˆε1_{Yt−^ξ _=ξ−ˆε}. Global existence and uniqueness of (Y^ξ, K^ξ) follows from the proof of Lemma 7.2.1. By comparison arguments and induction as in the proof of Lemma 7.3.5, one verifiesY_t^ξ ≤X_ˆτ+t^εn fort∈[0,∞). Indeed, [KS91, By constructionY^ξ depends onnandτ (throughξ), while the right-hand side of (7.15) does not. Thus one only needs to bound the probability of an (ε0−2ˆε)-displacement of diffusions Y^x with starting pointsx−2ˆε from a compact set, which are reflected (by (−ˆε)-jumps) at constant boundaries x−ε. By the arguments in the proof ofˆ Lemma 7.3.3 (here applied for Y^x which is reflected at a constant boundary), for δ=δ0>0 there existsN ∈Nwith the following property: for everyx∈[−M, M], the numberJ_δ^x:= inf{k|kεˆ≥K_δ^x}of jumps ofY^x until timeδis bounded by N−1 with probability at least 1−η/8.

Indeed, by (7.13), fixingδ >0,λ:= 1/δ, one gets for anyxthatP[J_δ^x≥ ⌈N(x)⌉]≤η/8 since for the event under consideration, the processY^x would have to move at least once (in at most N occasions) continuously from the point x−2ˆε to x−ε₀. Let

7 Approximating diffusion reflections at elastic boundaries

such that is holds for alln≥n0. Note thatδ1only depends onT (viaM andK) and onn0but not onn. Hence, for allδ∈(0, δ1], alln≥n0and all ˆτ ≤T we have

P[X_τ+δ^ε_ˆⁿ ≤X_ˆτ^εn−ε0]≤η

2. (7.18)

2) For the alternative second case X_τ+δ^ε_ˆⁿ ≥X_τ^ε_ˆⁿ+ε0, consider the solution (Yt)_t≥ˆτ

on [[ˆτ ,∞[[ of dYt=b(Yt) dt+σ(Yt) dWt withYτˆ=X_τ^ε_ˆⁿ. Using comparison results for continuous diffusions [KS91, Theorem 5.2.18] inductively over times [[τ_(k−1)ε^εn

n, τ_kε^εn

n[[, we findY_t≥X_t^εn for allt∈[[ˆτ ,∞[[, a.s. Hence, arguing like in the previous case

P[

X_τ+δ^ε_ˆⁿ ≥X_ˆτ^εn+ε₀,|X_ˆτ^εn| ≤M]

≤P[

Y_τ+δˆ ≥Y_τˆ+ε₀,|Y_ˆτ| ≤M]

≤ sup

−M≤y≤MP[

H^y→y+ε0 ≤δ]

. (7.19)

As in (7.17) we find aδ2>0 such that for allδ∈(0, δ2] the right side of (7.19) is bounded by η/4. Hence we haveP[X_τ+δ^ε_ˆⁿ ≥X_ˆτ^εn+ε0] ≤η/2, so with (7.18), Proposition 7.3.1 applies.

Now, to prove joint tightness of (X^εn, L^εn)_n, we can utilize the fact that both processes satisfy Aldous’ criterion and that their jump times and jump magnitudes are identical.

Lemma 7.3.7(Tightness of joint approximations). The sequence(X^εn, L^εn)n of c`adl`ag R²-valued processes defined by (7.4)and (7.5)is tight.

Proof. In view of Proposition 7.3.1, choose the spaceE:=R² equipped with Euclidean norm |·| and let Yⁿ := (X^εn, L^εn) ∈ D(

[0,∞), E)

. Then Y₀ⁿ= (g(0)−εn, εn) and JT(Yⁿ) =√

2εn form tight sequences inE andR, respectively. Furthermore, P[

|Y_τ+δˆⁿ −Y_ˆτⁿ| ≥ε0

]≤P[

|X_ˆτ+δ^εn −X_τ^ε_ˆⁿ| ≥ ε₀ 2

] +P[

|L^ε_ˆτ+δⁿ −L^ε_ˆτⁿ| ≥ ε₀ 2

] . HenceYⁿ also satisfies Aldous’s criterion and therefore is tight.

Tightness only implies weak convergence of a subsequence. It remains to show (in Lemma 7.3.9) that every limit point satisfies (7.1) and (7.2) and that uniqueness in law holds. The latter will follow from pathwise uniqueness results for SDEs with reflection, while for the former we apply results from [KP96] on weak converges of SDEs. For that purpose, note that the approximated local times form agoodsequence of semimartingales (cf. [KP96, Definition 7.3]), as shown in the following lemma.

Lemma 7.3.8. The sequence(L^εn)n is of uniformly controlled variation and thus good.

Proof. Letδ:= sup_nεn. Then all processesL^εn have jumps of size at most δ <∞. Fix someα >0. By tightness, there exists someC∈Rsuch thatP[L^ε_αⁿ> C]≤1/α. So the stopping timeτn,α:= inf{t≥0|L^ε_tⁿ > C} satisfiesP[τn,α≤α] =P[L^ε_αⁿ > C]≤1/α . Moreover, by monotonicity ofL^εn we haveE[∫t∧τn,α

0 d|L^εn|s

]

=E[L^ε_t∧τⁿ_n,α]≤C <∞. Hence (L^εn) is of uniformly controlled variation in the sense of [KP96, Definition 7.5].

So by [KP96, Theorem 7.10] it is agood sequence of semimartingales.

We have gathered all necessary results to prove convergence of our approximating diffusions and local times to the continuous counterpart.

128

7.3 Tightness and convergence Lemma 7.3.9 (Weak convergence of the approximations). The sequence(X^εn, L^εn)n

of c`adl`ag processes defined by (7.4)–(7.5)converges weakly to the unique continuous strong solution (X, L)of (7.1)–(7.2).

Proof. By Prokhorov’s theorem, tightness of (X^εn, L^εn, W)_n implies weak convergence of a subsequence to some limit point, (X^εnk, L^εnk, W)_k⇒( ˜X,L,˜ W˜)∈D(

[0,∞),R³) . Continuity of ( ˜X,L) is clear since˜ εn →0 is the maximum jump size. First we prove that ( ˜X,L) satisfies the asserted SDEs. Afterwards, we will prove uniqueness of the˜ limit point. To ease notation, let w.l.o.g. the subsequence (nk) be (n).

By [KP96, Theorem 8.1] we get that ( ˜X,L) satisfy (7.1) for the semimartingale ˜˜ W. That ˜W is a Brownian motion follows from standard arguments, cf. [NO10, proof of Theorem 1.9]. AsD(

[0,∞),R³)

is separable we find, by an application of the Skorokhod representation theorem, that ˜L is non-decreasing and ˜Xt≤g( ˜Lt) for allt ≥0, P-a.s.

because these properties already hold for (X^εn, L^εn).

To prove that ˜Lgrows only at timest with ˜Xt=g( ˜Lt), we have to approximate the indicator function by continuous functions. Forδ >0 define

h_δ(x, ℓ) :=

For δ ↘ 0 the functions hδ ↘ h0 converge pointwise monotonically. Continuity of hδ implies weak convergence (H^δ,n, L^εn)⇒ ( ˜H^δ,L). By Lemma 7.3.8, (L˜ ^εn) is a good sequence. So for everyδ >0 the stochastic integrals∫·

0H_s−^δ,ndL^ε_sⁿ⇒∫· By monotonicity of ˜L, d ˜Lt defines a random measure on [0,∞). Hence monotone convergence of ˜H_t^δ ↘H˜_t⁰yields d ˜Lt=h0( ˜Xt,L˜t) d ˜Lt.

Thus, we showed that (X^ε, L^ε) converges in distribution to a weak solution ( ˜X,L) of˜ the reflected SDE, i.e. it might be defined on a different probability space with its own Brownian motion. Note that ( ˜X,L) is continuous on [0,˜ ∞) and that ˜τ_∞:= sup_kτ˜_k=∞ a.s., where ˜τ_k:= inf{t >0| |X˜_t| ∨L˜_t> k}. To show the existence and uniqueness of a strong solution as stated in the theorem, we will use the results from [DI93]. Consider the domain ¯G:={(x, ℓ)∈R²|x≤g(ℓ), ℓ≥0}. We may interpret the process (X_t, L_t) as a continuous diffusion in ¯Gwith oblique reflection in direction (−1,+1) at the boundary, although the notion of a two-dimensional reflection seems unusual here, because (X, L) only varies in one dimension in the interior of G. The unbounded domain G can be exhausted by bounded domainsGk :={

(x, ℓ)∈G⏐

⏐|x|,|ℓ|< k}

, which might have a non-smooth boundary especially at (g(0),0), but still satisfy [DI93, Cond. (3.2)]. Hence, by [DI93, Cor. 5.2] the process (X, L) exists (up to explosion time) on the initial probability space and is (strongly) unique on [[0, τk[[ with exit timeτk:= inf{t >0| |Xt| ∨Lt> k}, for allk∈N. So (X, L) is unique until explosion timeτ∞:= sup_kτk. Moreover, by [DI93,

7 Approximating diffusion reflections at elastic boundaries

Theorem 5.1] we have the following pathwise uniqueness result: for any two continuous solutions (X¹, L¹) and (X², L²) with explosion timesτ_∞¹ and τ_∞², respectively defined on the same probability space with the same Brownian motion and the same initial condition, we have thatX¹=X² andL¹=L²on [[0, τ_k¹∧τ_k²]] for everyk∈Na.s. Using a known argument due to Yamada and Watanabe, ideas being as in [KS91, Ch. 5.3.D], one can bring the two (weak) solutions ( ˜X,L,˜ W˜) and (X, L, W) to a canonical space with a common Brownian motion. By pathwise uniqueness there, one concludes that τ_∞=∞a.s. (as ˜τ_∞=∞). Hence the strong solution (X, L) does not explode in finite time. In addition, we conclude uniqueness in law like in [KS91, Prop. 5.3.20] and thus any weak limit point of the approximating sequence (X^ε, L^ε) will have the same law as (X, L).

This convergence result can be strengthened as follows.

Corollary 7.3.10 (Convergence in probability). The sequence(X^εn, L^εn)n of c`adl`ag processes defined by (7.4)–(7.5)converges in probability to(X, L)defined by (7.1)–(7.2).

Proof. Following the proof of [KP91, Cor. 5.6], we will strengthen weak convergence (X^εn, L^εn)⇒(X, L) to convergence in probability. First, note that Lemma 7.3.9 implies weak convergence of the triple (X^εn, L^εn, W)⇒(X, L, W) by e.g. [SK85, Corollary 3.1].

Hence, for every bounded continuousF :D([0,∞);R²)→Rand every bounded continu-ousG:C([0,∞);R)→R, we have lim_n→∞E[F(X^εn, L^εn)G(W)] =E[F(X, L)G(W)]. Now, the previous equation even holds for all bounded measurableGbyL¹-approximation of measurable functions by continuous functions. By strong uniqueness of (X, L), there exists a measurable functionH:C([0,∞);R)→C([0,∞);R²) such that (X, L) =H(W).

In particular,G(W) :=F(H(W)) =F(X, L) is bounded and measurable, so we conclude

n→∞lim E[

(F(X^εn, L^εn)−F(X, L))²]

= lim

n→∞

(E[

F(X^εn, L^εn)²]

−2E[

F(X^εn, L^εn)F(X, L)] +E[

F(X, L)²])

= 0 and hence convergence in probability follows.

To show this, first considerD([0, T]) with Skorokhod-metricdT, instead ofD([0,∞)), by restricting paths on [0, T]. Fix an arbitraryη >0 and writeYⁿ := (X^εn, L^εn) and Y := (X, L). SinceP[∥Y∥_∞ ≤K]→1 andP[∥Yⁿ∥_∞≤K]→1 asK → ∞, we have thatY, Yⁿ are bounded with high probability.

By compactness of [0, T]×[−K, K]²and separability ofD([0, T]) (cf. e.g. [Bil99, Thm.

12.2]), there exists a finite covering ofD([0, T]; [−K, K]²) with ballsB_η/4(q). For each of these finitely manyq∈D([0, T]) letFq :D([0, T])→[0,1] be a continuous function with value 1 onBη/4(q) and 0 outside ofBη/2(q).

For two bounded pathsx, y∈D([0, T]; [−K, K]²) we have thatdT(x, y)> ηimplies (F_q(x)−F_q(y))²= 1 for at least one of the chosen q. We can therefore estimate the

probabilityP[d_T(Y, Yⁿ)≥η,∥Y∥ ∨ ∥Yⁿ∥ ≤K]≤∑

qE[(F_q(Y)−F_q(Yⁿ))²]. Since the finite sum on the right-hand side converges to 0 asn→ ∞, we get thatd_T(Y, Yⁿ)< η with high probability. By repeating this argument for multiple time horizonsT, we can also bound theD([0,∞);R²)-distanced(Y, Yⁿ) =∑∞

T=12^−T(1∧d_T(Y, Yⁿ))≤η with high probability.

Corollary 7.3.11 (Weak convergence of the inverse local times). For any ℓ >0, the sequence (τ_ℓ^εn)n from (7.6) converges in law to the inverse local timeτℓ defined by (7.3).

130

7.3 Tightness and convergence Proof. ConvergenceL^εn ⇒LimpliesL^ε_tⁿ⇒Lt at all continuity points ofL, i.e. at all points, henceP[

τ_ℓ^εn≤t]

=P[

L^ε_tⁿ ≥ℓ]

→P[Lt≥ℓ] =P[τℓ≤t]. This completes the proof of Theorem 7.2.2.

Bibliography

[AC00] Robert Almgren and Neil Chriss. Optimal execution of portfolio transactions.

J. Risk, 3(2):5–39, 2000.

[AFS10] Aur´elien Alfonsi, Antje Fruth, and Alexander Schied. Optimal execution strategies in limit order books with general shape functions.Quant. Finance, 10(2):143–157, 2010.

[AKS16] Aur´elien Alfonsi, Florian Kl¨ock, and Alexander Schied. Multivariate transient price impact and matrix-valued positive definite functions.Math. Oper. Res., 41(3):914–934, 2016.

[Alm12] Robert Almgren. Optimal trading with stochastic liquidity and volatility.

SIAM J. Financial Math., 3(1):163–181, 2012.

[AS10] Aur´elien Alfonsi and Alexander Schied. Optimal trade execution and absence of price manipulations in limit order book models. SIAM J. Financial Math., 1(1):490–522, 2010.

[ASS12] Aur´elien Alfonsi, Alexander Schied, and Alla Slynko. Order book resilience, price manipulation, and the positive portfolio problem. SIAM J. Financial Math., 3(1):511–533, 2012.

[AW95] Joseph Abate and Ward Whitt. Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput., 7(1):36–43, 1995.

[BB04] Peter Bank and Dietmar Baum. Hedging and portfolio optimization in financial markets with a large trader. Math. Finance, 14(1):1–18, 2004.

[BBF18a] Dirk Becherer, Todor Bilarev, and Peter Frentrup. Approximating diffusion reflections at elastic boundaries. Electron. Commun. Probab., 23:1–12, 2018.

[BBF18b] Dirk Becherer, Todor Bilarev, and Peter Frentrup. Optimal asset liquidation with multiplicative transient price impact. Appl. Math. Optim., 78(3):643–

676, 2018.

[BBF18c] Dirk Becherer, Todor Bilarev, and Peter Frentrup. Optimal liquidation under stochastic liquidity. Finance Stoch., 22(1):39–68, 2018.

[BBF19] Dirk Becherer, Todor Bilarev, and Peter Frentrup. Stability for gains from large investors’ strategies inM1/J1topologies. Bernoulli, 25(2):1105–1140, 2019.

[BC67] John Bather and Herman Chernoff. Sequential decisions in the control of a space-ship (finite fuel). J. Appl. Probab., 4(3):584–604, 1967.

Bibliography

[BE08] Erhan Bayraktar and Masahiko Egami. An analysis of monotone follower problems for diffusion processes. Math. Oper. Res., 33(2):336–350, 2008.

[Bil99] Patrick Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999.

[BL98] Dimitris Bertsimas and Andrew W. Lo. Optimal control of execution costs.

J. Financial Markets, 1(1):1–50, 1998.

[BLZ16] Bruno Bouchard, Gr´egoire Loeper, and Yiyi Zou. Almost-sure hedging with permanent price impact. Finance Stoch., 20(3):741–771, 2016.

[BR17] Tilmann Bl¨ummel and Thorsten Rheinl¨ander. Financial markets with a large trader. Ann. Appl. Probab., 27(6):3735–3786, 2017.

[BS02] Andrei N. Borodin and Paavo Salminen. Handbook of Brownian motion—

facts and formulae. Probability and its Applications. Birkh¨auser Verlag, Basel, second edition, 2002.

[CdL13] Rama Cont and Adrien de Larrard. Price dynamics in a Markovian limit order market. SIAM J. Financial Math., 4(1):1–25, 2013.

[C¸ JP04] Umut C¸ etin, Robert A. Jarrow, and Philip E. Protter. Liquidity risk and arbitrage pricing theory. Finance Stoch., 8(3):311–341, 2004.

[CL95] Louis K. C. Chan and Josef Lakonishok. The behavior of stock prices around institutional trades. J. Finance, 50(4):1147–1174, 1995.

[C¸ ST10] Umut C¸ etin, H. Mete Soner, and Nizar Touzi. Option hedging for small investors under liquidity costs. Finance Stoch., 14(3):317–341, 2010.

[CT04] Rama Cont and Peter Tankov. Financial Modelling with Jump Processes.

Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004.

[DAF14] Tiziano De Angelis and Giorgio Ferrari. A stochastic partially reversible in-vestment problem on a finite time-horizon: Free-boundary analysis. Stochas-tic Process. Appl., 124(12):4080 – 4119, 2014.

[DAFF17] Tiziano De Angelis, Salvatore Federico, and Giorgio Ferrari. Optimal bound-ary surface for irreversible investment with stochastic costs. Math. Oper.

Res., 42(4):1135–1161, 2017.

[DI93] Paul Dupuis and Hitoshi Ishii. SDEs with oblique reflection on nonsmooth domains. Ann. Probab., 21(1):554–580, 1993.

[DM82] Claude Dellacherie and Paul-Andr´e Meyer. Probabilities and Potential. B.

North-Holland, Amsterdam, 1982.

[DM04] Fran¸cois Dufour and Boris Miller. Singular stochastic control problems.

SIAM J. Control Optim., 43(2):708–730, 2004.

134

Bibliography [DS98] Freddy Delbaen and Walter Schachermayer. The fundamental theorem of asset pricing for unbounded stochastic processes.Math. Ann., 312(2):215–250, 1998.

[DY09] Min Dai and Fahuai Yi. Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem. J. Differential Equations, 246(4):1445–1469, 2009.

[DZ98] Mark H. A. Davis and Mihail Zervos. A pair of explicitly solvable singular stochastic control problems. Appl. Math. Optim., 38(3):327–352, 1998.

[EKK91] Nicole El Karoui and Ioannis Karatzas. A new approach to the Skorohod problem, and its applications. Stochastics Stochastics Rep., 34(1–2):57–82, 1991.

[FGL⁺04] J. Doyne Farmer, L´aszl´o Gillemot, Fabrizio Lillo, Szabolcs Mike, and Anindya Sen. What really causes large price changes? Quant. Finance, 4(4):383–397, 2004.

[FJ02] J. Doyne Farmer and Shareen Joshi. The price dynamics of common trading strategies. J. Econ. Behav. Organ., 49(2):149–171, 2002.

[FK19] Giorgio Ferrari and Torben Koch. An optimal extraction problem with price impact. Appl. Math. Optim., 2019. DOI: 10.1007/s00245-019-09615-9.

[FKTW12] Peter A. Forsyth, J. Shannon Kennedy, Sue T. Tse, and Heath Windcliff.

Optimal trade execution: A mean quadratic variation approach. J. Econom.

Dynam. Control, 36(12):1971–1991, 2012.

[FP14] Salvatore Federico and Huyˆen Pham. Characterization of the optimal boundaries in reversible investment problems. SIAM J. Control Optim., 52(4):2180–2223, 2014.

[FR19] Giorgio Ferrari and Neofytos Rodosthenous. Optimal control of debt-to-gdp ratio in an n-state regime switching economy. Technical report, Bielefeld Center for Mathematical Economics, 2019.

[Fre98] R¨udiger Frey. Perfect option hedging for a large trader. Finance Stoch., 2(2):115–141, 1998.

[FSU19] Antje Fruth, Torsten Sch¨oneborn, and Mikhail Urusov. Optimal trade execution in order books with stochastic liquidity. Mathematical Finance, 29(2):507–541, 2019.

[GF00] Israel M. Gelfand and Sergei V. Fomin. Calculus of Variations. Dover Books on Mathematics. Dover Publications, 2000.

[GH17] Paulwin Graewe and Ulrich Horst. Optimal trade execution with instan-taneous price impact and stochastic resilience. SIAM J. Control Optim., 55(6):3707–3725, 2017.

Bibliography

[GHS16] Paulwin Graewe, Ulrich Horst, and Eric S´er´e. Smooth solutions to portfolio liquidation problems under price-sensitive market impact. Stoch. Process.

Appl., 2016.

[GS13] Jim Gatheral and Alexander Schied. Dynamical models of market impact and algorithms for order execution. InHandbook on Systemic Risk, pages 579–602. Cambridge University Press, 2013.

[GSS12] Jim Gatheral, Alexander Schied, and Alla Slynko. Transient linear price impact and Fredholm integral equations. Math. Finance, 22(3):445–474, 2012.

[GZ15] Xin Guo and Mihail Zervos. Optimal execution with multiplicative price impact. SIAM J. Financial Math., 6(1):281–306, 2015.

[HH11] Vicky Henderson and David Hobson. Optimal liquidation of derivative portfolios. Math. Finance, 21(3):365–382, 2011.

[HHK92] Ayman Hindy, Chi-Fu Huang, and David Kreps. On intertemporal pref-erences in continuous time: The case of certainty. J. Math. Econom.,

[HHK92] Ayman Hindy, Chi-Fu Huang, and David Kreps. On intertemporal pref-erences in continuous time: The case of certainty. J. Math. Econom.,

Im Dokument Optimal Trading with Multiplicative Transient Price Impact for Non-Stochastic or Stochastic Liquidity (Seite 135-153)